chapter30

Chapter 30 Maxwell's Equations



Maxwell's equations are a group of four equations that lay the foundations for electricity and magnetism. The laws summarize decades of experimental work by Coulomb, Gauss, Biot-Savart, Ampere, and Faraday. It will be first beneficial to define some fundamental constants and variables that will show up when studying Maxwell's equations. Below

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The previous equations are expressed in differential form. However, they can also be expressed in an integral form, shown below.



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 * Gauss's Law Overview**

The first equation is Guass's Law. It states that the 'flux of the elctric field through any closed surface' is equal to the enclosed charge divided by the permeativity of free space. The permeativity of free space is a constant that is important to know when dealing with Maxwell's equations. The definition of the constant is defined below:

The unit of the permeativity constant is Farads/meter. Farads is the unit of measure that is used to measure capacitance. 'Guass's law implies that the electric field due to a point charge varies inversely as the square of the distance from the charge.'



The above diagram shows how many Gauss's law problems are solved. The surface integral of the electric field with respect to area typically reduces to simply (EA). Once one plugs in the appropriate electric charge and permeativity constant, a few algebraic steps will yield one the answer.

Before doing any problems, there are a few things one should know about Gauss's law. It deals with the electric field. The electric field inside a conductor is always equal to 0. Excess charge lies on the surface of the conductor. At every point on the surface of the conductor, the electric potential (V) is equal. This is known as an equipotential surface. Additionally, the electric field is perpendicular to the surface. These are important things to know.



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 * Gauss's Law for Magnetism**

The second of Maxwell's equations states that there are no magnetic monopoles. This is because the net magnetic flux is equal to 0. Some more explanation will help in understanding exactly what this equation is saying. The amount of magnetic field lines entering and leaving a specific area will always be the same (the net movement is 0). It means that there cannot be a north magnetic pole without a south magnetic pole. Poles cannot stand freely alone. Also, both magnetic poles have the same strength.

   __[]__ The above diagrams merely illustrate the aforementioned point. The lack of the existence of magnetic monopoles creates a net magentic flux of 0 through any surface.

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Faraday's law relates the induced electric field to the changing magnetic flux.
 * Faraday's Law**

We will only speak about the integral form although the differential form is also shown. The line integral of the electric field around a closed loop is equal to the negative rate of change of the magnetic flux through the enclosed area of the loop. In simpler terms, any change magnetic field induces a current.

An example of Faraday's law in action would be a magnet spinning next to a wire current attached to a lightbulb. Spinning the magnet changes the magnetic field which in turn induces a current. This can be proven by attaching the coil to a wire which is connected to a lightbulb. This lightbulb will produce light when the magnet spins.

Here is an online lab that will illustrate the above example with the lightbulb:  []

__Answer these questions after running the lab: __  __1. What happens to the RPM's as the water increases? __ __2. What happens to the strength of the light as the RPM's increase? __ __3. What kind of current is produced from the magnet's rotation? __ __4. What does the compass show? __


 * Ampere's Law with Maxwell's Correction**

Ampere's Law is the last of Maxwell's equations. It states that an electric current creates a magnetic field that is proportional to the electric current through the closed area. It also essentially states that magnetic fields are created/manipulatd by electrical currents (I) and changing magnetic fields. The first equation shown above represents that discovered by Ampere himself. However, at some point down the line he realized that his equation was not complete. Maxwell added what is called the displacement current to the end of what Ampere created. The final equation, written by both Maxwell and Ampere, looks like this:

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The discovery of this law lead Maxwell to see that light traveled with electromagnetic waves and calculated the speed of light.


 * CATEGORY ||  SCORE (1-4)  ||  POINTS (0-20)  ||
 * Content || 3 || 15 ||
 * Organization || 4 || 20 ||
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