Spectral resolution of the retarded potentials

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Presentation transcript:

Spectral resolution of the retarded potentials LL2 Section 64

Electromagnetic waves were spectrally resolved in section 49. Static fields were spectrally resolved in section 51. Now, fields of moving charges will be spectrally resolved.

The Fourier components of the potentials are Now, make a Fourier expansion of the sources of the fields, charge and current. Each Fourier component of is the source for the corresponding component of the field.

The retarded potentials were given in (62.9) There is a phase that depends on the distance from the source point to the field point Fourier component of scalar potential Divide out the factor that oscillates in time. Same recipe for the vector potential

The differential equation satisfied by the scalar potential for an arbitrary field is (62.3). This equation must b e satisfied for each Fourier component separately. Time dependence is gone.

Sub Fourier coefficient for charge density Into expression for Fourier coefficient of potential Restore prime

Periodic motion gives discrete frequencies w0. Period T = 2p/w0. Spectral resolution contains integer multiples of the fundamental: n w0.