User talk:Williamborg/Electromagnetics

Electromagnetic Waves

=Maxwell's Equations (Differential Formulation)= $$\nabla \cdot \mathbf{E} = \frac {\rho}{\epsilon_0} $$        (Equation 1)
 * Gauss's Law for Electric Fields

$$\nabla \cdot \mathbf{B} = 0 $$       (Equation 2)
 * Gauss's Law for Magnetic Fields

$$\nabla \times \mathbf{E} + \frac {\partial \mathbf{B}}{\partial t} = 0 $$       (Equation 3) $$\nabla \times \mathbf{B} - \mu_0 \epsilon_0 \frac {\partial \mathbf{E}}{\partial t} = \mu_0 \mathbf{J} $$        (Equation 4)
 * Faraday's Law
 * Ampere-Maxwell Law

Or alternately, defining a constant c such that $$c^2 = \frac {1}{\mu_0 \epsilon_0}$$ we can rewrite equation 4 in another form:

$$\nabla \times \mathbf{B} - \frac{1}{c^2} \frac {\partial \mathbf{E}}{\partial t} = \mu_0 \mathbf{J} $$        (Equation 4A)

There is a strong motivation for making this substitution, as we will shortly show that c is the speed of light in a vacuum.

=Electromagnetic Waves in a Vacuum = First let us solve Maxwell's Equations to determine what a solution of these equations look like in free space. We can simplify the solution of Maxwell's Equations by applying the properties of free space:
 * In free space the charge density ($$\rho$$) is zero.
 * Since there is no charge, the current density vector $$\mathbf J$$ is also zero.
 * The permittivity of free space is $$\epsilon_0$$ and the permeability of free space $$\mu_0$$. Both are constant throughout free space for all time.

The next step is to convert these equations to a form for which a solution is easy.

By taking the curl of the third equation , we can show:
 * $$\nabla \times \nabla \times \mathbf{E}= -\nabla^2 \mathbf{E}= -\frac{\partial}{\partial t} \nabla \times \mathbf{B}$$     (Equation 5)

Using Equation 4, subject to the condition that $$\mathbf J = 0$$, we know that:
 * $$\nabla \times \mathbf{B} = \mu_0\varepsilon_0  \frac{\partial \mathbf{E}}{\partial t}$$      (Equation 6)

Substituting Equation 6 into Equation 5, we can show:

-\nabla^2\mathbf{E}=-\frac{\partial}{\partial t}( \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t})$$      (Equation 7) or since $$\mu_0$$ and $$\epsilon_0$$ do not vary with time, $$ \nabla^2 \mathbf{E}-\mu_0\epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}= 0 $$     (Equation 8) $$\nabla^2 \mathbf{E}-\frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 $$     (Equation 9)

Similarly, taking the curl of Equation 4, it is possible to show: $$ \nabla^2 \mathbf{B}-\mu_0\epsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0 $$     (Equation 10)

$$ \nabla^2 \mathbf{B}-\frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0 $$      (Equation 11)

Solving the second order differential equations
By applying our knowledge of vector calculus, we found that in free space Maxwell's Equations convert into two equations, one for the electric field (Equation 9) and one for the magnetic field (Equation 10), which you may already recognize as second order differential equations, for which the solutions are vector wave equations.

Assume that a solution for the vector Electric Field ($$\mathbf{E}$$) exists and allow the the possibility that that solution is a complex function of space and time.