Electron Motion in Charged Sphere with Radiation · USAPhO 2016 #B2
Physics · electromagnetism
Problem
a. A spherical region of space of radius R has a uniform charge density and total charge +Q. An electron of charge −e is free to move inside or outside the sphere, under the influence of the charge density alone. For this first part ignore radiation effects.
i. Consider a circular orbit for the electron where r<R. Determine the period of the orbit T in terms of any or all of r, R, Q, e, and any necessary fundamental constants.
ii. Consider a circular orbit for the electron where r>R. Determine the period of the orbit T in terms of any or all of r, R, Q, e, and any necessary fundamental constants.
iii. Assume the electron starts at rest at r=2R. Determine the speed of the electron when it passes through the center in terms of any or all of R, Q, e, and any necessary fundamental constants.
b. Accelerating charges radiate. The total power P radiated by charge q with acceleration a is given by
P=Cξan
where C is a dimensionless numerical constant (which is equal to 1/6π), ξ is a physical constant that is a function only of the charge q, the speed of light c, and the permittivity of free space ϵ0, and n is a dimensionless constant. Determine ξ and n.
c. Consider the electron in the first part, except now take into account radiation. Assume that the orbit remains circular and the orbital radius r changes by an amount ∣Δr∣≪r.
i. Consider a circular orbit for the electron where r<R. Determine the change in the orbital radius Δr during one orbit in terms of any or all of r, R, Q, e, and any necessary fundamental constants. Be very specific about the sign of Δr.
ii. Consider a circular orbit for the electron where r>R. Determine the change in the orbital radius Δr during one orbit in terms of any or all r, R, Q, e, and any necessary fundamental constants. Be very specific about the sign of Δr.