Resolving Retarded Potentials and Fourier Components in Electromagnetic Fields
Explore the spectral resolution and Fourier expansions of retarded potentials, electromagnetic waves, and moving charges. Understand the differential equations and Fourier coefficients for scalar potentials in the context of electromagnetic field theory.
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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 0. Period T = 2/0. Spectral resolution contains integer multiples of the fundamental: n 0.
