question_answer

Two small spheres each of mass 'm' and each with charge q lie inside a non-conducting smooth hemispherical bowl of radius R placed on ground. The charge required for equilibrium at separation 2d is given by

A) \[\sqrt{\frac{16\pi {{\varepsilon }_{0}}{{d}^{3}}mg}{\sqrt{{{R}^{2}}-{{d}^{2}}}}}\]

B)        \[\sqrt{\frac{8\pi {{\varepsilon }_{0}}{{d}^{3}}mg}{\sqrt{{{R}^{2}}-{{d}^{2}}}}}\]

C) \[\sqrt{\frac{8\pi {{\varepsilon }_{0}}{{d}^{3}}mg}{\sqrt{{{R}^{2}}+{{d}^{2}}}}}\]

D)        \[\sqrt{\frac{16\pi {{\varepsilon }_{0}}{{d}^{3}}mg}{\sqrt{{{R}^{2}}+{{d}^{2}}}}}\]

Correct Answer: A

Solution :

[a]  \[{{F}_{e}}=N\,\,\sin \theta \] \[mg=N\cos \theta \] \[{{F}_{e}}=mg\,\tan \theta \] \[\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}^{2}}}{4{{d}^{2}}}=mg\frac{d}{\sqrt{{{R}^{2}}-{{d}^{2}}}}\] \[q=\sqrt{\frac{16\pi {{\varepsilon }_{0}}{{d}^{3}}mg}{\sqrt{{{R}^{2}}-{{d}^{2}}}}}\]