Maxwell's Equations (calculus/differential equations)
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Maxwell's Equations (integral and differential form) ....
Properties:
Maxwell's Equations
Introducing the del operator, that funny little triangle below. What is refers to is the partial derivative of the function in question with respect to a particular variable (x, y, or z)
In practice the del operator can be used in two ways depending on whether we are taking the dot product (or divergence) of a function or whether we are taking the cross product (or curl) of a function.
As can be seen depicted below the column on the left illustrates the integral form of Maxwell's Equations; while illustrated on the right are the differential form of Maxwell's Equations. It is important to understand that the solution of a given problem will call for the form of each equation used.
In order the equations are:
I apologize for the messiness, but I strive for the authenticity of the work. I want it to look like it came from my notebook to promote the fact that YOU can also do it!!!
As can be seen in the differential column the first two laws pertain to the divergence (dot product) of two variables in Gauss' Law and Gauss' Law in Magnetism. The second two laws, Faraday's law and Ampere-Maxwell's law use the curl (cross product) of two variables. We will get into more detail as to what each letter in the equation refers to as we solve problems using each of the laws.
Gauss' Law
Ok. Let's just get into the first of Maxwell's Equations: Gauss' Law. Right off the bat I should explain here that these laws are indispensable when it comes to solving physics problems pertaining to charged particles. In fact, because of the mathematical pursuit of these problems later physics mysteries like the speed of light owe their original investigation to the pursuit of these equations.
So just what is Gauss' Law and why is it important?
Well, just look at the picture below showing a positive charge a q+ surrounded by an area A. Area A is made up of a number of sub areas called delta A (triangle A) which when all summed up add to the total area A. That charge q+ is radiating an electric field E from it as the source. There are a number of field lines, called the electric flux represented by flux lines coming from the source, q+
What Gauss' law states is that the net electric flux through any closed Gaussian surface is equal to the net charge inside the surface divided by epsilon naught, or q/e0. That little funny looking e is not really an e but the greek letter epsilon. Looking at the expression below we designate phi sub c as equal to the closed integral of the product of the electric field and an element of the area called dA. This integral, however, is equivalent to the charge divided by epsilon. It's not really confusing, but the important part comes next. The important component of the electric field must be in the same direction as the area element in question. That is the reason for the dot product, or divergence, in the formula.
Now, let us see our first example of how Maxwell's Equations can be used to solve Physics Problems by looking at a Gauss' Law problem!
1st Gauss' Law Example problem ... Gauss Law Example 1.
Navigate below to go back to the start!
You can go to DE Ex2 here...D.E. Ex2.
You can go to DE Ex1 here ...D.E. Ex1.
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