A thin wire of negligible resistance and total length D is wound to form a thin cylindrical solenoid of length ℓ≪D. A conducting sphere of radius R≪ℓ is attached to each end of the solenoid. Initially there is no current in the wire, and the spheres have charges Q and −Q. Give all your answers in terms of R, ℓ, D, and the speed of light c=1/μ0ε0.
a. Assume this system can be modeled as an LC circuit. What is the angular frequency of its oscillations?
This system loses energy because it emits electromagnetic radiation. Consider an electric dipole consisting of charges ±q0cos(ωt) separated by distance d, whose dipole moment oscillates with amplitude p0=q0d. If d is much smaller than the wavelength λ of the radiation produced, then it can be shown that the power radiated is roughly (i.e. up to an order-one dimensionless factor)
P∼ε0c3ω4p02.
For the rest of the problem, your answers only need to be similarly rough estimates.
b. For this setup, the above formula applies if D≫D0. Find a rough estimate for D0.
c. Assuming D≫D0, estimate the number of oscillations that occurs until half the energy is lost.