Plane Electromagnetic Waves and Wave Propagation — John David Jackson
Physics — E&M · toroidal-coil-field-momentum
Problem
(a) Consider a circular toroidal coil of mean radius a and N turns, with a small uniform cross section of area A (both height and width small compared to a). The toroid has a current I flowing in it and there is a point charge Q located at its center. Calculate all the components of field momentum of the system; show that the component along the axis of the toroid is
(Pfield)z≈±4πa2μ0QINA
where the sign depends on the sense of the current flow in the coil. Assume that the electric field of the charge penetrates unimpeded into the region of nonvanishing magnetic field, as would happen for a toroid that is actually a set of N small nonconducting tubes inside which ionized gas moves to provide the current flow.
Check that the answer conforms to the approximation of Problem 6.5b.
(b) If Q=10−6 C (≈6×1012 electronic charges), I=1.0 A, N=2000, A=10−4 m2, a=0.1 m, find the electric field at the toroid in volts per meter, the magnetic induction in tesla, and the electromagnetic momentum in newton-seconds. Compare with the momentum of a 10 μg insect flying at a speed of 0.1 m/s.
[Note that the system of charge and toroid is at rest. Its total momentum must vanish. There must therefore be a canceling ``hidden'' mechanical momentum---see Problem 12.8.]