Difference between revisions of "Maxwell's Equations"

Latest revision as of 22:19, 12 March 2020

James Clerk Maxwell (b. 1831)

Maxwell's Equations 1861

In general, Maxwell's equations take the form:

$$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$

$$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$

$$\nabla \cdot \mathbf{B} = 0$$

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$

where $$\epsilon_0$$ is the permittivity of free space and $$\mu_0$$ is the permeability of free space.

In the example of an ideal vacuum with no charge or current, (i.e., $$\rho=0$$ and $$\mathbf{J}=0$$), these equations reduce to:

$$\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

$$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$

$$\nabla \cdot \mathbf{B} = 0$$

$$\nabla \cdot \mathbf{E} = 0$$

Note that the speed of light is:

$$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$

@@ Line 1: / Line 1: @@
-'''Joe Schmoe''' (b. xxxx)
+'''James Clerk Maxwell''' (b. 1831)
-'''''Title''''' xxxx
+'''''Maxwell's Equations''''' 1861
+In general, Maxwell's equations take the form:
-This formulation assumes no charge $$\rho=0$$ and $$J=0$$. One common example of these conditions is a vacuum.
+: $$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$
-: $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$$
 : $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
 : $$\nabla \cdot \mathbf{B} = 0$$
 : $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
+where $$\epsilon_0$$ is the permittivity of free space and $$\mu_0$$ is the permeability of free space.
+In the example of an ideal vacuum with no charge or current, (i.e., $$\rho=0$$ and $$\mathbf{J}=0$$), these equations reduce to:
+: $$\nabla \times \mathbf{B} =  \mu_0 \epsilon_0  \frac{\partial \mathbf{E}}{\partial t}$$
+: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
+: $$\nabla \cdot \mathbf{B} = 0$$
+: $$\nabla \cdot \mathbf{E} = 0$$
+Note that the speed of light is:
+: $$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$
 == Resources: ==
 *[https://en.wikipedia.org/wiki/Maxwell%27s_equations Maxwell's Equations]
 == Discussion: ==