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Electromagnetism

Of all the achievements in scientific history, it is arguably our mastery of electromagnetism that best defines the modern age. It is through our research of this fundamental force that the wealth of technology we see in our day-to-day life was developed. In fact, the reverse is also true, many of the insights into electromagnetism have come about due to the work of ingenious inventions and feats of engineering. So as electromagnetism is such an important force in our daily lives, it makes sense to get to grips with the fundamentals of such a fascinating topic, right? Well, that's exactly what we're going to do in this article, so read on to uncover more about the historical theories that came to define electromagnetism as a pillar of modern physics, as well as what these theories tell us about electromagnetic fields and their applications.

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Jetzt kostenlos anmeldenOf all the achievements in scientific history, it is arguably our mastery of electromagnetism that best defines the modern age. It is through our research of this fundamental force that the wealth of technology we see in our day-to-day life was developed. In fact, the reverse is also true, many of the insights into electromagnetism have come about due to the work of ingenious inventions and feats of engineering. So as electromagnetism is such an important force in our daily lives, it makes sense to get to grips with the fundamentals of such a fascinating topic, right? Well, that's exactly what we're going to do in this article, so read on to uncover more about the historical theories that came to define electromagnetism as a pillar of modern physics, as well as what these theories tell us about electromagnetic fields and their applications.

Electromagnetism is the study of the electromagnetic force and associated phenomena such as electricity and magnetism. The electromagnetic force is one of the four fundamental forces (or interactions) of nature and is responsible for the interactions between electrically charged particles such as protons and neutrons. As we shall see in the next section, the electromagnetic force is also responsible for light waves and is a field that connects many seemingly disparate areas of science, such as optics, electrical engineering, and physical chemistry.

The core concept of electromagnetism is the **e****lectromagnetic field, **this is a type of vector field which interacts with charged particles producing a force on them. This field can be understood to be composed of coupled electric and magnetic fields.

The excitation of one field produces excitations in the other, and these excitations can propagate throughout space as electromagnetic radiation. In a vacuum, the oscillating electric and magnetic fields are always perpendicular to each other, and are both perpendicular to the direction of the travel making electromagnetic waves transverse. This electromagnetic radiation is the source of all visible light, as well other forms of radiation such as radio waves and microwaves.

The existence of electric and magnetic phenomena have been known about since antiquity, due to things like *lightning* and naturally occurring magnetic ores known as *lodestone*. For most of the history, electricity and magnetism were the subjects of much research. However, it was not until the mid 19th century that physicists started to investigate the possibility that electricity and magnetism may, in fact, be two sides of the same phenomenon. The relationship between the two was first established by Danish physicist Hans Christian Oersted, who found that a current-carrying wire deflected a magnetic needle away from true north, hence showing that electricity produced magnetic forces. This spurred on much research, culminating with Maxwell's equations, which gave a complete mathematical description of electric and magnetic fields and their relationship. These equations are the fundamental laws of electromagnetism, and the entirety of classical electromagnetism can be derived from them. Let's take a closer look at these incredibly important equations.

The complete theoretical description of electromagnetism developed by James Clerk Maxwell in the mid-1860s is widely regarded as the greatest achievement in classical physics. Whilst it took some years for Maxwell's equations to be experimentally proven and widely accepted, their influence on modern physics is undeniable. Maxwell's electromagnetism role as the first fundamental field theory and for establishing a theoretical basis for a finite speed of light proved hugely influential on both *Quantum Field Theory* and *Einstein's Relativity, *the two pillars of modern physics.

Let's take a look at exactly what Maxwell's equations are and what they tell us about electromagnetic fields.

The first of Maxwell's equations was first formulated by the German Mathematician Carl Friedrich Gauss and concerns the amount of electric flux through an arbitrary surface.

**Electric Flux \(\Phi_E\) **is a quantity measuring how much of an electric field 'flows' through a surface.

For a constant electric field, the flux is given by

\[\Phi_E=\vec{E}\cdot\vec{A}\]

where \(\vec{A}=A\vec{n}\) is the surface vector, with the surface area as magnitude and the direction being perpendicular to the surface.

If the electric field is not constant across a surface, then a surface integral is used to add up the components of the electric field across each infinitesimal section of surface area

\[\Phi_E=\int_S\vec{E}\cdot\mathrm{d}\vec{A}.\]

**Gauss' Law** states that the electric flux through a surface is directly proportional to the amount of charge \(Q\) within the volume enclosed by the surface, regardless of how the charge is distributed throughout the volume.

Mathematically, it can be expressed as\[\Phi_E=\int_S\vec{E}\cdot\mathrm{d}\vec{A}=\frac{Q}{\epsilon_0}.\]

This flux can be understood using electric field lines, the number of field lines passing through a surface indicates the amount of flux.

The second of Maxwell's equations is a crucial statement about the flux of magnetic fields.

It states that for any surface, the magnetic flux through that surface must be zero:

\[\Phi_B=\int_S\vec{B}\cdot\mathrm{d}\vec{A}=0.\]

This is best interpreted in terms of 'magnetic field lines', as saying that the number of magnetic field lines entering a surface must be equal to the number of field lines exiting the surface.

This can be seen in the field lines around a bar magnet, these field lines are always closed loops and so, no matter where you choose to draw the surface, the number of field lines entering will be equal to the number of field lines leaving.

This law ensures that magnetic monopoles cannot exist in nature; unlike the electric field which has individual charges as its sources and act as electric monopoles, magnetic field poles must always come in 'North-South' pairs as far as we know.

The third of Maxwell's equations is a formulation of the empirical law of electromagnetic induction, first discovered by Michael Faraday.

It states that the rate of change of magnetic flux is equal to the Electromotive Force (EMF) propelling a charge around the loop. This EMF can be written as a loop integral of the electric field around the closed path followed by a charge

\[\begin{align}\int_{\partial S}\vec{E}\cdot\mathrm{d}\vec{l}&=-\frac{\mathrm{d}}{\mathrm{d}t}\int_{S}\vec{B}\cdot\mathrm{d}\vec{A}\\&=-\frac{\mathrm{d}\Phi_B}{\mathrm{d}t},\end{align}\]

where \(\partial S\) is the loop enclosing the surface \(S\).

This law is especially important in electromagnetism, as it quantifies how changing magnetic fields induce changes in electric fields and vice versa.

The fourth and final Maxwell equation relates the magnitude of the induced magnetic field along a loop to the current flowing through the loop.

Note that this mathematical loop is simply imaginary and needn't refer to any sort of physical loop.

The origin of this law is in the work of French physicist Andre Ampère when investigating the magnetic force between two current carrying wires. Maxwell generalised this by including a term to account for the magnetic field produced by a changing electric flux:

\[\begin{align}\int_{\partial S}\vec{B}\cdot\mathrm{d}\vec{l}=\mu_0I+\mu_0\epsilon_0\frac{\mathrm{d}\Phi_E}{\mathrm{d}t}.\end{align}\]

The great achievement of Maxwell's equations was in demonstrating that light was, in fact, a consequence of oscillating electric and magnetic fields propagating through space. Maxwell found that by manipulating the equations the magnetic and electric fields satisfied standard wave equations, with the wave speed equal to \[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}=3\times10^8\,\mathrm{m}\,\mathrm{s}^{-1}.\] This value was already known at the time to be the speed of light in a vacuum. Maxwell's equations also give a physical explanation for how light propagates; an initial oscillation in the magnetic field induces an oscillation in the electric field as per Faraday's Law, which in turn induces an oscillation in the magnetic field as per the Ampere-Maxwell Law. These oscillations back and forth between the two fields can then propagate infinitely through a vacuum as electromagnetic radiation.

Let's look at some example problems applying all the laws mentioned above.

Using Gauss's Law for Electric fields, derive Coulomb's Law and find an expression for Coulomb's constant \(k\).

To derive Coulomb's Law, we need to consider two charges \(q_1,q_2\) separated by some distance \(r\). By definition, the force experienced by \(q_2\) is determined by the Electric field \(\vec{E}_1\) produced by \(q_1\) as\[F=q_2E_1.\]

We can find an expression for this force by considering an imaginary sphere of radius \(r\) which encloses \(q_1\) but not \(q_2\).

Gauss's Law for Electric fields states that the flux \(\Phi_E\) of the electric field out of this sphere, found by integrating the field over the surface, is given by\[\Phi_e=\int_S\vec{E_1}\cdot\mathrm{d}\vec{A}=\frac{q_1}{\epsilon_0}.\]

If we assume that the electric field is spherically symmetric given, which is valid for stationary charges, then we can take the field vector out of the integrand

\[\int_S\vec{E_1}\cdot\mathrm{d}\vec{A}=\vec{E_1}\int_S\mathrm{d}\vec{A}.\]

The integral is then simply integrating over the surface vector, and is equal to the surface area of the sphere\[\vec{E_1}\int_S\mathrm{d}\vec{A}=\vec{E_1}4\pi r^2\vec{r}\]

where \(\vec{r}\) is the unit radial vector pointing away from the charge. Plugging this into Gauss's Law gives\[\begin{align}\vec{E_1}4\pi r^2\vec{r}&=\frac{q_1}{\epsilon_0}\\\vec{E_1}&=\frac{q_1}{4\pi\epsilon_0r^2}\vec{r}.\end{align}\]Applying this to the definition of the force gives\[F=q_2\vec{E_1}=\frac{q_1q_2}{4\pi\epsilon_0r^2}\vec{r}\]

which is Coulomb's law with \(k=\frac{1}{4\pi\epsilon}.\)

Consider a time dependent magnetic field defined by the function \(\vec{B}(t)=B\sin\left(2\pi t\right)\vec{z}\). If a circular loop of radius \(r=0.1\,\mathrm{m}\) is placed in the field such that its radial vector \(\vec{r}\) is at an angle of \(\theta=45^{\circ}\,\mathrm{deg}\) with the magnetic field direction \(\vec{z}\). What will the value of the induced EMF \(\mathcal{E}\) be?

Faraday's law tells us that the EMF induced by an oscillating magnetic field is proportional to the rate of change of the magnetic flux.

\[\mathcal{E}=-\frac{\mathrm{d}}{\mathrm{d}t}\int_{S}\vec{B}\cdot\mathrm{d}\vec{A}\]

Let's first find an expression for the magnetic flux. \[\begin{align}\Phi_B&=\int_{S}\vec{B}\cdot\mathrm{d}\vec{A}\\\\&=\int_SB\sin\left(2\pi t\right)\vec{z}\cdot\vec{A}\end{align}\]

From the definition of the dot product and the angle given in the question, we know\[\vec{z}\cdot\mathrm{d}\vec{A}=\cos\left(45\right)\mathrm{d}A=\frac{\sqrt{2}}{2}.\]

Note that the magnetic field is spatially independent, and so we can take it outside the integrand.

\[\Phi_B=\frac{\sqrt{2}}{2}B\sin\left(2\pi t\right)\int_{S}\mathrm{d}\vec{A}\]

The integrand now just gives the surface area enclosed by the circular loop, which is \(\pi r^2=\frac{\pi}{100}\,\mathrm{m}^2\):

\[\Phi_B=\frac{\sqrt{2}\pi}{200}B\sin\left(2\pi t\right).\]

To find the EMF, we need to take the derivative with respect to time\[\mathcal{E}=\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}=\frac{\sqrt{2}\pi^2}{100}B\cos\left(2\pi t\right).\]

Whilst there are four fundamental forces of nature, it is only two of them that can be directly observed in our day-to-day lives. These are the electromagnetic force and gravity, and the study of these two forces has been central to physics for most of the history. These two forces share many similarities, and as gravity is often a more intuitive force, it can help to use it as an analogy when learning about electromagnetism. Let's take a look at some of the similarities of these two fundamental forces, before looking at the key differences which distinguish them.

Both forces have an infinite range, unlike the other fundamental forces (Strong and Weak Forces).

Whilst the forces have an infinite range, the strength of the forces follows a \(\frac{1}{r^2}\) law, meaning that the strength of the force reduces greatly with distance. This can be seen in Newton's Law of Gravitation as well as Coulomb's Law for Electric Forces\[F_{\text{G}}=\frac{Gm_1m_2}{r^2},\,F_{\text{E}}=\frac{kq_1q_2}{r^2}.\]

As can be seen in the equation for electric and gravitational forces, the effect of both Gravity and Electromagnetism produced by, and exerted on, a particle is defined by a specific property of the particle. This is mass \(m\) for Gravity and charge \(q\) for Electromagnetism, if a particle has mass and/or charge it will be affected by these fundamental forces.

However, it's important to also look at the differences between gravity and electromagnetism, as much of the structure of the universe relies on the unique properties of electromagnetism.

Gravity is a solely attractive force, whilst electromagnetic forces can be both attractive or repulsive depending on the signs of the charges. It's the repulsive electromagnetic force that is partially responsible for preventing all matter from collapsing in on itself under its own weight!

Mass is only ever a positive quantity, so all matter experiences the same kind of attractive force under gravity, just to varying strengths. However, electric charge can be either positive or negative, with like opposite charges being attracted towards one another whilst like charges repel each other.

Electromagnetism is a significantly stronger force than gravity, as can be seen by comparing the coupling constants of each force. The gravitational constant \(G=6.67\times10^{-11}\,\mathrm{N}\,\mathrm{m}^2\,\mathrm{kg}^{-2}\) is around \(20\) orders of magnitude smaller than Coulomb's constant \(k=9\times10^9\,\mathrm{N}\,\mathrm{m}^2\,\mathrm{C}^{-2}\). This disparity in strengths is what allows even relatively weak magnets to pick up objects despite the gravitational pull of the earth.

Gravity is a constant force that is only dependent on the mass of an object. However, when considering magnetic forces, the strength of the force is also dependent on the velocity of the charges.

Electromagnetism is absolutely crucial to our modern world and the technologies we use every day. It's thanks to our understanding of electromagnetism that we have been able to harness the power of electricity to power our homes and our schools and develop technologies such as computers and smartphones.

One particularly ingenious use of electromagnetism is in our ability to manipulate electromagnetic radiation in order to transfer information. For example, radio waves and microwaves produce the cellular data and Wi-Fi needed for your smartphone to access the internet. Radio transmitters use time-dependent alternating currents with composed of accelerating charges. Accelerating electric charges produce oscillating magnetic fields due to Ampere's Law. If the frequency of these oscillations is high enough, these magnetic fields will produce coupled oscillating electric fields and propagate away from the transmitter as electromagnetic radio waves. When these electromagnetic waves reach the antennae of a receiver, the oscillating magnetic field produces an oscillating EMF in the antennae (as per Faraday's Law) which re-produces the initial alternating current. This oscillating current can then be decoded to provide the information broadcasted in the radio signal, pretty incredible, right?

- Electromagnetism is the study of the electromagnetic force, one of the four fundamental forces or interactions of nature, responsible for interactions between charged particles.
- The electromagnetic field is made up of coupled electric and magnetic fields, with changes in one field inducing a change in the other.
- The precise mathematical theory of electromagnetism was developed by James Clerk Maxwell, and is summarized into four equations known as Maxwell's equations.
- Electromagnetism shares some key traits with gravity, such as an inverse square law for the force

- Fig. 1 - Lightning Pritzerbe 01 (MK) (https://commons.wikimedia.org/wiki/File:Lightning_Pritzerbe_01_(MK).jpg) by Mathias Krumbholz (https://commons.wikimedia.org/wiki/User:Leviathan1983) is licenced under CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/deed.en)
- Fig. 2 - Portrait of James Clerk Maxwell(https://commons.wikimedia.org/wiki/File:James_clerk_maxwell.jpg) is under Public Domain.
- Fig. 3 - Electric field lines around point charge, StudySmarter Originals.
- Fig. 4 - Mafnetic field (https://commons.wikimedia.org/wiki/File:Mafnetic_field.png) by Rajiv1840478 is licenced under CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)
- Fig. 5 - Gauss Law diagram, StudySmarter Originals.
- Fig. 6 - Antenna on the top of the building, Umag, Istria, Croatia (https://commons.wikimedia.org/wiki/File:Antenna_on_the_top_of_the_building,_Umag,_Istria,_Croatia.jpg) by Michal Klajban (https://commons.wikimedia.org/wiki/User:Podzemnik) is licenced under CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)

Electromagnetism is the series of interactions and phenomenons which cover electrical charges, magnetic fields and electrical fields.

It is one of the most studied fields in science and engineering nowadays.

Radar and the phone are two real applications of electromagnetism.

The fundamental equations of classical electromagnetism are known as...?

Maxwell's equations.

Electric flux is defined as the amount of electric field flowing ... to a surface.

Perpendicular.

Which of these equations is the equation for Gauss' Law for electric fields?

\(\Phi_E=\displaystyle\int_S\vec{E}\cdot\mathrm{d}\vec{A}=\frac{Q}{\epsilon_0}\).

Use Gauss' Law to establish the amount of charge enclosed within a Gaussian square with side of length \(2\,\mathrm{m}\) if a constant field of \(E=10\,\mathrm{J}\,\mathrm{C}^{-1}\) emanates from the center.

\(3.54\times10^{-10}\,\mathrm{C}\).

State Gauss' Law for Magnetism.

The magnetic flux is always equal to zero.

What does Gauss' Law for Magnetism imply about magnetic monopoles.

They are impossible in nature.

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