Classical Electrodynamics · Problem 5.31

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · accelerator-bending-magnet-inductance

Problem

An accelerator bending magnet consists of NN turns of superconducting cable whose current configuration can be described approximately by the axial current density Jz(ρ,ϕ)=(NI2R)cos⁡ϕ δ(ρ−R)J_z(\rho, \phi) = \left(\frac{NI}{2R}\right)\cos\phi\,\delta(\rho - R) The right circular current cylinder is centered on the axis of a hollow iron cylinder of inner radius R′R' (R′>RR' > R). The relative dimensions (RR, R′R' a few centimeters and a magnet length of several meters) permit the use of a two-dimensional approximation, at least away from the ends of the magnet. Assume that the relative permeability of the iron can be taken as infinite. [Then the outer radius of the iron is irrelevant.] (a) Show that the magnetic field inside the current sheath is perpendicular to the axis of the cylinder in the direction defined by ϕ=±π/2\phi = \pm\pi/2 and has the magnitude B0=(μ0NI4R)[1+R2R′2]B_0 = \left(\frac{\mu_0 NI}{4R}\right)\left[1 + \frac{R^2}{R'^2}\right] (b) Show that the magnetic energy inside r=Rr = R is augmented (and that outside diminished) relative to the values in the absence of the iron. (Compare part b of Problem 5.30.) (c) Show that the inductance per unit length is dLdz=(πμ0N28)[1+R2R′2]\frac{dL}{dz} = \left(\frac{\pi\mu_0 N^2}{8}\right)\left[1 + \frac{R^2}{R'^2}\right]

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