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A charged particle with charge $q$ and mass $m$ starts with an initial kinetic energy $K$ at the middle of a uniformly charged spherical region of total charge $Q$ and radius $R$ . $q$ and $Q$ have opposite signs. The spherically charged region is not free to move . The value of $K_0$ is such that the particle will just reach the boundary of the spherically charged region. How much time does it take for the particle to reach the boundary of the region.
$t = \frac{\pi }{2}\sqrt {\frac{{4\pi { \in _0}m{R^3}}}{{qQ}}} $
$t = \frac{\pi }{2}\sqrt {\frac{{2\pi { \in _0}m{R^3}}}{{qQ}}} $
$t = \frac{\pi }{4}\sqrt {\frac{{2\pi { \in _0}m{R^3}}}{{qQ}}} $
$t = \frac{\pi }{4}\sqrt {\frac{{4\pi { \in _0}m{R^3}}}{{qQ}}} $
Solution
Charge $\mathrm{q}$ executes $\mathrm{SHM}$ along diameter about centre as mean position.
$\overrightarrow{\mathrm{F}}_{\mathrm{q}}=\frac{-\mathrm{k} \mathrm{q} \mathrm{Q} \overrightarrow{\mathrm{x}}}{\mathrm{R}^{3}}$
$\mathrm{T}=2 \pi \sqrt{\frac{4 \pi \varepsilon_{0} \mathrm{mR}^{3}}{9 Q}}$
Time taken to move from centre to boundry
$=\frac{T}{4}=\frac{\pi}{2} \sqrt{\frac{4 \pi \varepsilon_{0} m R^{3}}{q Q}}$
