Difference between revisions of "Maxwell's Equations"

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'''James Clerk Maxwell''' (b. 1831)


: $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$$
'''''Maxwell's Equations''''' 1861
 
In general, Maxwell's equations take the form:
 
: $$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$
: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
: $$\nabla \cdot \mathbf{B} = 0$$
: $$\nabla \cdot \mathbf{B} = 0$$
: $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
: $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
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:
In the example of an ideal vacuum with no charge or current, (i.e., $$\rho=0$$ and $$\mathbf{J}=0$$), these equations reduce to:


: $$\nabla \times \mathbf{B} = +\frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$$
: $$\nabla \times \mathbf{B} = \mu_0 \epsilon_0  \frac{\partial \mathbf{E}}{\partial t}$$
: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
: $$\nabla \cdot \mathbf{B} = 0$$
: $$\nabla \cdot \mathbf{B} = 0$$
: $$\nabla \cdot \mathbf{E} = 0$$
: $$\nabla \cdot \mathbf{E} = 0$$
Note that the speed of light is:
: $$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$


== Resources: ==
== Resources: ==
*[https://en.wikipedia.org/wiki/Maxwell%27s_equations Maxwell's Equations]
*[https://en.wikipedia.org/wiki/Maxwell%27s_equations Maxwell's Equations]
== Discussion: ==
== Discussion: ==

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:

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

Resources:

Discussion: