The magnetic flux density, B$,$ of a planet such as the Earth can be treated by assuming the planet is a uniformly magnetised sphere.

$($a$)$ Using this approximation, sketch both B and the magnetic field strength H

inside and outside the planet. Justify the choices you make in the sketch via reference

to boundary conditions or otherwise.

$($b$)$ The strength of the Earth$$s magnetic field may be measured by observing the motion of a suitably balanced compass needle displaced from true North. Assuming a

needle of moment of inertia I and magnetic moment m$,$ write down an expression for

the magnetic field strength in terms of observable quantities.