The absolute measurement of resistance

This apparatus enables a measurement of resistance to be made that uses only fundamental units.
A copper disc is mounted on an axle inside a long solenoid (Figure 1).

The solenoid carries a current and there is therefore a uniform magnetic field of flux density B along its axis. The disc is rotated by a motor and the e.m.f. generated between its rim and its axle is balanced by that produced between the ends of the resistance R to be measured. The current through the solenoid also runs through R, as shown.
The speed of rotation of the disc is varied until the galvanometer shows no deflection. The number of revolutions of the disc per second (r) is measured with a stroboscope.

Let the axle diameter be 2b and the disc diameter be 2a.

Then area swept out by the disc per second
     = π(a² - b²)r = dA/dt

However, B = m_oNI/L and E = d(BA)/dt

Therefore

E = IR = [Np(a² - b²)rI]/L    and so:

The experiment is difficult to perform, as the e.m.f.s generated are usually very small. In addition friction between the contacts and the disc causes thermoelectric e.m.f.s that may not be negligible compared with those produced by induction. Allowance should also be made for the magnetic field of the Earth and for the fact that the field of the solenoid is not quite uniform over the area of the disc.

Example problem
Example Calculate the e.m.f produced by a disc rotating at 20 revs per second inside a solenoid of 1000 turns and length 1 m carrying a current of 1 A.

The radii of the disc and axle are 2 cm and 0.25 cm respectively

E.m.f. generated = 4πx10^-7x1000xπx3.34x10^-4 = 1.36x10^-6 V = 1.36 μV