Conservation of Charge

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Conservation of Charge
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Lorentz Transformation of Current
Example Lorentz transform from charges rest
frame
Total charge is invariant!
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Lorentz Transformation of a Sheet of Charge
Sheet of charge
Length contraction
Moving charge
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Lorentz Transformation of Sheet of Current
Sheet of current
See problem sets
Where does this come from?
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Maxwells Equations
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On the Way to the Wave Equation
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Choice of Gauge
Freedom to choose
Lorentz gauge
0
The Wave Equation!!!
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GaugeTransformation
Gauge Transformation
We want
We have
Therefore
Just a solution to Laplaces equation.
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Lorentz Gauge 4 Vectors
Lorentz Condition
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Lorentz Transforms of Vector Potential
But you must transform the positions also!
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Single Charge Example
Potential in charges rest frame
Potential in moving frame
Change to moving frame coordinates
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Single Charge Example (continued)
Evaluated at t0.
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E Field Direction
Direction of E
Always radial!
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E Field Magnitude
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E Field Magnitude
Ex
Ex on x axis
Increasing b
x
Ey
Ey on y axis
Increasing b
y
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Wave Equation
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Electric and Magnetic Fields
4
n 1
m 1
4
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Transformation of Field Tensor
But remember, you must change the positions also!
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Maxwells Charge Source Equations
Maxwells Source Equations
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Example Sheet of Charge
Sheet of charge
Transformed fields
In moving frame
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Example Sheet of Current
Sheet of current
Transformed fields
In moving frame
Gausss and Amperes Laws
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Maxwells Monopole Source Equations
Dual Tensor
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Gauge Transformations
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Summary of EM Equations
Conservation of Charge
Lorentz Condition
Wave Equation for Potentials
Field Equations
Maxwells Charge Equations
Maxwells Monopole Equations
Gauge Transformations