1. Lienard Wierchert Potential1 1
Plane Waves &
Lienard Wierchert Potential
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. 2 2
Plane Waves 2
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3. Maxwell’s Equations in a Vacuum
d'Alembertian:
Full solution is a plane wave:
E1 and E2 are complex, the relative phase determines the polarization. 3
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4. Poynting Vector
The E & B field are oriented normal to the direction of propagation:
The Poynting vector describes the energy flow per unit area of the fields:
And the time average of the Poynting vector for a plane wave:
If E and H are complex, the time average can be written simply: 4
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5. Lienard Wierchert Potentials:5 5
Lienard Wierchert Potentials:
Relativistic 5
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6. Fields at the Speed of Light
The plane wave illustrates the speed at which electromagnetic information is propagated.
The fields from a moving charge radiate outward at speed c.
See Java Demo. 6
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7. Retarded Time
A point in space is affected by the E-fields originating from a source in the past, at a time called the retarded time.
In the past, the position of the source will have changed and that will in turn impact the time that should be evaluated. 7
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8. Retarded Time & Light Cone8
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9. Potentials for Retarded Time9
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10. 10 10
Moving Point Charge 10
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11. Solve by change of variables:
Moving Point Charge
Solve by change of variables: 11
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12. Moving Point Charge 12
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13. Moving Point Charge Potentials
We have derived:
A similar derivation shows that: 13
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14. Alternate Derivation With some algebra, it can be shown: 14 14 14 14
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15. Lienard-Wierchert Potentials
By using the instantaneous velocity of the particle at tret, the expression we derived can be applied to any particle movement, not just straight line movement. 15
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16. Fields from a Point Charge16
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17. Power Radiated from Accelerating Charge
Radiation in reference frame in which Beta is small:
Integrate over Poynting for radiation: 17
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18. Power Radiated from Accelerating Charge
Total power radiated:
Linear acceleration:
Circular acceleration:
Can also be written 18
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