Maxwell's Equations: Definition & Application

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  • 0:03 What are Maxwell's Equations?
  • 2:04 Applications of Maxwell
  • 3:19 Example
  • 4:40 Lesson Summary
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this video, you will be able to explain what Maxwell's equations are, the basic principles behind each one, and what benefits they have led to in society. A short quiz will follow.

What Are Maxwell's Equations?

Maxwell's equations are a series of four partial differential equations that describe the force of electromagnetism. They were derived by mathematician James Clerk Maxwell, who first published them in 1861 and in 1862. Individually, the four equations are named Gauss' law, Gauss' law for magnetism, Faraday's law and Ampere's law. The equations look like this:

Four Maxwell equations
four maxwell equations

While using these equations involves integrating (calculus), we can still talk about what each law represents conceptually, and how they're used:

  • Gauss' law relates the distribution of electric charge to the field that charge creates. If you know the shape of the object and, therefore, how the charge is distributed, you can use Gauss' law to figure out an expression for the electric field. This is generally used when there's a degree of symmetry, making the equation simpler.
  • Gauss' law for magnetism says that magnetic monopoles do not exist. It's really more of a statement than something we might use to derive expressions. Charges exist as positive or negative. But in magnetism, whenever you have a south pole, you also have a north pole - there are no single, or monopoles, as yet discovered.
  • Faraday's law says that any change to the magnetic environment of a coil of wire will cause a voltage to be induced in the coil. If the magnetic field strength changes, or the magnet moves, or the coil moves, or the coil is rotated - any of these things will create a voltage in the coil.
  • Ampere's law says that the magnetic field created by an electric current is proportional to the size of that electric current, with a constant of proportionality equal to the permeability of free space. Stationary charges produce electric fields, proportional to the magnitude of that charge. But moving charges produce magnetic fields, proportional to the current (the charge and movement).

Applications of Maxwell

The uses and applications of Maxwell's equations are just too many to count. By understanding electromagnetism we're able to create images of the body using MRI scanners in hospitals; we've created magnetic tape, generated electricity, and built computers. Any device that uses electricity or magnets is on a fundamental level built upon the original discovery of Maxwell's equations.

While using Maxwell's equations often involves calculus, there are simplified versions of the equations we can study. These versions only work in certain circumstances, but can be useful and save a lot of trouble. Let's look at one of these - the simplified version of Faraday's law.

As a reminder, Faraday's law says that any change to the magnetic environment of a coil of wire will cause a voltage to be induced in the coil. And we can quantify those changes in a simple equation. Doing so gives you this equation below, where N is the number of turns on the coil of wire, delta BA is the change in the magnetic field times the area of the coil of wire, and delta t is the time over which that change occurs.

faraday law equation

This equation will give you the voltage produced in the coil. If anything changes the values of B or A, a voltage will be produced.


Let's go through an example of how to use the simplified version of Faraday's law. A coil of wire is placed in an external magnetic field of strength 0.1 teslas. The coil has 50 turns on it, and a cross-sectional area of 0.05 meters squared. If the field strength is changed to 0.4 teslas gradually, over a period of 3 seconds, what voltage will be induced in the coil of wire?

First of all, we should write down what we know: the initial magnetic field, Bi, equals 0.1; the number of turns, N, is equal to 50; the area, A, equals 0.05; the final magnetic field, Bf, equals 0.4; and the time, t, equals 3.

Since the equation has delta B in it, we need to find the change in field. The change in field will be the difference between the initial and final values, which is 0.4 minus 0.1, and that equals 0.3.

Finally, we just plug our numbers into the equation and solve. 50 multiplied by 0.3 multiplied by 0.05 (since the area didn't change) divided by 3 equals 0.25 volts. And that's it; that's our answer.

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