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. (a) `(1)/(K)[mg - (qR^(2))/(2l)(dB)/(dt)]`
B. (b) `(1)/(K)[mg + (qR^(2))/(l)(dB)/(dt)]`
C. ( c) `(1)/(K)[mg + (2qR^(2))/(l)(dB)/(dt)]`
D. (d) `(1)/(K)[mg + (qR^(2))/(2l)(dB)/(dt)]`

Answer

D

Explanation

(d) `E2pil = pir^(2)((dB)/(dt)), E = (R^(2))/(2l)((dB)/(dt))`
`qE + mg = Kx`
`rarr` `x = (qR^(2))/(K2l)((dB)/(dt)) + (mg)/(K), x = (1)/(K)[(mg + (qR^(2))/(2l)(dB)/(dt)]`