Question

There is a horizontal cylindrical uniform but time-varying magnetic field increasing at a constant rate $dB/dt$ as shown in figure. A charged particle having charge $q$ and mass $m$ is kept in equilibrium, at the top of a 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 :

• A. $\frac{1}{K}[mg-\frac{q{R}^2}{2l}\frac{dB}{dt}]$
• B. $\frac{1}{K}[mg+\frac{q{R}^2}{l}\frac{dB}{dt}]$
• C. $\frac{1}{K}[mg+\frac{2q{R}^2}{l}\frac{dB}{dt}]$
• D. $\frac{1}{K}[mg+\frac{q{R}^2}{2l}\frac{dB}{dt}]$

Answer 1

The correct option is D $\frac{1}{K}[mg+\frac{q{R}^2}{2l}\frac{dB}{dt}]$

Induced electric field $\int E\cdot dl=-\frac{d\left(AB\right)}{dt}\Rightarrow E\left(2\pi l\right)=-\left(\pi R^2\right)\frac{dB}{dt}$

When your right thumb points in the direction opposite to the change in B your fingers curl in the direction of the induced electric field.

$E=\frac{R^2}{2l}\frac{dB}{dt}...\left(downward\right)$

Electric force $F=qE=\frac{{qR}^2}{2l}\frac{dB}{dt}...\left(downward\right)$

Weight $W=mg...\left(downward\right)$

Compression $x=\frac{F+W}{K}=\frac{1}{K}[mg+\frac{{qR}^2}{2l}\frac{dB}{dt}]$