The radiation from a point charge we have studied in the textbook arises from its spatial motion. However, the time depe

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The Radiation From A Point Charge We Have Studied In The Textbook Arises From Its Spatial Motion However The Time Depe 1 (88.18 KiB) Viewed 53 times

The radiation from a point charge we have studied in the textbook arises from its spatial motion. However, the time dependence can stem from the temporal fluctuation in the carried charge. (a) Suppose we have a time-dependent charge distribution pſr,t) = 20$(3)(r) cos(wt), (1) such that the electric potential V(r, t) satisfies the following inhomogeneous wave equation 1 a2 v2V ət2 (2) EO We can solve Eq. (2) by invoking the complex amplitude ở that obeys V = Re[Ve-iw] as we have studied in Ch. 9. Find the wave equation of by regarding p as Re[908(3) (r)e-iwt]. (b) The wave equation of Ổ can be solved by the Green's function of the operator V2 + . Alternatively, we can perform the Fourier transformation to convert it into the momentum space, then we can use the contour integral to directly solve ï. The final result would be ỹ ./ \ انسان 401 (3) 4πεο |r| Prove that Eq. (3) satisfies your wave equation of Ŭ in (a). Is this result reasonable? (hint: what is its static limit?) In addition, what is the shape of the wavefront in this case?