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Physics: High School18 chapters | 211 lessons

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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.

**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:

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).

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.

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.

**Maxwell's equations** are a series of four partial differential equations that describe the force of electromagnetism. The equations look like this:

**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. **Gauss' law for magnetism** says that magnetic monopoles do not exist.

**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. And **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.

The uses and applications of Maxwell's equations are too many to count. By understanding electromagnetism, we are able to create images of the body using MRI scanners in hospitals; we've created magnetic tape, generated electricity, and built computers.

We can simplify Faraday's law and quantify the induced voltage it describes. Doing so gives us 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:

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

You will have the ability to do the following after watching this video lesson:

- Identify the four Maxwell equations
- Describe each law that makes up Maxwell's equations
- Recall the simplified equation for Faraday's law

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Physics: High School18 chapters | 211 lessons

- Magnetic Force: Definition, Poles & Dipoles 6:09
- What is a Magnetic Field? 6:47
- How Magnetic Fields Are Created 6:19
- How Magnetic Forces Affect Moving Charges 6:13
- Electromagnetic Induction: Definition & Variables that Affect Induction 7:06
- Electromagnetic Induction: Conductor to Conductor & Transformers 7:43
- Electric Motors & Generators: Converting Between Electrical and Mechanical Energy 6:01
- Faraday's Law of Electromagnetic Induction: Equation and Application 8:12
- The Biot-Savart Law: Definition & Examples 6:19
- Ampere's Law: Definition & Examples 5:58
- Maxwell's Equations: Definition & Application 6:22
- Go to Magnetism in Physics

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