Maxwell's Equations: Difference between revisions
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''' | '''James Clerk Maxwell''' (b. 1831) | ||
''''' | '''''Maxwell's Equations''''' 1861 | ||
In general, Maxwell's equations take the form: | |||
: <math>\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)</math> | |||
: <math>\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}</math> | |||
: <math>\nabla \cdot \mathbf{B} = 0</math> | |||
: <math>\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}</math> | |||
where <math>\epsilon_0</math> is the permittivity of free space and <math>\mu_0</math> is the permeability of free space. | |||
In the example of an ideal vacuum with no charge or current, (i.e., | In the example of an ideal vacuum with no charge or current, (i.e., <math>\rho=0</math> and <math>\mathbf{J}=0</math>), these equations reduce to: | ||
: | : <math>\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}</math> | ||
: | : <math>\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}</math> | ||
: | : <math>\nabla \cdot \mathbf{B} = 0</math> | ||
: | : <math>\nabla \cdot \mathbf{E} = 0</math> | ||
Note that the speed of light is: | |||
: <math>c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}</math> | |||
== 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 16:45, 19 February 2023
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]