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A bullet of mass $m$ and charge $q$ is fired towards a solid uniformly charged sphere of radius $R$ and total charge $+ q$. If it strikes the surface of sphere with speed $u$, find the minimum speed $u$ so that it can penetrate through the sphere. (Neglect all resistance forces or friction acting on bullet except electrostatic forces)
$\frac{q}{{\sqrt {2\pi {\varepsilon _0}mR} }}$
$\frac{q}{{\sqrt {4\pi {\varepsilon _0}mR} }}$
$\frac{q}{{\sqrt {8\pi {\varepsilon _0}mR} }}$
$\frac{{\sqrt 3 \,\,q}}{{\sqrt {4\pi {\varepsilon _0}mR} }}$
Solution
Maximum potential is at the surface of the sphere
$\frac{1}{2} m u^{2}=v_{c}-v_{s}$
$=\frac{3}{2} \frac{k q^{2}}{R}-\frac{k q^{2}}{R}$
$\frac{1}{2} m u^{2}=\frac{k q^{2}}{2 R}$
$u^{2}=\frac{k q^{2}}{m R}$
$u=\frac{k q}{\sqrt{m R}}=\frac{q}{\sqrt{4 \pi \varepsilon_{0} m R}}$
