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JEE Main & Advanced Physics Electrostatics & Capacitance Question Bank Critical Thinking

  • question_answer
    If on the concentric hollow spheres of radii \[r\] and \[R(>r)\] the charge \[Q\] is distributed such that their surface densities are same then the potential at their common centre is  [IIT 1981; MP PMT 2003]

    A)                    \[\frac{Q({{R}^{2}}+{{r}^{2}})}{4\pi {{\varepsilon }_{0}}(R+r)}\]

    B)                                      \[\frac{QR}{R+r}\]

    C)                    Zero                                 

    D)            \[\frac{Q(R+r)}{4\pi {{\varepsilon }_{0}}({{R}^{2}}+{{r}^{2}})}\]

    Correct Answer: D

    Solution :

               \[{{q}_{1}}+{{q}_{2}}=Q\] and \[\frac{{{q}_{1}}}{4\pi {{r}^{2}}}=\frac{{{q}_{2}}}{4\pi {{R}^{2}}}\] (given) \[{{q}_{1}}=\frac{Q{{r}^{2}}}{{{R}^{2}}+{{r}^{2}}}\] and \[{{q}_{2}}=\frac{Q{{R}^{2}}}{{{R}^{2}}+{{r}^{2}}}\] Potential at common centre \[\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{Q{{r}^{2}}}{({{R}^{2}}+{{r}^{2}})r}+\frac{Q{{R}^{2}}}{({{R}^{2}}+{{r}^{2}})R} \right]=\frac{Q(R+r)}{4\pi {{\varepsilon }_{0}}({{R}^{2}}+{{r}^{2}})}\]


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