91Ó°ÊÓ

Modern magnetometers for measuring B-fields are described in Comment C12.2. In the older vibration magnetometer, a small permanent dipole was allowed to perform small angular oscillations in a horizontal plane first in a standard field \(\mathbf{B}_{0}\) and then in the unknown \(\mathrm{B}\). If the periods of oscillation were, respectively, \(T_{0}\) and \(T\), show that \(B=B_{0} T_{6}^{2} T^{2}\),

Two flat coils each of 10 turns have mean radii of 20 and \(2 \mathrm{~cm}\). Find an approximate value for the force between them if they are coaxial, \(10 \mathrm{~cm}\) apart and if each carries \(5 \mathrm{~A}\) in the same sense.

Take a small rectangular current loop of sides \(\mathrm{d} x\), dy situated at the origin in the xy plane and carrying a current \(I\) round its four sides. Find the B-field at a. field point P a great distance from the loop along the \(x\) axis. Show that the field has the form \(\mu_{0} m /\left(4 \pi x^{3}\right)\), where \(m\) is its dipole moment. (Note that all four sides produce a field.)

A magnetic dipole of moment \(m\) is situated at the origin of polar coordinates and lies along the \(\theta=0\) axis. Find the magnetic flux across the base of a cone of side \(r\) and semi-vertical angle \(\theta\) which has its apex at the origin and its axis along the direction of \(\mathbf{m}\).

A particle with charge \(e\) and mass \(m\) is projected with linear momentum \(p\) at right angles to a uniform magnetic field \(B\). What are the magnetic dipole moment and angular momentum possessed by the particle as a result of its subsequent motion?

A sphere of radius \(a\) is magnetized uniformly parallel to a diameter so, that its magnetic moment per unit volume is \(M\). Find the B-field at a distance \(r\) from the centre for \(r>a\) and \(r