Maxwell's Equations: Difference between revisions

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In general, Maxwell's equations take the form:
In general, Maxwell's equations take the form:


: $$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$
: <math>\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)</math>
: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
: <math>\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}</math>
: $$\nabla \cdot \mathbf{B} = 0$$
: <math>\nabla \cdot \mathbf{B} = 0</math>
: $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
: <math>\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}</math>


where $$\epsilon_0$$ is the permittivity of free space and $$\mu_0$$ is the permeability of free space.
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., $$\rho=0$$ and $$\mathbf{J}=0$$), these equations reduce to:
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:


: $$\nabla \times \mathbf{B} =  \mu_0 \epsilon_0  \frac{\partial \mathbf{E}}{\partial t}$$
: <math>\nabla \times \mathbf{B} =  \mu_0 \epsilon_0  \frac{\partial \mathbf{E}}{\partial t}</math>
: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
: <math>\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}</math>
: $$\nabla \cdot \mathbf{B} = 0$$
: <math>\nabla \cdot \mathbf{B} = 0</math>
: $$\nabla \cdot \mathbf{E} = 0$$
: <math>\nabla \cdot \mathbf{E} = 0</math>


Note that the speed of light is:
Note that the speed of light is:


: $$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$
: <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]

Resources:[edit]

Discussion:[edit]