Braking up - Electromagnetic Angular Momentum · USAPhO 2020 #A1

Physics · electromagnetism

Problem

An infinitely long wire with linear charge density −λ-\lambda lies along the z-axis. An infinitely long insulating cylindrical shell of radius aa is concentric with the wire and can rotate freely about the z-axis. The shell has moment of inertia per unit length II. Charge is uniformly distributed on the shell, with surface charge density λ2πa\frac{\lambda}{2\pi a}. The system is immersed in an external magnetic field B0z^B_0 \hat{z}, and is initially at rest. Starting at t=0t = 0, the external magnetic field is slowly reduced to zero over a time T≫a/cT \gg a/c, where cc is the speed of light. a. Find an expression of the final angular velocity ω\omega of the cylinder in terms of the symbols given and other constants. b. You may be surprised that the expression you find above is not zero! However, the electric and magnetic fields can have angular momentum. Analogous to the "regular" angular momentum definition, the EM field angular momentum per unit volume at a displacement r⃗\vec{r} from the axis of rotation is: L⃗(r⃗)=r⃗×P⃗(r⃗).\vec{L}(\vec{r}) = \vec{r} \times \vec{P}(\vec{r}). P⃗(r⃗)\vec{P}(\vec{r}) is a vector analogous to momentum, given by P⃗(r⃗)=α⋅(E⃗(r⃗)×B⃗(r⃗)).\vec{P}(\vec{r}) = \alpha \cdot (\vec{E}(\vec{r}) \times \vec{B}(\vec{r})). where α\alpha is some proportionality constant. Find an expression for α\alpha in terms of given variables and fundamental constants.

Browse the problem library