Maxwell's Equations

James Clerk Maxwell (b. 1831)

Maxwell's Equations 1861

In general, Maxwell's equations take the form:

[math]\displaystyle{ \nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right) }[/math]

[math]\displaystyle{ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} }[/math]

[math]\displaystyle{ \nabla \cdot \mathbf{B} = 0 }[/math]

[math]\displaystyle{ \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} }[/math]

where [math]\displaystyle{ \epsilon_0 }[/math] is the permittivity of free space and [math]\displaystyle{ \mu_0 }[/math] is the permeability of free space.

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

[math]\displaystyle{ \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} }[/math]

[math]\displaystyle{ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} }[/math]

[math]\displaystyle{ \nabla \cdot \mathbf{B} = 0 }[/math]

[math]\displaystyle{ \nabla \cdot \mathbf{E} = 0 }[/math]

Note that the speed of light is:

[math]\displaystyle{ c = \frac{1}{\sqrt{\epsilon_0 \mu_0}} }[/math]

Resources:

• Maxwell's Equations