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KVPY Sample Paper KVPY Stream-SX Model Paper-11

  • question_answer
    A solid sphere of radius \[{{R}_{1}}\] and volume charge density \[\rho =\frac{{{\rho }_{0}}}{r}\] is enclosed bt a hollow sphere of radius \[{{R}_{2}}\] with negative surface charge density \[\sigma ,\]such that the total charge in the system is zero. \[{{\rho }_{0}}\] is a positive constant and \[r\]is the distance from the Centre of the sphere. The ratio \[{{R}_{2}}/{{R}_{1}}\] is

    A) \[\frac{\sigma }{{{\rho }_{0}}}\]

    B) \[\sqrt{\frac{2\sigma }{{{\rho }_{0}}}}\]

    C) \[\sqrt{\frac{{{\rho }_{0}}}{2\sigma }}\]            

    D) \[\frac{{{\rho }_{0}}}{\sigma }\]

    Correct Answer: C

    Solution :

    \[{{q}_{1}}=\int\limits_{0}^{{{R}_{1}}}{\rho \times 4\pi {{r}^{2}}dr}\]\[=4\pi \int\limits_{0}^{{{R}_{1}}}{\frac{{{\rho }_{0}}}{r}\times {{r}^{2}}dr=2\pi {{\rho }_{0}}{{R}_{1}}^{2}}\]
    And \[{{q}_{2}}=-\sigma \times 4\pi {{R}^{2}}_{2}\]
    Given\[{{q}_{1}}+{{q}_{2}}=0\]
    Or, \[2\pi {{\rho }_{0}}{{R}_{1}}^{_{2}}-\sigma \times 4\pi {{R}_{2}}^{2}=0\]  
    \[\therefore \frac{{{R}_{2}}}{{{R}_{1}}}=\sqrt{\frac{{{\rho }_{0}}}{2\sigma }}\]


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