Figure 6.20 shows a metal rod PQ resting on the smooth rails AB and positioned between the poles of a permanent magnet. The rails, the rod, and the magnetic field are in three mutual perpendicular directions. A galvanometer G connects the rails through a switch K. Length of the rod = 15 cm, B = 0.50 T, resistance of the closed loop containing the rod = 9.0 m`Omega`. Assume the field to be uniform.
(a) Suppose K is open and the rod is moved with a speed of 12 cm `s^(-1)` in the direction shown. Give the polarity and magnitude of the induced emf.

(b) Is there an excess charge built up at the ends of the rods when K is open? What if K is closed?
(c) With K open and the rod moving uniformly, there is no net force on the electrons in the rod PQ even though they do experience magnetic force due to the motion of the rod. Explain.
(d) What is the retarding force on the rod when K is closed?
(e) How much power is required (by an external agent) to keep the rod moving at the same speed (=12 cm `s^(-1)`) when K is closed? How much power is required when K is open?
(f ) How much power is dissipated as heat in the closed circuit? What is the source of this power?
(g) What is the induced emf in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?

Explanation

`|epsi|= vBl= 0.12xx0.50xx0.15= 9.0 mV,`
P positive end and Q negative end.
(b) Yes. When K is closed, the excess charge is maintained by the continuous flow of current.
(c) Magnetic force is cancelled by the electric force set-up due to the excess charge of opposite signs at the ends of the rod.
(d) Retarding force = `IBl`
`(9mV)/(9m Omega) xx 0.5 T xx 0.15 m`
`= 75 xx 10^(-3) N `
(e) Power expended by an external agent against the above retarding force to keep the rod moving uniformly at 12 cm `s^(-1)`
`=7.5xx10^(-3) xx 12 xx 10^(-12) = 9.0 xx 10^(-3) W`
When K is open , no power is expended
(f) `I^2 R = 1 xx 1 xx 9 xx 10^(-3) = 9.0 xx 10^(-3) W`
The source of this power is the power provided by the external agent as calculated above.
(g) Zero, motion of the rod does not cut across the field lines. [Note:length of PQ has been considered above to be equal to the spacing between the rails.]