Question

Two charges with the same magnitude but opposite signs, +q and −q, are placed on opposite sides of a circle with radius a; thus, they are separated by a distance d = 2a. Now, assume that the circle is in the x−y plane and the two charges orbit around the z-axis through the center of the circle with a common frequency \omega . a.) Express the electric dipole moment ~p of such a system with two circulating opposite charges as a function of time using Cartesian coordinates. b.) With d ≪ c/\omega and for points far away from the moving charges, i.e., r ≫ d, the average radiated power can be computed for the general case via the average Poynting vector S~ and one finds hdPi = hS~i · dA~ = µ0 16\pi 2c h( ¨~p \times rˆ) 2 i dΩ Use your results from part a.) to compute the angular distribution of the radiated power for the above system of the two circulating charges. c.) How does this result compare to the power radiated by a simple oscillating electric dipole?

          Two charges with the same magnitude but opposite signs, +q and −q, are placed on
opposite sides of a circle with radius a; thus, they are separated by a distance d = 2a.
Now, assume that the circle is in the x−y plane and the two charges orbit around
the z-axis through the center of the circle with a common frequency \omega .
a.) Express the electric dipole moment ~p of such a system with two circulating
opposite charges as a function of time using Cartesian coordinates.
b.) With d ≪ c/\omega  and for points far away from the moving charges, i.e., r ≫ d, the
average radiated power can be computed for the general case via the average
Poynting vector S~ and one finds
hdPi = hS~i · dA~ =
µ0
16\pi 
2c
h(
¨~p \times  rˆ)
2
i dΩ
Use your results from part a.) to compute the angular distribution of the
radiated power for the above system of the two circulating charges.
c.) How does this result compare to the power radiated by a simple oscillating
electric dipole?

Added by Rosario A.

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