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  • There is a horizontal cylindrical uniform but time varying magnetic field increasing at a constant rate <img alt="\mathrm{\frac{dB}{dt}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%

There is a horizontal cylindrical uniform but time varying magnetic field increasing at a constant rate \mathrm{\frac{dB}{dt}} as shown. 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 figure. Them compression in the spring will be

Option: 1

\mathrm{\frac{1}{\mathrm{~K}}\left[\mathrm{mg}-\frac{\mathrm{qR}^2}{2 \ell} \frac{\mathrm{dB}}{\mathrm{dt}}\right]}


Option: 2

\mathrm{\frac{1}{\mathrm{~K}}\left[\mathrm{mg}+\frac{\mathrm{qR}^2}{\ell} \frac{\mathrm{dB}}{\mathrm{dt}}\right]}


Option: 3

\mathrm{\frac{1}{\mathrm{~K}}\left[\mathrm{mg}+\frac{2 \mathrm{qR}^2}{\ell} \frac{\mathrm{dB}}{\mathrm{dt}}\right]}


Option: 4

\mathrm{\frac{1}{\mathrm{~K}}\left[\mathrm{mg}+\frac{\mathrm{qR}^2}{2 \ell} \frac{\mathrm{dB}}{\mathrm{dt}}\right].}


Answers (1)

best_answer

\mathrm{\begin{aligned} &\mathrm{E} 2 \pi \ell=\pi \mathrm{R}^2\left(\frac{\mathrm{dB}}{\mathrm{dt}}\right)\\ &\begin{aligned} & \mathrm{E}=\frac{\mathrm{R}^2}{2 \ell}\left(\frac{\mathrm{dB}}{\mathrm{dt}}\right) \\ & \mathrm{qE}+\mathrm{mg}=\mathrm{Kx} \\ & \mathrm{x}=\frac{\mathrm{q} \cdot \mathrm{R}^2}{\mathrm{~K} 2 \ell}\left(\frac{\mathrm{dB}}{\mathrm{dt}}\right)+\frac{\mathrm{mg}}{\mathrm{K}} \\ & \mathrm{x}=\frac{1}{\mathrm{~K}}\left[\mathrm{mg}+\frac{\mathrm{qR}}{2 \ell} \frac{\mathrm{dB}}{\mathrm{dt}}\right] \end{aligned} \end{aligned}}

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