Maxwell's Equations: Definition, Derivation, How To Remember (W/ Examples)

Solving the mysteries of electromagnetism has been one of the greatest accomplishments of physics to date, and the lessons learned are fully encapsulated in Maxwell's equations.

James Clerk Maxwell gives his name to these four elegant equations, but they are the culmination of decades of work by many physicists, including Michael Faraday, Andre-Marie Ampere and Carl Friedrich Gauss – who give their names to three of the four equations – and many others. While Maxwell himself only added a term to one of the four equations, he had the foresight and understanding to collect the very best of the work that had been done on the topic and present them in a fashion still used by physicists today.

For many, many years, physicists believed electricity and magnetism were separate forces and distinct phenomena. But through the experimental work of people like Faraday, it became increasingly clear that they were actually two sides of the same phenomenon, and Maxwell's equations present this unified picture that is still as valid today as it was in the 19th century. If you're going to study physics at higher levels, you absolutely need to know Maxwell's equations and how to use them.

Maxwell's equations are as follows, in both the differential form and the integral form. (Note that while knowledge of differential equations is helpful here, a conceptual understanding is possible even without it.)

​Gauss' Law for Electricity​

Differential form:

\(\bm{∇∙E} = \frac{ρ}{ε_0}\)

Integral form:

\(\int \bm{E ∙} d\bm{A} = \frac{q}{ε_0}\)

​No Monopole Law / Gauss' Law for Magnetism​

Differential form:

\(\bm{∇∙B} = 0\)

Integral form:

\(\int \bm{B ∙} d\bm{A} = 0\)

​Faraday's Law of Induction​

Differential form:

\(\bm{∇ × E} = − \frac{∂\bm{B}}{∂t}\)

Integral form:

\(\int \bm{E∙ }d\bm{s}= − \frac{∂\phi_B}{ ∂t}\)

​Ampere-Maxwell Law / Ampere's Law​

Differential form:

\(\bm{∇ × B} = \frac{J}{ ε_0 c^2} + \frac{1}{c^2} \frac{∂E}{∂t}\)

Integral form:

\(\int \bm{B ∙} d\bm{s} = μ_0 I + \frac{1}{c^2} \frac{∂}{∂t} \int \bm{E ∙ }d\bm{A}\)

Maxwell's equations use a pretty big selection of symbols, and it's important you understand what these mean if you're going to learn to apply them. So here's a run-down of the meanings of the symbols used:

​B​ = magnetic field

​E​ = electric field

​ρ​ = electric charge density

​_ε₀_​ = permittivity of free space = 8.854 × 10^-12 m^-3 kg^-1 s⁴ A²

​q​ = total electric charge (net sum of positive charges and negative charges)

​𝜙​_B = magnetic flux

​J​ = current density

​I​ = electric current

​c​ = speed of light = 2.998 × 10⁸ m/s

​μ​₀ = permeability of free space = 4π × 10⁻⁷ N / A²

Additionally, it's important to know that ∇ is the del operator, a dot between two quantities (​**​X​ ∙ ​Y​​) shows a scalar product, a bolded multiplication symbol between two quantities is a vector product (​​X​ × ​Y​​), that the del operator with a dot is called the "divergence" (e.g., ∇ ∙​​ X​​ = divergence of ​​X​​ = div ​​X​​) and a del operator with a scalar product is called the curl (e.g., ∇ ​×​ ​​Y​​ = curl of ​​Y​​ = curl ​​Y​​). Finally, the ​​A​​ in d​​A​​ means the surface area of the closed surface you're calculating for (sometimes written as d​​S​**​), and the ​s​ in d​s​ is a very small part of the boundary of the open surface you're calculating for (although this is sometimes d​l​, referring to an infinitesimally small line component).

The first equation of Maxwell's equations is Gauss' law, and it states that the net electric flux through a closed surface is equal to the total charge contained inside the shape divided by the permittivity of free space. This law can be derived from Coulomb's law, after taking the important step of expressing Coulomb's law in terms of an electric field and the effect it would have on a test charge.

The second of Maxwell's equations is essentially equivalent to the statement that "there are no magnetic monopoles." It states that the net magnetic flux through a closed surface will always be 0, because magnetic fields are always the result of a dipole. The law can be derived from the Biot-Savart law, which describes the magnetic field produced by a current element.

The third equation – Faraday's law of induction – describes how a changing magnetic field produces a voltage in a loop of wire or conductor. It was originally derived from an experiment. However, given the result that a changing magnetic flux induces an electromotive force (EMF or voltage) and thereby an electric current in a loop of wire, and the fact that EMF is defined as the line integral of the electric field around the circuit, the law is easy to put together.

The fourth and final equation, Ampere's law (or the Ampere-Maxwell law to give him credit for his contribution) describes how a magnetic field is generated by a moving charge or a changing electric field. The law is the result of experiment (and so – like all of Maxwell's equations – wasn't really "derived" in a traditional sense), but using ​Stokes' theorem​ is an important step in getting the basic result into the form used today.

To be frank, especially if you aren't exactly up on your vector calculus, Maxwell's equations look quite daunting despite how relatively compact they all are. The best way to really understand them is to go through some examples of using them in practice, and Gauss' law is the best place to start. Gauss' law is essentially a more fundamental equation that does the job of Coulomb's law, and it's pretty easy to derive Coulomb's law from it by considering the electric field produced by a point charge.

Calling the charge ​q​, the key point to applying Gauss' law is choosing the right "surface" to examine the electric flux through. In this case, a sphere works well, which has surface area ​A​ = 4π​r​², because you can center the sphere on the point charge. This is a huge benefit to solving problems like this because then you don't need to integrate a varying field across the surface; the field will be symmetric around the point charge, and so it will be constant across the surface of the sphere. So the integral form:

\(\int \bm{E ∙} d\bm{A} = \frac{q}{ε_0}\)

Can be expressed as:

\(E × 4πr^2 = \frac{q}{ε_0}\)

Note that the ​**​E​**​ for the electric field has been replaced with a simple magnitude, because the field from a point charge will simply spread out equally in all directions from the source. Now, dividing through by the surface area of the sphere gives:

\(E = \frac{q}{4πε_0r^2}\)

Since the force is related to the electric field by ​E​ = ​F​/​q​, where ​q​ is a test charge, ​F​ = ​qE​, and so:

\(F = \frac{q_1q_2}{4πε_0r^2}\)

Where the subscripts have been added to differentiate the two charges. This is Coulomb's law stated in standard form, shown to be a simple consequence of Gauss' law.

Faraday's law allows you to calculate the electromotive force in a loop of wire resulting from a changing magnetic field. A simple example is a loop of wire, with radius ​r​ = 20 cm, in a magnetic field that increases in magnitude from ​B​_i = 1 T to ​B​_f = 10 T in the space of ∆​t​ = 5 s – what is the induced EMF in this case? The integral form of the law involves the flux:

\(\int \bm{E∙ }d\bm{s}= − \frac{∂\phi_B}{ ∂t}\)

which is defined as:

\(ϕ = BA \cos (θ)\)

The key part of the problem here is finding the rate of change of flux, but since the problem is fairly straightforward, you can replace the partial derivative with a simple "change in" each quantity. And the integral really just means the electromotive force, so you can rewrite Faraday's law of induction as:

\(\text{EMF} = − \frac{∆BA \cos (θ)}{∆t}\)

If we assume the loop of wire has its normal aligned with the magnetic field, ​θ​ = 0° and so cos (​θ​) = 1. This leaves:

\(\text{EMF} = − \frac{∆BA}{∆t}\)

The problem can then be solved by finding the difference between the initial and final magnetic field and the area of the loop, as follows:

\(\begin{aligned}
\text{EMF} &= − \frac{∆BA}{∆t} \
&= − \frac{(B_f – B_i) × πr^2}{∆t} \
&= − \frac{(10 \text{ T}- 1 \text{ T}) × π × (0.2 \text{ m})^2}{5 \text{ s}} \
&= − 0.23 \text{ V}
\end{aligned}\)

This is only a small voltage, but Faraday's law is applied in the same way regardless.

The Ampere-Maxwell law is the final one of Maxwell's equations that you'll need to apply on a regular basis. The equation reverts to Ampere's law in the absence of a changing electric field, so this is the easiest example to consider. You can use it to derive the equation for a magnetic field resulting from a straight wire carrying a current ​I​, and this basic example is enough to show how the equation is used. The full law is:

\(\int \bm{B ∙} d\bm{s} = μ_0 I + \frac{1}{c^2} \frac{∂}{∂t} \int \bm{E ∙ }d\bm{A}\)

But with no changing electric field it reduces to:

\(\int \bm{B ∙} d\bm{s} = μ_0 I\)

Now, as with Gauss' law, if you choose a circle for the surface, centered on the loop of wire, intuition suggests that the resulting magnetic field will be symmetric, and so you can replace the integral with a simple product of the circumference of the loop and the magnetic field strength, leaving:

\(B × 2πr = μ_0 I\)

Dividing through by 2π​r​ gives:

\(B = \frac{μ_0 I}{2πr}\)

Which is the accepted expression for the magnetic field at a distance ​r​ resulting from a straight wire carrying a current.

When Maxwell assembled his set of equations, he began finding solutions to them to help explain various phenomena in the real world, and the insight it gave into light is one of the most important results he obtained.

Because a changing electric field generates a magnetic field (by Ampere's law) and a changing magnetic field generates an electric field (by Faraday's law), Maxwell worked out that a self-propagating electromagnetic wave might be possible. He used his equations to find the wave equation that would describe such a wave and determined that it would travel at the speed of light. This was a "eureka" moment of sorts; he realized that light is a form of electromagnetic radiation, working just like the field he imagined!

An electromagnetic wave consists of an electric field wave and a magnetic field wave oscillating back and forth, aligned at right angles to each other. The oscillation of the electric part of the wave generates the magnetic field, and the oscillating of this part in turn produces an electric field again, on and on as it travels through space.

Like any other wave, an electromagnetic wave has a frequency and a wavelength, and the product of these is always equal to ​c​, the speed of light. Electromagnetic waves are all around us, and as well as visible light, other wavelengths are commonly called radio waves, microwaves, infrared, ultraviolet, X-rays and gamma rays. All of these forms of electromagnetic radiation have the same basic form as explained by Maxwell's equations, but their energies vary with frequency (i.e., a higher frequency means a higher energy).

So, for a physicist, it was Maxwell who said, "Let there be light!"