Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 314 / 3 / a

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Solution

With no dependence on the azimuthal or axial coordinates, Gauss's law for magnetism in cylindrical coordinates gives

\frac{1}{R} \frac{d ( R B _{R} )}{d R} = 0, B_{R} = \frac{C}{R} .

Regularity at the axis forces

C = 0

, so

B_{R} = 0

. The radial component of the magnetostatic Ampère-Maxwell equation is

j_{R} = \frac{c}{4 π} (\nabla \times B)_{R} = \frac{c}{4 π} (\frac{1}{R} \partial_{ϕ} B_{z} - \partial_{z} B_{ϕ}) = 0.

Consequently

B_{R} = j_{R} = 0.

This is also the starting point of cylindrical magnetostatic pressure balance.

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