There is a horizontal cylindrical uniform but time-varying nagnetic field increasing at a constant rate `dB//dt` as shown in Fig. 3.173. A charged particle having charge `q` and mass `m` is kept in euilibrium, at the top of a spring of spring constant `K`, in such a way that it is on the horizontal line passing through the center of the magnetic field as shown in the figure. The compression in the spring will be

Options

A. `(1)/(K) [ mg - (q R ^(2))/( 2l ) (dB)/(dt) ]`
B. `(1)/(K) [ mg + ( q R ^(2))/( l ) (dB)/(dt) ]`
C. `(1)/(K) [ mg + ( 2 q R ^(2))/( l ) (dB)/( dt) ]`
D. `(1)/(K) [ mg + (q R ^(2))/( 2l) (dB)/( dt)]`

Answer

D

Explanation

`E 2 pi l = pi. R ^(2) ((dB)/( dt))`
`E = ( R ^(2))/( 2l ) ((dB)/(dt))`
`d E + mg = K x`
`x = ( q. R ^(2))/( K 2l ) ((d B)/( dt))+ ( mg)/( K )`
`x = (1)/( K ) [ mg + ( q R ^(2))/( 2l) ((dB)/( dt))].`