electromagnetic force experiment

and solve for the permeability of free space…

Returning to formula for the magnetic force on a current carrying wire leads to the following definition of magnetic field strength and its unit, the tesla.

Third right/left hand rule

Cathode ray tube: color television (color monitor), oscilloscope,

mass spectrometer

space weather, aurora, Van Allen radiation belts

electric motor

electromagnetic rail gun

nuclear magnetic resonance?

The National MagLab is funded by the National Science Foundation and the State of Florida.

Interactive Tutorials

These demonstrations about laws and tools associated with electricity and magnetism allow you to adjust variables at and to visualize invisible forces — which makes them almost better than the real thing.

Two columns list page

Alternating Current

Every time you plug something into the electricity in your house, you are utilizing the power of alternating current (AC.)

Arc lamps were the first type of electric light, so brilliant the lamps were used for lighthouses and street lights.

Barkhausen Effect

The Barkhausen effect makes the concept of magnetic domains audible.

Bullet Speed

This tutorial takes a shot at explaining how circuits can be used to measure things beyond the capacity of human senses.

A capacitor is similar to a battery, but a few key differences make them crucial additions to many machines.

Cathode Ray Electromagnetic Deflection Basics

Discover how cathode rays behave in a magnetic field.

Cathode Ray Tube Television

For decades, the Cathode Ray Tube was used for video displays from televisions to computer screens. 

Compasses in Magnetic Fields

The invention of the magnetic compass radically changed the way humans navigated from place to place. Travelers could orient themselves even when the …

Contracting Helix

This device demonstrates how parallel wires attract because of the magnetic fields they generate.

Current Flow

This tutorial illustrates the flow of electricity through a circuit and how that flow is impacted by resistors in the circuit.

Daniell Cell

English chemist John Frederick Daniell came up with a twist on the simple voltaic cell.

This simple direct current (DC) motor has been created by pairing a permanent magnet and an electromagnet. The permanent magnet is called a stator bec…

Deionization

The magnets here at the lab generate massive amounts of heat. To cool them off, we need massive amounts of water. But first, we have to take the ions …

Diamagnetism and Paramagnetism

Certain metals exhibit a strong response to a magnetic field. But everything reacts to magnetic fields in some way.

Electricity Meter

The newest electric meters rely on different techniques to measure usage. But power to many homes and businesses is still monitored by traditional met…

Electromagnetic Induction

When a permanent magnet is moved inside of a copper wire coil, electrical current flows inside of the wire. This important physics phenomenon is calle…

Electromotive Force in Inductors

Electromotive Force is an important phenomenon that impacts the way electrons flow through a conductor.

Electrostatic Generator

Though simple by today's standards, the early electrostatic generators were a great milestone in humankind's understanding of electricity.

Electrostatic Repulsion from Van de Graaff Generator

A fun way to illustrate electrostatic forces from a Van de Graaff generator.

Faraday Motor

Just a year after electromagnetism was discovered, the great scientist Michael Faraday figured out how to turn it into motion.

Faraday’s Ice Pail

Out of a humble ice pail the great experimentalist Michael Faraday created a device to demonstrate key principles of attraction, repulsion and electro…

Foucault's Disk

In 1855, a French physicist created a device that illustrated how eddy currents work.

Galvanometer

A galvanometer detects and measures small amounts of current in an electrical circuit.

Guitar Pickup

This simple device transforms the mechanical energy of the vibrating guitar strings into electrical energy.

Hall Effect

When a magnetic field is applied to a flowing current, it creates a weak but measurable voltage. This is the Hall effect.

Heat Resistance

Metals conduct electricity because their atoms have free electrons that can move between them. As those free electrons move through the metal conducto…

Ignition Coil

Start your engines and learn about the ignition coil, a key to operating your car.

Inductive Pendulum

Get the swing of electromagnetic induction with this device.

Inductive Reactance

Like resistance, reactance slows down an electrical current. This phenomenon occurs only in AC circuits.

Kelvin Water Dropper

The legendary Lord Kelvin made electricity from water with his water dropper.

Lodge's Experiment

Sir Oliver Lodge's experiment demonstrating the first tunable radio receiver was an important stepping stone on the path toward the invention of a pra…

Lorentz Force

A wire fashioned into a pendulum moves inside a magnetic field, demonstrating the Lorentz force.

Magnetic Core Memory

Magnetic core memory was developed in the late 1940s and 1950s, and remained the primary way in which early computers read, wrote and stored data unti…

Magnetic Domains

Why can some materials be turned into magnets? It’s all thanks to magnetic domains.

Magnetic Field Around a Wire

Whenever current travels through a conductor, a magnetic field is generated.

Magnetic Field of a Solenoid

You can create a stronger, more concentrated magnetic field by taking wire and forming it into a coil called a solenoid.

Magnetic Resonance Imaging (MRI)

Magnetic Resonance Imaging machines, commonly known as MRIs, are awesome diagnostic tools for medical applications and research. Relying on strong sup…

Magnetic Shunt

Magnetic shunts are often used to adjust the amount of flux in the magnetic circuits found in most electrical motors.

Mass Spectra

Mass spectrum reveals how many isotopes of a given element are to be found in a material.

Mass Spectrometer (Dual Sector)

Mass spectrometers are instruments that give scientists insight into the composition of complex materials. These spectrometers can analyze materials a…

Mass Spectrometer (Single Sector)

Mass spectrometers are instruments that give scientists information on the composition of a material. Mass spectrometers can pick apart complex substa…

How does a microwave heat your food? Water interacting with high-frequency electromagnetic waves.

Ørsted's Compass

In 1820, Hans Christian Ørsted discovered the relationship between electricity and magnetism in this very simple experiment.

Parallel Wires

A pair of parallel wires serves to illustrate a principle that French scientist André-Marie Ampère was the first to comprehend.

Pixii Machine

This “magneto-electric machine” was the first to turn motion into electricity.

Simple Electrical Cell

The simple electrical cell explained here is the most basic type of "wet" cell and demonstrates the fundamental chemistry behind batteries.

Tape Recorder

Two heads — or even three — are better than one when it comes to understanding how tape recorders harness electromagnetic induction.

Transformers

Transformers are devices that transfer a voltage from one circuit to another circuit via induction.

Transmission Lines

Electricity goes through some ups and downs on its way from the power plant to your house. Here's how it works.

Van de Graaff Generator

The Van de Graaff generator is a popular tool for teaching the principles of electrostatics. You might remember it as the thing that made your hair st…

Voltaic Pile

Italian scientist Alessandro Volta was the first to recognize key principles of electrochemistry, and applied those principles to the creation of the …

Wheatstone Bridge

This circuit is most commonly used to determine the value of an unknown resistance to an electrical current.

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1 Electromagnetism

1–1 Electrical forces

Consider a force like gravitation which varies predominantly inversely as the square of the distance, but which is about a billion-billion-billion-billion times stronger. And with another difference. There are two kinds of “matter,” which we can call positive and negative. Like kinds repel and unlike kinds attract—unlike gravity where there is only attraction. What would happen?

A bunch of positives would repel with an enormous force and spread out in all directions. A bunch of negatives would do the same. But an evenly mixed bunch of positives and negatives would do something completely different. The opposite pieces would be pulled together by the enormous attractions. The net result would be that the terrific forces would balance themselves out almost perfectly, by forming tight, fine mixtures of the positive and the negative, and between two separate bunches of such mixtures there would be practically no attraction or repulsion at all.

There is such a force: the electrical force. And all matter is a mixture of positive protons and negative electrons which are attracting and repelling with this great force. So perfect is the balance, however, that when you stand near someone else you don’t feel any force at all. If there were even a little bit of unbalance you would know it. If you were standing at arm’s length from someone and each of you had one percent more electrons than protons, the repelling force would be incredible. How great? Enough to lift the Empire State Building? No! To lift Mount Everest? No! The repulsion would be enough to lift a “weight” equal to that of the entire earth!

With such enormous forces so perfectly balanced in this intimate mixture, it is not hard to understand that matter, trying to keep its positive and negative charges in the finest balance, can have a great stiffness and strength. The Empire State Building, for example, swings less than one inch in the wind because the electrical forces hold every electron and proton more or less in its proper place. On the other hand, if we look at matter on a scale small enough that we see only a few atoms, any small piece will not, usually, have an equal number of positive and negative charges, and so there will be strong residual electrical forces. Even when there are equal numbers of both charges in two neighboring small pieces, there may still be large net electrical forces because the forces between individual charges vary inversely as the square of the distance. A net force can arise if a negative charge of one piece is closer to the positive than to the negative charges of the other piece. The attractive forces can then be larger than the repulsive ones and there can be a net attraction between two small pieces with no excess charges. The force that holds the atoms together, and the chemical forces that hold molecules together, are really electrical forces acting in regions where the balance of charge is not perfect, or where the distances are very small.

You know, of course, that atoms are made with positive protons in the nucleus and with electrons outside. You may ask: “If this electrical force is so terrific, why don’t the protons and electrons just get on top of each other? If they want to be in an intimate mixture, why isn’t it still more intimate?” The answer has to do with the quantum effects. If we try to confine our electrons in a region that is very close to the protons, then according to the uncertainty principle they must have some mean square momentum which is larger the more we try to confine them. It is this motion, required by the laws of quantum mechanics, that keeps the electrical attraction from bringing the charges any closer together.

There is another question: “What holds the nucleus together”? In a nucleus there are several protons, all of which are positive. Why don’t they push themselves apart? It turns out that in nuclei there are, in addition to electrical forces, nonelectrical forces, called nuclear forces, which are greater than the electrical forces and which are able to hold the protons together in spite of the electrical repulsion. The nuclear forces, however, have a short range—their force falls off much more rapidly than $1/r^2$. And this has an important consequence. If a nucleus has too many protons in it, it gets too big, and it will not stay together. An example is uranium, with 92 protons. The nuclear forces act mainly between each proton (or neutron) and its nearest neighbor, while the electrical forces act over larger distances, giving a repulsion between each proton and all of the others in the nucleus. The more protons in a nucleus, the stronger is the electrical repulsion, until, as in the case of uranium, the balance is so delicate that the nucleus is almost ready to fly apart from the repulsive electrical force. If such a nucleus is just “tapped” lightly (as can be done by sending in a slow neutron), it breaks into two pieces, each with positive charge, and these pieces fly apart by electrical repulsion. The energy which is liberated is the energy of the atomic bomb. This energy is usually called “nuclear” energy, but it is really “electrical” energy released when electrical forces have overcome the attractive nuclear forces.

We may ask, finally, what holds a negatively charged electron together (since it has no nuclear forces). If an electron is all made of one kind of substance, each part should repel the other parts. Why, then, doesn’t it fly apart? But does the electron have “parts”? Perhaps we should say that the electron is just a point and that electrical forces only act between different point charges, so that the electron does not act upon itself. Perhaps. All we can say is that the question of what holds the electron together has produced many difficulties in the attempts to form a complete theory of electromagnetism. The question has never been answered. We will entertain ourselves by discussing this subject some more in later chapters.

As we have seen, we should expect that it is a combination of electrical forces and quantum-mechanical effects that will determine the detailed structure of materials in bulk, and, therefore, their properties. Some materials are hard, some are soft. Some are electrical “conductors”—because their electrons are free to move about; others are “insulators”—because their electrons are held tightly to individual atoms. We shall consider later how some of these properties come about, but that is a very complicated subject, so we will begin by looking at the electrical forces only in simple situations. We begin by treating only the laws of electricity—including magnetism, which is really a part of the same subject.

We have said that the electrical force, like a gravitational force, decreases inversely as the square of the distance between charges. This relationship is called Coulomb’s law. But it is not precisely true when charges are moving—the electrical forces depend also on the motions of the charges in a complicated way. One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect. That is why we call the subject “electromagnetism.”

There is an important general principle that makes it possible to treat electromagnetic forces in a relatively simple way. We find, from experiment, that the force that acts on a particular charge—no matter how many other charges there are or how they are moving—depends only on the position of that particular charge, on the velocity of the charge, and on the amount of charge. We can write the force $\FLPF$ on a charge $q$ moving with a velocity $\FLPv$ as \begin{equation} \label{Eq:II:1:1} \FLPF=q(\FLPE+\FLPv\times\FLPB). \end{equation} We call $\FLPE$ the electric field and $\FLPB$ the magnetic field at the location of the charge. The important thing is that the electrical forces from all the other charges in the universe can be summarized by giving just these two vectors. Their values will depend on where the charge is, and may change with time . Furthermore, if we replace that charge with another charge, the force on the new charge will be just in proportion to the amount of charge so long as all the rest of the charges in the world do not change their positions or motions. (In real situations, of course, each charge produces forces on all other charges in the neighborhood and may cause these other charges to move, and so in some cases the fields can change if we replace our particular charge by another.)

We know from Vol. I how to find the motion of a particle if we know the force on it. Equation ( 1.1 ) can be combined with the equation of motion to give \begin{equation} \label{Eq:II:1:2} \ddt{}{t}\biggl[\frac{m\FLPv}{(1-v^2/c^2)^{1/2}}\biggr]= \FLPF=q(\FLPE+\FLPv\times\FLPB). \end{equation} So if $\FLPE$ and $\FLPB$ are given, we can find the motions. Now we need to know how the $\FLPE$’s and $\FLPB$’s are produced.

One of the most important simplifying principles about the way the fields are produced is this: Suppose a number of charges moving in some manner would produce a field $\FLPE_1$, and another set of charges would produce $\FLPE_2$. If both sets of charges are in place at the same time (keeping the same locations and motions they had when considered separately), then the field produced is just the sum \begin{equation} \label{Eq:II:1:3} \FLPE=\FLPE_1+\FLPE_2. \end{equation} This fact is called the principle of superposition of fields. It holds also for magnetic fields.

This principle means that if we know the law for the electric and magnetic fields produced by a single charge moving in an arbitrary way, then all the laws of electrodynamics are complete. If we want to know the force on charge $A$ we need only calculate the $\FLPE$ and $\FLPB$ produced by each of the charges $B$, $C$, $D$, etc., and then add the $\FLPE$’s and $\FLPB$’s from all the charges to find the fields, and from them the forces acting on charge $A$. If it had only turned out that the field produced by a single charge was simple, this would be the neatest way to describe the laws of electrodynamics. We have already given a description of this law (Chapter  28 , Vol. I) and it is, unfortunately, rather complicated.

It turns out that the forms in which the laws of electrodynamics are simplest are not what you might expect. It is not simplest to give a formula for the force that one charge produces on another. It is true that when charges are standing still the Coulomb force law is simple, but when charges are moving about the relations are complicated by delays in time and by the effects of acceleration, among others. As a result, we do not wish to present electrodynamics only through the force laws between charges; we find it more convenient to consider another point of view—a point of view in which the laws of electrodynamics appear to be the most easily manageable.

1–2 Electric and magnetic fields

First, we must extend, somewhat, our ideas of the electric and magnetic vectors, $\FLPE$ and $\FLPB$. We have defined them in terms of the forces that are felt by a charge. We wish now to speak of electric and magnetic fields at a point even when there is no charge present. We are saying, in effect, that since there are forces “acting on” the charge, there is still “something” there when the charge is removed. If a charge located at the point $(x,y,z)$ at the time $t$ feels the force $\FLPF$ given by Eq. ( 1.1 ) we associate the vectors $\FLPE$ and $\FLPB$ with the point in space $(x,y,z)$. We may think of $\FLPE(x,y,z,t)$ and $\FLPB(x,y,z,t)$ as giving the forces that would be experienced at the time $t$ by a charge located at $(x,y,z)$, with the condition that placing the charge there did not disturb the positions or motions of all the other charges responsible for the fields.

Following this idea, we associate with every point $(x,y,z)$ in space two vectors $\FLPE$ and $\FLPB$, which may be changing with time. The electric and magnetic fields are, then, viewed as vector functions of $x$, $y$, $z$, and $t$. Since a vector is specified by its components, each of the fields $\FLPE(x,y,z,t)$ and $\FLPB(x,y,z,t)$ represents three mathematical functions of $x$, $y$, $z$, and $t$.

It is precisely because $\FLPE$ (or $\FLPB$) can be specified at every point in space that it is called a “field.” A “field” is any physical quantity which takes on different values at different points in space. Temperature, for example, is a field—in this case a scalar field, which we write as $T(x,y,z)$. The temperature could also vary in time, and we would say the temperature field is time-dependent, and write $T(x,y,z,t)$. Another example is the “velocity field” of a flowing liquid. We write $\FLPv(x,y,z,t)$ for the velocity of the liquid at each point in space at the time $t$. It is a vector field.

Returning to the electromagnetic fields—although they are produced by charges according to complicated formulas, they have the following important characteristic: the relationships between the values of the fields at one point and the values at a nearby point are very simple. With only a few such relationships in the form of differential equations we can describe the fields completely. It is in terms of such equations that the laws of electrodynamics are most simply written.

There have been various inventions to help the mind visualize the behavior of fields. The most correct is also the most abstract: we simply consider the fields as mathematical functions of position and time. We can also attempt to get a mental picture of the field by drawing vectors at many points in space, each of which gives the field strength and direction at that point. Such a representation is shown in Fig.  1–1 . We can go further, however, and draw lines which are everywhere tangent to the vectors—which, so to speak, follow the arrows and keep track of the direction of the field. When we do this we lose track of the lengths of the vectors, but we can keep track of the strength of the field by drawing the lines far apart when the field is weak and close together when it is strong. We adopt the convention that the number of lines per unit area at right angles to the lines is proportional to the field strength . This is, of course, only an approximation, and it will require, in general, that new lines sometimes start up in order to keep the number up to the strength of the field. The field of Fig.  1–1 is represented by field lines in Fig.  1–2 .

1–3 Characteristics of vector fields

There are two mathematically important properties of a vector field which we will use in our description of the laws of electricity from the field point of view. Suppose we imagine a closed surface of some kind and ask whether we are losing “something” from the inside; that is, does the field have a quality of “outflow”? For instance, for a velocity field we might ask whether the velocity is always outward on the surface or, more generally, whether more fluid flows out (per unit time) than comes in. We call the net amount of fluid going out through the surface per unit time the “flux of velocity” through the surface. The flow through an element of a surface is just equal to the component of the velocity perpendicular to the surface times the area of the surface. For an arbitrary closed surface, the net outward flow —or flux —is the average outward normal component of the velocity, times the area of the surface: \begin{equation} \label{Eq:II:1:4} \text{Flux}=(\text{average normal component})\cdot(\text{surface area}). \end{equation} \begin{equation} \label{Eq:II:1:4} \text{Flux}= \begin{pmatrix} \text{average}\\[-.75ex] \text{normal}\\[-.75ex] \text{component} \end{pmatrix} \cdot \begin{pmatrix} \text{surface}\\[-.75ex] \text{area} \end{pmatrix}. \end{equation}

In the case of an electric field, we can mathematically define something analogous to an outflow, and we again call it the flux, but of course it is not the flow of any substance, because the electric field is not the velocity of anything. It turns out, however, that the mathematical quantity which is the average normal component of the field still has a useful significance. We speak, then, of the electric flux —also defined by Eq. ( 1.4 ). Finally, it is also useful to speak of the flux not only through a completely closed surface, but through any bounded surface. As before, the flux through such a surface is defined as the average normal component of a vector times the area of the surface. These ideas are illustrated in Fig.  1–3 .

There is a second property of a vector field that has to do with a line, rather than a surface. Suppose again that we think of a velocity field that describes the flow of a liquid. We might ask this interesting question: Is the liquid circulating? By that we mean: Is there a net rotational motion around some loop? Suppose that we instantaneously freeze the liquid everywhere except inside of a tube which is of uniform bore, and which goes in a loop that closes back on itself as in Fig.  1–4 . Outside of the tube the liquid stops moving, but inside the tube it may keep on moving because of the momentum in the trapped liquid—that is, if there is more momentum heading one way around the tube than the other. We define a quantity called the circulation as the resulting speed of the liquid in the tube times its circumference. We can again extend our ideas and define the “circulation” for any vector field (even when there isn’t anything moving). For any vector field the circulation around any imagined closed curve is defined as the average tangential component of the vector (in a consistent sense) multiplied by the circumference of the loop (Fig.  1–5 ): \begin{equation} \label{Eq:II:1:5} \text{Circulation}=(\text{average tangential component})\cdot(\text{distance around}). \end{equation} \begin{equation} \label{Eq:II:1:5} \text{Circulation}= \begin{pmatrix} \text{average}\\[-.75ex] \text{tangential}\\[-.75ex] \text{component} \end{pmatrix} \cdot \begin{pmatrix} \text{distance}\\[-.75ex] \text{around} \end{pmatrix} \end{equation} You will see that this definition does indeed give a number which is proportional to the circulation velocity in the quickly frozen tube described above.

With just these two ideas—flux and circulation—we can describe all the laws of electricity and magnetism at once. You may not understand the significance of the laws right away, but they will give you some idea of the way the physics of electromagnetism will be ultimately described.

1–4 The laws of electromagnetism

The first law of electromagnetism describes the flux of the electric field: \begin{equation} \label{Eq:II:1:6} \text{The flux of $\FLPE$ through any closed surface}= \frac{\text{the net charge inside}}{\epsO}, \end{equation} \begin{equation} \label{Eq:II:1:6} \begin{pmatrix} \text{Flux of $\FLPE$}\\[-.5ex] \text{through any}\\[-.5ex] \text{closed surface} \end{pmatrix} = \frac{\begin{pmatrix} \text{net charge}\\[-.5ex] \text{inside} \end{pmatrix} }{\epsO}, \end{equation} where $\epsO$ is a convenient constant. (The constant $\epsO$ is usually read as “epsilon-zero” or “epsilon-naught”.) If there are no charges inside the surface, even though there are charges nearby outside the surface, the average normal component of $\FLPE$ is zero, so there is no net flux through the surface. To show the power of this type of statement, we can show that Eq. ( 1.6 ) is the same as Coulomb’s law, provided only that we also add the idea that the field from a single charge is spherically symmetric. For a point charge, we draw a sphere around the charge. Then the average normal component is just the value of the magnitude of $\FLPE$ at any point, since the field must be directed radially and have the same strength for all points on the sphere. Our rule now says that the field at the surface of the sphere, times the area of the sphere—that is, the outgoing flux—is proportional to the charge inside. If we were to make the radius of the sphere bigger, the area would increase as the square of the radius. The average normal component of the electric field times that area must still be equal to the same charge inside, and so the field must decrease as the square of the distance—we get an “inverse square” field.

If we have an arbitrary stationary curve in space and measure the circulation of the electric field around the curve, we will find that it is not, in general, zero (although it is for the Coulomb field). Rather, for electricity there is a second law that states: for any surface $S$ (not closed) whose edge is the curve $C$, \begin{equation} \label{Eq:II:1:7} \text{Circulation of $\FLPE$ around $C$}=-\ddt{}{t}(\text{flux of $\FLPB$ through $S$}). \end{equation} \begin{equation} \label{Eq:II:1:7} \begin{pmatrix} \text{Circulation of $\FLPE$}\\[-.5ex] \text{around $C$} \end{pmatrix} =-\ddt{}{t}\begin{pmatrix} \text{flux of $\FLPB$}\\[-.5ex] \text{through $S$} \end{pmatrix}. \end{equation}

We can complete the laws of the electromagnetic field by writing two corresponding equations for the magnetic field $\FLPB$: \begin{equation} \label{Eq:II:1:8} \text{Flux of $\FLPB$ through any closed surface}=0. \end{equation} \begin{equation} \label{Eq:II:1:8} \begin{pmatrix} \text{Flux of $\FLPB$}\\[-.5ex] \text{through any}\\[-.5ex] \text{closed surface} \end{pmatrix} =0. \end{equation} For a surface $S$ bounded by the curve $C$, \begin{align} c^2(\text{circulation of $\FLPB$ around $C$})=&\ddt{}{t}(\text{flux of $\FLPE$ through $S$})\notag\\ \label{Eq:II:1:9} &+\frac{\text{flux of electric current through $S$}}{\epsO}. \end{align} \begin{gather} \label{Eq:II:1:9} c^2 \begin{pmatrix} \text{circulation of $\FLPB$}\\[-.5ex] \text{around $C$} \end{pmatrix} =\\[1.5ex] \ddt{}{t} \begin{pmatrix} \text{flux of $\FLPE$}\\[-.5ex] \text{through $S$} \end{pmatrix} +\frac{ \begin{pmatrix} \text{flux of}\\[-.75ex] \text{electric current}\\[-.5ex] \text{through $S$} \end{pmatrix} }{\epsO}.\notag \end{gather}

The constant $c^2$ that appears in Eq. ( 1.9 ) is the square of the velocity of light. It appears because magnetism is in reality a relativistic effect of electricity. The constant $\epsO$ has been stuck in to make the units of electric current come out in a convenient way.

Equations ( 1.6 ) through ( 1.9 ), together with Eq. ( 1.1 ), are all the laws of electrodynamics 1 . As you remember, the laws of Newton were very simple to write down, but they had a lot of complicated consequences and it took us a long time to learn about them all. These laws are not nearly as simple to write down, which means that the consequences are going to be more elaborate and it will take us quite a lot of time to figure them all out.

We can illustrate some of the laws of electrodynamics by a series of small experiments which show qualitatively the interrelationships of electric and magnetic fields. You have experienced the first term of Eq. ( 1.1 ) when combing your hair, so we won’t show that one. The second part of Eq. ( 1.1 ) can be demonstrated by passing a current through a wire which hangs above a bar magnet, as shown in Fig.  1–6 . The wire will move when a current is turned on because of the force $\FLPF=q\FLPv\times\FLPB$. When a current exists, the charges inside the wire are moving, so they have a velocity $\FLPv$, and the magnetic field from the magnet exerts a force on them, which results in pushing the wire sideways.

When the wire is pushed to the left, we would expect that the magnet must feel a push to the right. (Otherwise we could put the whole thing on a wagon and have a propulsion system that didn’t conserve momentum!) Although the force is too small to make movement of the bar magnet visible, a more sensitively supported magnet, like a compass needle, will show the movement.

How does the wire push on the magnet? The current in the wire produces a magnetic field of its own that exerts forces on the magnet. According to the last term in Eq. ( 1.9 ), a current must have a circulation of $\FLPB$—in this case, the lines of $\FLPB$ are loops around the wire, as shown in Fig.  1–7 . This $\FLPB$-field is responsible for the force on the magnet.

Equation ( 1.9 ) tells us that for a fixed current through the wire the circulation of $\FLPB$ is the same for any curve that surrounds the wire. For curves—say circles—that are farther away from the wire, the circumference is larger, so the tangential component of $\FLPB$ must decrease. You can see that we would, in fact, expect $\FLPB$ to decrease linearly with the distance from a long straight wire.

Now, we have said that a current through a wire produces a magnetic field, and that when there is a magnetic field present there is a force on a wire carrying a current. Then we should also expect that if we make a magnetic field with a current in one wire, it should exert a force on another wire which also carries a current. This can be shown by using two hanging wires as shown in Fig.  1–8 . When the currents are in the same direction, the two wires attract, but when the currents are opposite, they repel.

In short, electrical currents, as well as magnets, make magnetic fields. But wait, what is a magnet, anyway? If magnetic fields are produced by moving charges, is it not possible that the magnetic field from a piece of iron is really the result of currents? It appears to be so. We can replace the bar magnet of our experiment with a coil of wire, as shown in Fig.  1–9 . When a current is passed through the coil—as well as through the straight wire above it—we observe a motion of the wire exactly as before, when we had a magnet instead of a coil. In other words, the current in the coil imitates a magnet. It appears, then, that a piece of iron acts as though it contains a perpetual circulating current. We can, in fact, understand magnets in terms of permanent currents in the atoms of the iron. The force on the magnet in Fig.  1–7 is due to the second term in Eq. ( 1.1 ).

Where do the currents come from? One possibility would be from the motion of the electrons in atomic orbits. Actually, that is not the case for iron, although it is for some materials. In addition to moving around in an atom, an electron also spins about on its own axis—something like the spin of the earth—and it is the current from this spin that gives the magnetic field in iron. (We say “something like the spin of the earth” because the question is so deep in quantum mechanics that the classical ideas do not really describe things too well.) In most substances, some electrons spin one way and some spin the other, so the magnetism cancels out, but in iron—for a mysterious reason which we will discuss later—many of the electrons are spinning with their axes lined up, and that is the source of the magnetism.

Since the fields of magnets are from currents, we do not have to add any extra term to Eqs. ( 1.8 ) or ( 1.9 ) to take care of magnets. We just take all currents, including the circulating currents of the spinning electrons, and then the law is right. You should also notice that Eq. ( 1.8 ) says that there are no magnetic “charges” analogous to the electrical charges appearing on the right side of Eq. ( 1.6 ). None has been found.

The first term on the right-hand side of Eq. ( 1.9 ) was discovered theoretically by Maxwell and is of great importance. It says that changing electric fields produce magnetic effects. In fact, without this term the equation would not make sense, because without it there could be no currents in circuits that are not complete loops. But such currents do exist, as we can see in the following example. Imagine a capacitor made of two flat plates. It is being charged by a current that flows toward one plate and away from the other, as shown in Fig.  1–10 . We draw a curve $C$ around one of the wires and fill it in with a surface which crosses the wire, as shown by the surface $S_1$ in the figure. According to Eq. ( 1.9 ), the circulation of $\FLPB$ around $C$ (times $c^2$) is given by the current in the wire (divided by $\epsO$). But what if we fill in the curve with a different surface $S_2$, which is shaped like a bowl and passes between the plates of the capacitor, staying always away from the wire? There is certainly no current through this surface. But, surely, just changing the location of an imaginary surface is not going to change a real magnetic field! The circulation of $\FLPB$ must be what it was before. The first term on the right-hand side of Eq. ( 1.9 ) does, indeed, combine with the second term to give the same result for the two surfaces $S_1$ and $S_2$. For $S_2$ the circulation of $\FLPB$ is given in terms of the rate of change of the flux of $\FLPE$ between the plates of the capacitor. And it works out that the changing $\FLPE$ is related to the current in just the way required for Eq. ( 1.9 ) to be correct. Maxwell saw that it was needed, and he was the first to write the complete equation.

With the setup shown in Fig.  1–6 we can demonstrate another of the laws of electromagnetism. We disconnect the ends of the hanging wire from the battery and connect them to a galvanometer which tells us when there is a current through the wire. When we push the wire sideways through the magnetic field of the magnet, we observe a current. Such an effect is again just another consequence of Eq. ( 1.1 )—the electrons in the wire feel the force $\FLPF=q\FLPv\times\FLPB$. The electrons have a sidewise velocity because they move with the wire. This $\FLPv$ with a vertical $\FLPB$ from the magnet results in a force on the electrons directed along the wire, which starts the electrons moving toward the galvanometer.

Suppose, however, that we leave the wire alone and move the magnet. We guess from relativity that it should make no difference, and indeed, we observe a similar current in the galvanometer. How does the magnetic field produce forces on charges at rest? According to Eq. ( 1.1 ) there must be an electric field. A moving magnet must make an electric field. How that happens is said quantitatively by Eq. ( 1.7 ). This equation describes many phenomena of great practical interest, such as those that occur in electric generators and transformers.

The most remarkable consequence of our equations is that the combination of Eq. ( 1.7 ) and Eq. ( 1.9 ) contains the explanation of the radiation of electromagnetic effects over large distances. The reason is roughly something like this: suppose that somewhere we have a magnetic field which is increasing because, say, a current is turned on suddenly in a wire. Then by Eq. ( 1.7 ) there must be a circulation of an electric field. As the electric field builds up to produce its circulation, then according to Eq. ( 1.9 ) a magnetic circulation will be generated. But the building up of this magnetic field will produce a new circulation of the electric field, and so on. In this way fields work their way through space without the need of charges or currents except at their source. That is the way we see each other! It is all in the equations of the electromagnetic fields.

1–5 What are the fields?

We now make a few remarks on our way of looking at this subject. You may be saying: “All this business of fluxes and circulations is pretty abstract. There are electric fields at every point in space; then there are these ‘laws.’ But what is actually happening? Why can’t you explain it, for instance, by whatever it is that goes between the charges.” Well, it depends on your prejudices. Many physicists used to say that direct action with nothing in between was inconceivable. (How could they find an idea inconceivable when it had already been conceived?) They would say: “Look, the only forces we know are the direct action of one piece of matter on another. It is impossible that there can be a force with nothing to transmit it.” But what really happens when we study the “direct action” of one piece of matter right against another? We discover that it is not one piece right against the other; they are slightly separated, and there are electrical forces acting on a tiny scale. Thus we find that we are going to explain so-called direct-contact action in terms of the picture for electrical forces. It is certainly not sensible to try to insist that an electrical force has to look like the old, familiar, muscular push or pull, when it will turn out that the muscular pushes and pulls are going to be interpreted as electrical forces! The only sensible question is what is the most convenient way to look at electrical effects. Some people prefer to represent them as the interaction at a distance of charges, and to use a complicated law. Others love the field lines. They draw field lines all the time, and feel that writing $\FLPE$’s and $\FLPB$’s is too abstract. The field lines, however, are only a crude way of describing a field, and it is very difficult to give the correct, quantitative laws directly in terms of field lines. Also, the ideas of the field lines do not contain the deepest principle of electrodynamics, which is the superposition principle. Even though we know how the field lines look for one set of charges and what the field lines look like for another set of charges, we don’t get any idea about what the field line patterns will look like when both sets are present together. From the mathematical standpoint, on the other hand, superposition is easy—we simply add the two vectors. The field lines have some advantage in giving a vivid picture, but they also have some disadvantages. The direct interaction way of thinking has great advantages when thinking of electrical charges at rest, but has great disadvantages when dealing with charges in rapid motion.

The best way is to use the abstract field idea. That it is abstract is unfortunate, but necessary. The attempts to try to represent the electric field as the motion of some kind of gear wheels, or in terms of lines, or of stresses in some kind of material have used up more effort of physicists than it would have taken simply to get the right answers about electrodynamics. It is interesting that the correct equations for the behavior of light were worked out by MacCullagh in 1839. But people said to him: “Yes, but there is no real material whose mechanical properties could possibly satisfy those equations, and since light is an oscillation that must vibrate in something , we cannot believe this abstract equation business.” If people had been more open-minded, they might have believed in the right equations for the behavior of light a lot earlier than they did.

In the case of the magnetic field we can make the following point: Suppose that you finally succeeded in making up a picture of the magnetic field in terms of some kind of lines or of gear wheels running through space. Then you try to explain what happens to two charges moving in space, both at the same speed and parallel to each other. Because they are moving, they will behave like two currents and will have a magnetic field associated with them (like the currents in the wires of Fig.  1–8 ). An observer who was riding along with the two charges, however, would see both charges as stationary, and would say that there is no magnetic field. The “gear wheels” or “lines” disappear when you ride along with the object! All we have done is to invent a new problem. How can the gear wheels disappear?! The people who draw field lines are in a similar difficulty. Not only is it not possible to say whether the field lines move or do not move with charges—they may disappear completely in certain coordinate frames.

What we are saying, then, is that magnetism is really a relativistic effect. In the case of the two charges we just considered, travelling parallel to each other, we would expect to have to make relativistic corrections to their motion, with terms of order $v^2/c^2$. These corrections must correspond to the magnetic force. But what about the force between the two wires in our experiment (Fig.  1–8 ). There the magnetic force is the whole force. It didn’t look like a “relativistic correction.” Also, if we estimate the velocities of the electrons in the wire (you can do this yourself), we find that their average speed along the wire is about $0.01$ centimeter per second. So $v^2/c^2$ is about $10^{-25}$. Surely a negligible “correction.” But no! Although the magnetic force is, in this case, $10^{-25}$ of the “normal” electrical force between the moving electrons, remember that the “normal” electrical forces have disappeared because of the almost perfect balancing out—because the wires have the same number of protons as electrons. The balance is much more precise than one part in $10^{25}$, and the small relativistic term which we call the magnetic force is the only term left. It becomes the dominant term.

It is the near-perfect cancellation of electrical effects which allowed relativity effects (that is, magnetism) to be studied and the correct equations—to order $v^2/c^2$—to be discovered, even though physicists didn’t know that’s what was happening. And that is why, when relativity was discovered, the electromagnetic laws didn’t need to be changed. They—unlike mechanics—were already correct to a precision of $v^2/c^2$.

1–6 Electromagnetism in science and technology

Let us end this chapter by pointing out that among the many phenomena studied by the Greeks there were two very strange ones: that if you rubbed a piece of amber you could lift up little pieces of papyrus, and that there was a strange rock from the land of Magnesia which attracted iron. It is amazing to think that these were the only phenomena known to the Greeks in which the effects of electricity or magnetism were apparent. The reason that these were the only phenomena that appeared is due primarily to the fantastic precision of the balancing of charges that we mentioned earlier. Study by scientists who came after the Greeks uncovered one new phenomenon after another that were really some aspect of these amber and/or lodestone effects. Now we realize that the phenomena of chemical interaction and, ultimately, of life itself are to be understood in terms of electromagnetism.

At the same time that an understanding of the subject of electromagnetism was being developed, technical possibilities that defied the imagination of the people that came before were appearing: it became possible to signal by telegraph over long distances, and to talk to another person miles away without any connections between, and to run huge power systems—a great water wheel, connected by filaments over hundreds of miles to another engine that turns in response to the master wheel—many thousands of branching filaments—ten thousand engines in ten thousand places running the machines of industries and homes—all turning because of the knowledge of the laws of electromagnetism.

Today we are applying even more subtle effects. The electrical forces, enormous as they are, can also be very tiny, and we can control them and use them in very many ways. So delicate are our instruments that we can tell what a man is doing by the way he affects the electrons in a thin metal rod hundreds of miles away. All we need to do is to use the rod as an antenna for a television receiver!

From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.

• We need only to add a remark about some conventions for the sign of the circulation. ↩

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20. Electromagnetic force experiment

Experiment to demonstrate the force on a charged particle through a magnetic field

GCSE Keywords: Electromagnetic force, Charge, Particle, Magnetic field, Moving, Motor effect, Fleming's left hand rule

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• Intro Lab - Build an Electromagnet

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In this hands-on electronics experiment, you will build an electromagnet and learn about electromagnetism including the relationship of magnetic polarity to current flow.

Project overview.

In this project, you will build and test the electromagnet circuit illustrated in Figure 1.  Electromagnetism has many applications, including:

• Electric motors
• Computer printer mechanisms
• Magnetic media write heads (tape recorders and disk drives)

Figure 1. Electromagnet circuit for generating a magnetic field from an electric current.

Parts and materials.

• 6 V battery
• Magnetic compass
• Small permanent magnet
• Spool of 28-gauge magnet wire
• Large bolt, nail, or steel rod
• Electrical tape

Magnet wire is a term for thin-gauge copper wire with enamel insulation instead of rubber or plastic insulation. Its small size and very thin insulation allow for many turns to be wound in a compact coil. Keep in mind that you will need enough magnet wire to wrap hundreds of turns around the bolt, nail, or other rod-shaped steel forms.

Another thing, make sure to select a bolt, nail, or rod that is magnetic. Stainless steel, for example, is non-magnetic and will not function for the purpose of an electromagnet coil! The ideal material for this experiment is soft iron, but any commonly available steel will suffice.

Learning Objectives

• Application of the left-hand rule
• Electromagnet construction

Instructions

Step 1:  Wrap a single layer of electrical tape around the steel bar (or bolt or mail) to protect the wire from abrasion.

Step 2:  Proceed to wrap several hundred turns of wire around the steel bar, making the coil as even as possible. It is okay to overlap wire, and it is okay to wrap in the same style that a fishing reel wraps the line around the spool. The only rule you must follow is that all turns must be wrapped around the bar in the same direction (no reversing from clockwise to counter-clockwise!).

I find that a drill press works as a great tool for coil winding: clamp the rod in the drill’s chuck as if it were a drill bit, then turn the drill motor on at a slow speed and let it do the wrapping! This allows you to feed wire onto the rod in a very steady, even manner.

Step 3:  After you’ve wrapped several hundred turns of wire around the rod, wrap a layer or two of electrical tape over the wire coil to secure the wire in place.

Step 4:  Scrape the enamel insulation off the ends of the coil wires to expose the wire for connection to jumper leads

Step 5: Connect the coil to a battery, as illustrated in Figure 1 and defined in the circuit schematic of Figure 2.

Figure 2.  Schematic diagram of the electromagnet circuit.

Step 6:  When the electric current goes through the coil, it will produce a strong magnetic field with one pole at each end of the rod. This phenomenon is known as electromagnetism. With the electromagnet energized (connected to the battery), use the magnetic compass to identify the north and south poles of the electromagnet. 

Step 7: Place a permanent magnet near one pole and note whether there is an attractive or repulsive force.

Step 8:  Reverse the orientation of the permanent magnet and repeat steps 7 and 8. Note the difference in force caused by changing the polarity of the applied voltage and the direction of the current flow. 

Inductive Kickback

You might notice a significant spark whenever the battery is disconnected from the electromagnet coil, much greater than the spark produced if the battery is short-circuited. This spark results from a high-voltage surge created whenever current is suddenly interrupted through the coil.

The effect is called inductive kickback  and can deliver a small but harmless electric shock. To avoid receiving this shock, do not place your body across the break in the circuit when de-energizing. Use one hand at a time when un-powering the coil, and you’ll be perfectly safe.

Related Content

Learn more about the fundamentals behind this project in the resources below.

• Magnetism and Electromagnetism
• Electromagnetism

Worksheets:

• Basic Electromagnetism and Electromagnetic Induction Worksheet
• Intermediate Electromagnetism and Electromagnetic Induction Worksheet
• Advanced Electromagnetism and Electromagnetic Induction Worksheet
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Electromagnetic Forces (A)

3. electromagnetic forces (a) #.

The primary objective of this experiment is to study the interaction between electrical and mechanical systems that are magnetically coupled. The electromechanical relay (solenoid) studied in Experiment 2 will be analyzed to predict its steady-state force-versus-position and force-versus-current characteristics. In the follow-up lab, computer simulation will be used to predict its dynamic force-versus-time characteristics. The analytical and simulated results will be compared with laboratory measurements.

3.1. Introduction #

In this lab, we will consider the cylindrical solenoid depicted in Fig. 3.1 . In the previous experiment, it was shown that the self-inductance can be approximated as

Here, \(K\) and \(k_0\) are assumed to be known constants and \(L_l\) is the leakage inductance. In the prelab, we will establish an expression for the force versus position and set forth the electrical and mechanical equations in state-space form. Hopefully, our experimental measurements will confirm the general form of this equation.

Fig. 3.1 Simplified cutaway view of cylindrical solenoid. #

3.2. Prelab #

For an electromechanical device whose inductance versus position is given by (3.1) , express the electromagnetic force as a function of current and position.

Using the \(K\) and \(k_0\) values (in metric units) determined in Experiment 2, plot \(f_e\) versus \(i\) for \(0 < i < \qty{200}{\mA}\) and \(x = \qty{0.0032}{\m}\) . Plot \(f_e\) versus \(x\) for \(0 < x < \qty{0.0127}{\m}\) and \(i = \qty{200}{\mA}\) .

3.3. In the Laboratory #

You must wear eye protection for this lab.

3.3.1. Force versus Position #

Our first objective will be to measure the force-versus-position characteristics with a constant current. You will need to measure current in the solenoid. A voltage signal proportional to the current is available on one of the BNC connectors of the power amplifier.

Energize the solenoid coil with \(\qty{12}{\V}\) (DC). Use \(R\) from previous experiment, calculate expected current. Measure coil current and compare with expected value. Provide reasonable explanation for any difference. Connect the lower BNC output to the oscilloscope to read the current on an open circuit. Record this value, which will be subtracted from future DC current measurements.

Position the solenoid plunger at \(x = 0\) (the set screw should be all the way out). Add weights to the solenoid plunger until plunger pulls out. Record the mass that is required to just pull the solenoid plunger out to nearest \(\qty{10}{\g}\) . Either round up or down but be consistent from one measurement to next.

Be careful not to drop the weights on your (or your lab partner’s) foot. You may use the lab chairs to prevent dropping weights on the floor.

Repeat the measurements with plunger position varied from \(0\) to \(\qty{1/2}{\inch}\) at \(\qty{1/32}{\inch}\) intervals (1 screw turn = \(\qty{1/32}{\inch}\) ). Ensure that the screw is touching the plunger before counting turns. If the minimum ( \(\qty{50}{\g}\) ) level is reached, then stop.

Calculate the plunger force using \(F=mg\) where \(g = \qty{9.8}{\meter\per\second\squared}\) and \(m\) is the external mass. This is approximately equal to the magnitude of the electromagnetic force developed by the solenoid.

3.3.2. Force versus Current #

Our next objective will be to measure force versus current at a fixed position.

Apply a DC voltage to the coil so that \(i=\qty{60}{\mA}\) . Use Ohm’s law to determine the required voltage using the value of \(R\) . Measure the current using the oscilloscope to be sure. If necessary, adjust the applied voltage to achieve the desired current. Position plunger at \(x = \qty{1/8}{\inch}\) . Add weight to coil plunger until plunger pulls out. Again, be careful not to drop the weights.

Repeat measurements at the same position for coil currents from \(60\) to \(\qty{105}{\mA}\) in \(5\) - \(\mA\) intervals. Record the maximum weight values as the current varies.

3.4. Postlab #

Plot, on the same axis, the measured and calculated electromagnetic force versus current with \(x = \qty{1/8}{\inch}.\) Compare the results and explain discrepancies.

The most common problem involves mixing Standard International (SI) and English units. It is best to convert all measured data to SI units before making any comparisons.

Plot, on the same axis, the measured and calculated electromagnetic force-versus-position characteristics with constant \(i\) . Compare the results and explain discrepancies.

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Electromagnetic Force

Electromagnetism is a branch of physics that establishes the relationship between electricity and magnetism. A magnetic field affects moving charges, resulting in an induced current. Moving charges produce magnetic fields. Therefore, the perception that magnetism and electricity are closely intertwined is correct.

What is Electromagnetic Force

The electromagnetic force is the force of interaction between electrically charged particles, like electrons and protons, either stationary or moving. It consists of two distinct forces – electric force and magnetic force . It is one of the four fundamental forces of nature, including gravitational force , strong nuclear force , and weak nuclear force . Like the other fundamental forces, it has an infinite range. The electromagnetic force has a strength of 1/137 relative to the strong nuclear force . However, it is 10 36 times stronger than the gravitational force.

How does Electromagnetic Force Work?

The electric force originates from the interaction between charged particles, whether they are stationary or moving. Opposite charges attract each other while like charges repel. Charged particles give rise to an electric field . However, when the particles start to move, they generate a magnetic field which gives rise to a magnetic force. The electromagnetic force is manifested when the electric and magnetic fields interact with the charged particles. It is an exchange force, and the particles that carry the force are known as photons.

Who Discovered Electromagnetic Force?

In 1820, Danish physicist Hans Christian Oersted was the first to observe that the needle of a magnetic compass deflects when placed near a current-carrying wire. Later that year, he published his results in a pamphlet.

Electromagnetic Force Examples

An effect of the electromagnetic force is that it is responsible for most interactions in nature. Here are some of its examples.

• Electrons moving around a nucleus
• The molecules of water
• Electromagnetic waves (E.g., light, radio waves, X-rays, and UV-rays)
• A current passing through a wire

Electromagnetic Force Equation

How to calculate electromagnetic force.

The electromagnetic force is a combination of the electric and the magnetic force. Hence, the equation is given by Lorentz force, whose formula is

Where q is the charge of the particle, \(\vec{v}\) is the velocity , \( \vec{E} \) is the electric field and \( \vec{B}\) is the magnetic field.

Applications of Electromagnetic Force

The electromagnetic force is applied in electromagnets, which find their way into many devices.

• Hard-drives
• Microphones
• Home security system
• Solenoid and toroid

Difference between Gravitational Force and Electromagnetic Force

Ans. An electromagnet is a temporary magnet that can be magnetized by passing an electric current. A permanent magnet has a permanent magnetism.

Ans. The difference between electrostatic and electromagnetic forces is that the former refers to the force between charges which are not moving relative to each other. On the other hand, electromagnetic force refers to electrostatic forces and other forces between moving charges and magnetic fields.

Ans. An electric generator is used to convert kinetic energy into electrical energy through electromagnetic induction . In this process, a turning shaft rotates a coil between two pole pieces of a magnet that generates an electric current.

• Electromagnetic Force – Energyeducation.ca
• Electromagnetic Force – Physics.info
• What is Electromagnetic Force? – Sciencing.com
• Introduction to the Electromagnetic Force – Wikilectures.eu
• Electromagnetic Forces and Fields – Cliffsnotes.com
• The Electromagnetic Force – Thestargarden.co.uk
• The Electromagnetic Force – Webs.morningside.edu
• Electromagnetic Force – Ecuip.lib.uchicago.edu
• Electromagnetic Force – Emandpplabs.nscee.edu
• The Four Fundamental Forces of Nature – Space.com
• Electric and Magnetic Forces – Web.mit.edu
• What is Electromagnetic Force vs. Gravitational Force – Radiation-dosimetry.org
• Electromagnetism – Nde-ed.org

Article was last reviewed on Friday, May 3, 2024

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• Published: 24 May 2024

Magnetic-field tuning of the Casimir force

• Yichi Zhang 1 , 2 , 3 ,
• Hui Zhang   ORCID: orcid.org/0000-0001-9749-2555 1 , 2 , 4 ,
• Xiuxia Wang 5 ,
• Yiheng Wang   ORCID: orcid.org/0009-0002-0840-0394 1 , 2 , 3 ,
• Yuchen Liu 1 , 2 , 3 ,
• Shu Li 1 , 2 , 3 ,
• Tianyi Zhang 1 , 2 ,
• Chuang Fan 1 , 2 , 3 &
• Changgan Zeng   ORCID: orcid.org/0000-0001-8630-845X 1 , 2 , 4  

Nature Physics ( 2024 ) Cite this article

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• Characterization and analytical techniques
• Quantum mechanics

The quantum fluctuation-induced Casimir force can be either attractive or repulsive, depending on the dielectric permittivities and magnetic permeabilities of the materials involved. However, it is challenging to manipulate the dielectric permittivities of most materials using external fields. In contrast, the magnetic permeabilities of ferrofluids can be readily tuned by magnetic fields, which opens up the possibility of magnetic-field tuning of the Casimir force. Here, we demonstrate that this tuning can be achieved for a gold sphere and a silica plate immersed in water-based ferrofluids. Our theoretical calculations predict that, by varying the magnetic field, separation distance and ferrofluid volume fraction, the Casimir force can be tuned from attractive to repulsive over a wide range of parameters in this system. Experimentally, we develop a cantilever designed to conduct measurements within water-based ferrofluids. Using this setup, we observe the predicted transitions. These findings may lead to the development of switchable micromechanical devices based on the Casimir effect.

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Casimir, H. B. G. On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet. 51 , 793–795 (1948).

Google Scholar  

Casimir, H. B. G. & Polder, D. The influence of retardation on the London–van der Waals forces. Phys. Rev. 73 , 360–372 (1948).

Article   ADS   Google Scholar  

Lifshitz, E. M. The theory of molecular attractive forces between solids. Sov. Phys. JETP 2 , 73–83 (1956).

Lamoreaux, S. K. Demonstration of the Casimir force in the 0.6 to 6 μm range. Phys. Rev. Lett. 78 , 5–8 (1998).

Mohideen, U. & Roy, A. Precision measurement of the Casimir force from 0.1 to 0.9 μm. Phys. Rev. Lett. 81 , 4549–4552 (1998).

Bressi, G., Carugno, G., Onofrio, R. & Ruoso, G. Measurement of the Casimir force between parallel metallic surfaces. Phys. Rev. Lett. 88 , 041804 (2002).

Rodriguez, A. W., Capasso, F. & Johnson, S. G. The Casimir effect in microstructured geometries. Nat. Photonics 5 , 211–221 (2011).

Zhao, R. et al. Stable Casimir equilibria and quantum trapping. Science 364 , 984–987 (2019).

Feiler, A. A., Bergström, L. & Rutland, M. W. Superlubricity using repulsive van der Waals forces. Langmuir 24 , 2274–2276 (2008).

Article   Google Scholar  

Woods, L. M. et al. Materials perspective on Casimir and van der Waals interactions. Rev. Mod. Phys. 88 , 045003 (2016).

Article   ADS   MathSciNet   Google Scholar  

Tang, L. et al. Measurement of non-monotonic Casimir forces between silicon nanostructures. Nat. Photonics 11 , 97–101 (2017).

Hertlein, C. et al. Direct measurement of critical Casimir forces. Nature 451 , 172–175 (2007).

Schmidt, F. et al. Tunable critical Casimir forces counteract Casimir–Lifshitz attraction. Nat. Phys. 19 , 271–278 (2023).

van Zwol, P. J., Palasantzas, G. & De Hosson, J. T. M. Influence of dielectric properties on van der Waals/Casimir forces in solid–liquid systems. Phys. Rev. B 79 , 195428 (2009).

van Zwol, P. J. & Palasantzas, G. Repulsive Casimir forces between solid materials with high-refractive-index intervening liquids. Phys. Rev. A 81 , 062502 (2010).

Hutter, J. L. & Bechhoefer, J. Manipulation of van der Waals forces to improve image resolution in atomic-force microscopy. J. Appl. Phys. 73 , 4123–4129 (1993).

Milling, A., Mulvaney, P. & Larson, I. Direct measurement of repulsive van der Waals interactions using an atomic force microscope. J. Colloid Int. Sci. 180 , 460–465 (1996).

Lee, S.-W. & Sigmund, W. M. Repulsive van der Waals forces for silica and alumina. J. Colloid Int. Sci. 243 , 365–369 (2001).

Munday, N., Capasso, F. & Parsegian, V. A. Measured long-range repulsive Casimir-Lifshitz forces. Nature 457 , 170–173 (2009).

Ma, J., Zhao, Q. & Meng, Y. Magnetically controllable Casimir force based on a superparamagnetic metametamaterial. Phys. Rev. B 89 , 075421 (2014).

Hu, Q., Ye, Y., Zhao, Q. & Meng, Y. Casimir repulsion in superparamagnetic metamaterial constructed by non-monodisperse nanoparticles. J. Phys. Condens. Matter 30 , 084003 (2018).

Klimchitskaya, G. L., Mostepanenko, V. M., Nepomnyashchaya, E. K. & Velichko, E. N. Impact of magnetic nanoparticles on the Casimir pressure in three-layer systems. Phys. Rev. B 99 , 045433 (2019).

Klimchitskaya, G. L., Mostepanenko, V. M. & Velichko, E. N. Effect of agglomeration of magnetic nanoparticles on the Casimir pressure through a ferrofluid. Phys. Rev. B 100 , 035422 (2019).

Banishev, A. A., Chang, C.-C., Klimchitskaya, G. L., Mostepanenko, V. M. & Mohideen, U. Measurement of the gradient of the Casimir force between a nonmagnetic gold sphere and a magnetic nickel plate. Phys. Rev. B 85 , 195422 (2012).

Banishev, A. A., Klimchitskaya, G. L., Mostepanenko, V. M. & Mohideen, U. Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate. Phys. Rev. Lett. 110 , 137401 (2013).

Banishev, A. A., Klimchitskaya, G. L., Mostepanenko, V. M. & Mohideen, U. Casimir interaction between two magnetic metals in comparison with nonmagnetic test bodies. Phys. Rev. B 88 , 155410 (2013).

Kole, M. & Khandekar, S. Engineering applications of ferrofluids: a review. J. Magn. Magn. Mater. 527 , 168222 (2021).

Bordag, M., Klimchitskaya, G. L., Mohideen, U. & Mostepanenko, V. M. Advances in the Casimir Effect (Oxford Univ. Press, 2015).

Ivanov, O. et al. Magnetic properties of polydisperse ferrofluids: a critical comparison between experiment, theory, and computer simulation. Phys. Rev. E 75 , 061405 (2007).

Le Cunuder, A., Petrosyan, A., Palasantzas, G., Svetovoy, V. & Ciliberto, S. Measurement of the Casimir force in a gas and in a liquid. Phys. Rev. B 98 , 201408(R) (2018).

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Acknowledgements

Y.Z., H.Z., X.W., Y.W., Y.L., S.L., T.Z., C.F. and C.Z. were supported by the National Key Research and Development Program of China (grant no. 2023YFA1406300), the Innovation Program for Quantum Science and Technology (grant no. 2021ZD0302800), the National Natural Science Foundation of China (grant nos. 92165201, 92265201 and 12074357), Anhui Provincial Key Research and Development Project (grant no. 2023z04020008), the CAS Project for Young Scientists in Basic Research (grant no. YSBR-046), the Geek Center Project of USTC and the Fundamental Research Funds for the Central Universities (grant nos. WK9990000118 and WK2310000104). Part of this work was carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.

Author information

Authors and affiliations.

CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics and Department of Physics, University of Science and Technology of China, Hefei, China

Yichi Zhang, Hui Zhang, Yiheng Wang, Yuchen Liu, Shu Li, Tianyi Zhang, Chuang Fan & Changgan Zeng

International Center for Quantum Design of Functional Materials (ICQD), Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, China

School of the Gifted Young, University of Science and Technology of China, Hefei, China

Yichi Zhang, Yiheng Wang, Yuchen Liu, Shu Li & Chuang Fan

Hefei National Laboratory, Hefei, China

Hui Zhang & Changgan Zeng

Center for Micro- and Nanoscale Research and Fabrication, University of Science and Technology of China, Hefei, China

Xiuxia Wang

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Contributions

C.Z. and H.Z. designed and supervised the work. Y.Z. and H.Z. performed the experiments with assistance from X.W., Y.L., S.L., T.Z. and C.F. Y.Z. performed the theoretical calculation and analysis. C.Z., Y.Z. and H.Z. analysed the data and wrote the manuscript with assistance from Y.W. and C.F. All authors contributed to the scientific discussion and manuscript revisions.

Corresponding authors

Correspondence to Hui Zhang or Changgan Zeng .

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The authors declare no competing interests.

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Nature Physics thanks Daniel Dantchev, Giovanni Volpe and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended data fig. 1 calculated dielectric permittivities of materials used in this work..

a , Calculated dielectric permittivities of Au, SiO 2 , H 2 O, Fe 3 O 4 and kerosene respectively. The inset is a zoom-in plot of the dielectric permittivities with the Matsubara energy in the range between 100 eV and 500 eV. b , Calculated dielectric permittivities of water-based ferrofluids with different volume fractions. The dielectric permittivities of Au and SiO 2 are also shown for comparison.

Extended Data Fig. 2 Magnetization curve of 4.9% ferrofluid measured by vibrating sample magnetometer.

The square dots are the measured results. The orange line is the theoretical fitting by first-order modified mean-field model (see Supplementary Note 3 ). No apparent hysteresis is observed.

Extended Data Fig. 3 Calculated Casimir force between a gold sphere and a silica plate in ferrofluid as a function of separation distance and magnetic field.

a-d , 3.5%, 4.9%, 5.5% and 9% volume fractions, respectively. Iso-lines are plotted in bright dashed lines to show the magnitude of the Casimir force.

Source data

Extended data fig. 4 measured casimir force in 5.5% water-based ferrofluid between gold-coated sphere and sio 2 plate..

Measurements data at μ 0 H  = 0 and 8.6 mT are averaged from eleven and six measurements respectively, with error bars indicating the standard deviation. The fitting results according to the Lifshitz theory are presented by solid lines.

Supplementary information

Supplementary information.

Supplementary Sections 1–6 and Figs. 1–5.

Source Data Fig. 2

Statistical source data.

Source Data Fig. 3

Source data fig. 4, source data extended data fig./table 3, source data extended data fig./table 4, rights and permissions.

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Zhang, Y., Zhang, H., Wang, X. et al. Magnetic-field tuning of the Casimir force. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02521-0

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Received : 22 November 2023

Accepted : 22 April 2024

Published : 24 May 2024

DOI : https://doi.org/10.1038/s41567-024-02521-0

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Effect of punch type on microstructure and mechanical properties of aluminum alloy structures prepared by electromagnetic riveting

• Original Article
• Published: 15 June 2024
• Volume 24 , article number  179 , ( 2024 )

Cite this article

• Xu Zhang 1 ,
• Jiawei Zhang 1 ,
• Yingyu Wang 1 ,
• Huakun Deng 1 ,
• Congwen Duan 4 ,
• Guangyao Li 3 ,
• Junjia Cui 3 &
• Dongying Dong   ORCID: orcid.org/0009-0005-1636-8998 2  

In this work, a special electromagnetic riveting punch with a convex die was proposed to improve deformation uniformity of rivets. Electromagnetic riveting experiments with the special and conventional flat punch were employed to contrastively investigate microstructure evolution and mechanical properties. Results showed that the width of adiabatic shear bands in the flat headers was narrower than that in the concave headers. The deformation uniformity of the concave headers was better. Microhardness distributions in concave rivet headers were more uniform than that in flat rivet headers. The plastic flow toward rivet hole was larger for the concave headers, resulted in a greater interference-fit strengthening effect of riveted structures. The difference in shear loads of riveted structures with the flat and concave headers was not significant. The riveted structures with concave headers had better vibration absorption effect. The shear friction effect on the fracture surface was more pronounced. The maximum pull-out loads of riveted structures with the flat and concave headers were respectively 13.4 kN and 13.9 kN, showing that riveted structures with concave headers had good resistance to pull-off. By changing the geometric shape of the punch, the quality of the riveted structures has been improved.

Special punch was used for electromagnetic riveting of aluminum alloy structures.

The microstructure distributions of different riveted header were investigated.

The effect of microstructure distributions on interference-fit values was analyzed.

Microhardness and mechanical properties of riveted header were studied.

The strengthening mechanism of special punch on the riveted header was revealed.

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The data supporting this study’s findings are available from the corresponding author upon reasonable request.

Czerwinski F. Current trends in automotive lightweighting strategies and materials. Materials. 2021;14(21):6631.

Article   Google Scholar  

Zhang W, Xu J. Advanced lightweight materials for automobiles: a review. Mate Design. 2022;221: 110994.

Cui GH, Yang CL, Zhang X, et al. Microstructure evolution and mechanical properties of AA2195-T8 Al-Li alloy back gas-shielded VPPA weld. J Manuf Process. 2023;106:288–302.

Xu L, Wang XY, Pan JL, et al. Bond behavior between steel-FRP composite bars and engineered cementitious composites in pullout conditions. Eng Struct. 2024;299: 117086.

Zhu Y, Zhang XY, Kong WY, et al. Experimental and numerical investigations on shear behavior of high-strength bolted connections after impact. J Build Eng. 2023;79: 107872.

Yang Z, Jiang RS, Zou YJ. Riveting damage behavior and mechanical performance assessments of CFRP/CFRP single-lap gasket-riveted joints. Eng Fail Anal. 2023;149: 107253.

Huang CQ, Ji YN, Cui XH, et al. Deformation behavior and mechanical properties of 5052-O aluminum alloy joints formed by high-speed clinching. Arch Civ Mech Eng. 2023;23:152.

Blanchot V, Daidie A. Riveted assembly modelling: study and numerical characterisation of a riveting process. J Mater Process tech. 2006;180:201–9.

Hossein CS. Effect of variations in the riveting process on the quality of riveted joints. Int J Adv Manuf Technol. 2008;39:1144–55.

Aniello R, Valerio A, Andrea S, et al. A comparative numerical-experimental investigation on the tensile behaviour of bonded, rivetted and hybrid composite joints configurations. Compos Struct. 2023;318: 117114.

Min JY, Li YQ, Li JJ, et al. Mechanics in frictional penetration with a blind rivet. J Mater Process tech. 2015;222:268–79.

Chen NJ, Ducloux R, Pecquet C, et al. Numerical and experimental studies of the riveting process. Int J Mater Form. 2011;4:45–54.

Chen NJ, Luo HY, Wan M, et al. Experimental and numerical studies on failure modes of riveted joints under tensile load. J Mater Process tech. 2015;214:2049–58.

Li G, Shi GQ. Effect of the riveting process on the residual stress in fuselage lap joints. Canadian Aeronaut Space J. 2004;50(2):91–105.

Wang C, Du ZP, Cheng AG, et al. Influence of process parameters and heat treatment on self-piercing riveting of high-strength steel and die-casting aluminium. J Mater Res Technol. 2023;26:8857–78.

Korbel A. Effect of aircraft rivet installation process and production variables on residual stress, clamping force and fatigue behaviour of thin sheet riveted lap joints. Thin Wall Struct. 2022;181: 110041.

Duan JR, Chen C. Effect of edge riveting on the failure mechanism and mechanical properties of self-piercing riveted aluminium joints. Eng Fail Anal. 2023;150: 107305.

Zhang Y, Lei B, Wang T, et al. Fatigue failure mechanism and estimation of aluminum alloy self-piercing riveting at different load levels. Eng Fract Mech. 2023;291: 109583.

Cao ZQ, Zuo YJ. Electromagnetic riveting technique and its applications. Chin J Aeronaut. 2020;33:5–15.

Cui JJ, Dong DY, Zhang X, et al. Influence of thickness of composite layers on failure behaviors of carbon fiber reinforced plastics/aluminium alloy electromagnetic riveted lap joints under high-speed loading. Int J Impact Eng. 2018;115:1–9.

Hartmann J, Meeke C. Automated Wing Panel Assembly for the A340–600. SAE Technical Paper Series; 2000-01-3015.

Gong C, Wang SL, Jiang M. The development and characteristics of electromagnetic riveting technology. Appl Mech Mater. 2014;568–570:1694–7.

Wang YHX, Cao ZQ, Zheng G, et al. Analysis of TB2 rivet forming quality and mechanical properties of thin sheet riveted joints based on different riveting methods. Eng Fail Anal. 2023;154: 107666.

Zhang X, Zhang MY, Sun LQ, et al. Numerical simulation and experimental investigations on TA1 titanium alloy rivet in electromagnetic riveting. Arch Civ Mech Eng. 2018;18:887–901.

Jiang H, Cong YJ, Zhang JS, et al. Fatigue response of electromagnetic riveted joints with different rivet dies subjected to pull-out loading. Int J Fatigue. 2019;129: 105238.

Qin YF, Jiang H, Cong YJ, et al. Rivet die design and optimization for electromagnetic riveting of aluminium alloy joints. Eng Opt. 2020;4:1–19.

Google Scholar  

Mucha J, Witkowski W. Mechanical behavior and failure of riveting joints in tension and shear tests. Strength Mater. 2015;47(5):755–69.

Duan LM, Feng XR, Cui JJ, et al. Effect of angle deflection on deformation characteristic and mechanical property of electromagnetic riveted joint. Int J Adv Manuf Tech. 2021;112:3529–43.

Liao YX, Sun H, Wu GS, et al. Effect of rivet arrangement on fatigue performance of electromagnetic riveted joint with Φ10 mm diameter rivet. Int J Fatigue. 2023;176: 107892.

Zhang X, Yu HP, Su H, et al. Experimental evaluation on mechanical properties of a riveted structure with electromagnetic riveting. Int J Adv Manuf Tech. 2016;83:2071–82.

Li M, Tian W, Hu JS, et al. Study on shear behavior of riveted lap joints of aircraft fuselage with different hole diameters and squeeze forces. Eng Fail Anal. 2021;127: 105499.

Deng JH, Tang C, Fu MW, et al. Effect of discharge voltage on the deformation of Ti Grade 1 rivet in electromagnetic riveting. Mat Sci Eng A. 2014;591:26–32.

Zhang X, Yu HP, Li J, et al. Microstructure investigation and mechanical property analysis in electromagnetic riveting. Int J Adv Manuf Tech. 2015;78:613–23.

Dong DY, Sun LQ, Wang Q, et al. Influence of electromagnetic riveting process on microstructures and mechanical properties of 2A10 and 6082 Al riveted structures. Arch Civ Mech Eng. 2019;19:1284–94.

Jiang H, Sun LQ, Dong DY, et al. Microstructure and mechanical property evolution of CFRP/Al electromagnetic riveted lap joint in a severe condition. Eng Struct. 2019;180:181–91.

Gong CP, Fan ZS, Cheng L, et al. Predictions of the total electromagnetic repulsion force in electromagnetic riveting process: numerical analysis model and experiments. J Manuf Process. 2021;69:656–70.

QJ 782A-2005. People’s Republic of China aerospace industry standard: general technical requirements for riveting. 2nd ed. Beijing: National Defense Science and Technology Industry Committee; 2005.

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Acknowledgements

This project is supported by the Foundation for National Natural Science Foundation of China (52005055); Excellent Youth Funding of Hunan Provincial Education Department (22B0340).

This study was funded by Foundation for National Natural Science Foundation of China (52005055); Excellent Youth Funding of Hunan Provincial Education Department (22B0340).

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College of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha, 410114, China

Xu Zhang, Jiawei Zhang, Yingyu Wang & Huakun Deng

College of Energy and Power Engineering, Changsha University of Science and Technology, Changsha, 410114, China

Dongying Dong

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, China

Guangyao Li & Junjia Cui

Hebei Key Lab of Power Plant Flue Gas Multi-Pollutants Control, Department of Environmental Science and Engineering, North China Electric Power University, Baoding, 071003, China

Congwen Duan

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Correspondence to Dongying Dong .

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Zhang, X., Zhang, J., Wang, Y. et al. Effect of punch type on microstructure and mechanical properties of aluminum alloy structures prepared by electromagnetic riveting. Arch. Civ. Mech. Eng. 24 , 179 (2024). https://doi.org/10.1007/s43452-024-00985-8

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Received : 02 September 2023

Revised : 27 May 2024

Accepted : 01 June 2024

Published : 15 June 2024

DOI : https://doi.org/10.1007/s43452-024-00985-8