A point charge q, of mass m, is attached to a spring of constant k. At time t = 0 it is given a kick, so its initial energy is Now it oscillates, gradually radiating away this energy.

(a) Confirm that the total energy radiated is equal to U0. Assume the radiation damping is small, so you can write the equation of motion as 

(b) Suppose now we have two such oscillators, and we start them off with identical kicks. Regardless of their relative positions and orientations, the total energy radiated must be 2U0. But what if they are right on top of each other, so it’s equivalent to a single oscillator with twice the charge; the Larmor formula says that the power radiated is four times as great, suggesting that the total will be 4U₀. Find the error in this reasoning, and show that the total is actually 2U₀, as it should be.

Transcribed Image Text:

U₁ = 1/2mv ²