Difference between revisions of "Maxwell's Equations"

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These are the Maxwell's Equations we will be using to solve for regions "I" and "II" in our approximation of the Michelson interferometer.
 
These are the Maxwell's Equations we will be using to solve for regions "I" and "II" in our approximation of the Michelson interferometer.
  
−
Gauss' Law:
+
{|align=center
 +
|Gauss' Law:
 +
|Gauss' Law for Magnetism:
 +
|-
 +
|<math>\boldsymbol{\nabla \cdot E} = 0 </math>
 +
|<math>\boldsymbol{\nabla \cdot B} = 0</math>
 +
|-
 +
|height="20"|&nbsp;||&nbsp;
 +
|-
 +
|Faradays's Law:
 +
|Ampere's Law:
 +
|-
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|width="400"|<math>\boldsymbol{\nabla \times E} + \frac{\partial \boldsymbol{B}}{\partial t}= 0</math>
 +
|width="400"|<math>\boldsymbol{\nabla \times B} - \mu_0\epsilon_0\frac{\partial \boldsymbol{E}}{\partial t}= 0 </math>
 +
|}
  
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<math>\boldsymbol{\nabla \cdot E} = 0 </math>
+
== In the Presence of Charges and Dielectric Media ==
  
−
Gauss' Law for Magnetism:
+
{|align=center
 +
|Gauss' Law:
 +
|Gauss' Law for Magnetism:
 +
|-
 +
|<math>\boldsymbol{\nabla \cdot D} = \rho </math>
 +
|<math>\boldsymbol{\nabla \cdot B} = 0</math>
 +
|-
 +
|height="20"|&nbsp;||&nbsp;
 +
|-
 +
|Faradays's Law:
 +
|Ampere's Law:
 +
|-
 +
|width="400"|<math>\boldsymbol{\nabla \times E} + \frac{\partial \boldsymbol{B}}{\partial t}= 0</math>
 +
|width="400"|<math>\boldsymbol{\nabla \times H} - \frac{\partial \boldsymbol{D}}{\partial t}= \boldsymbol{j} </math>
 +
|}
  
−
<math>\boldsymbol{\nabla \cdot B} = 0</math>
 
  
−
Faradays's Law:
+
Where <math>\boldsymbol{D} = \epsilon_0 \boldsymbol{E}</math> and <math>\boldsymbol{B} = \mu_0 \boldsymbol{H}</math>.
−
 
 
−
<math>\boldsymbol{\nabla \times E} + \frac{\partial \boldsymbol{B}}{\partial t}= 0</math>
 
−
 
 
−
Ampere's Law:
 
−
 
 
−
<math>\boldsymbol{\nabla \times B} - \mu_0\epsilon_0\frac{\partial \boldsymbol{E}}{\partial t}= 0 </math>
 
−
 
 
−
<font color="red">Need to add possibly derivation of wave equation and definitely Maxwell's equation in presence.  Need also to introduce D and H and relate them to E and B.</font>
 
−
 
 
−
Gauss' Law:
 
−
 
 
−
<math>\boldsymbol{\nabla \cdot D} = \rho </math>
 
−
 
 
−
Gauss' Law for Magnetism:
 
−
 
 
−
<math>\boldsymbol{\nabla \cdot B} = 0</math>
 
−
 
 
−
Faradays's Law:
 
−
 
 
−
<math>\boldsymbol{\nabla \times E} + \frac{\partial \boldsymbol{B}}{\partial t}= 0</math>
 
−
 
 
−
Ampere's Law:
 
−
 
 
−
<math>\boldsymbol{\nabla \times H} - \frac{\partial \boldsymbol{D}}{\partial t}= \boldsymbol{j} </math>
 
−
 
 
−
 
 
−
Back to [[Mapping diamond surfaces using interference]]
 

Latest revision as of 02:52, 6 April 2007

In Free Space

These are the Maxwell's Equations we will be using to solve for regions "I" and "II" in our approximation of the Michelson interferometer.

Gauss' Law: Gauss' Law for Magnetism:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\nabla \cdot E} = 0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\nabla \cdot B} = 0}
   
Faradays's Law: Ampere's Law:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\nabla \times E} + \frac{\partial \boldsymbol{B}}{\partial t}= 0} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\nabla \times B} - \mu_0\epsilon_0\frac{\partial \boldsymbol{E}}{\partial t}= 0 }

In the Presence of Charges and Dielectric Media

Gauss' Law: Gauss' Law for Magnetism:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\nabla \cdot D} = \rho }  
   
Faradays's Law: Ampere's Law:
   


Where   and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{B} = \mu_0 \boldsymbol{H}} .

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