Difference between revisions of "Maxwell's Equations"
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'''James Clerk Maxwell''' (b. 1831) | |||
: $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \ | '''''Maxwell's Equations''''' 1861 | ||
In general, Maxwell's equations take the form: | |||
: $$\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} = | : $$\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$ | ||
: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$ | : $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$ | ||
: $$\nabla \cdot \mathbf{B} = 0$$ | : $$\nabla \cdot \mathbf{B} = 0$$ | ||
: $$\nabla \cdot \mathbf{E} = 0$$ | : $$\nabla \cdot \mathbf{E} = 0$$ | ||
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:
- $$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$