question_answer

The capacitance of a spherical condenser is \[1\mu F\]. If the spacing between the two spheres is\[1\,mm\], then the radius of the outer sphere is                                    [CPMT 1989]

A)                    \[30cm\]

B)                                      \[6m\]

C)                    \[5cm\]                           

D)            \[3m\]

Correct Answer: D

Solution :

Given : (b ? a) = 1 ´ 10?3 m     ..... (i) and \[C=4\pi {{\varepsilon }_{0}}\left( \frac{ab}{b-a} \right)=1\times {{10}^{-6}}\] Þ \[1\times {{10}^{-6}}=\frac{1}{9\times {{10}^{9}}}\left( \frac{ab}{{{10}^{-3}}} \right)\]                  Þ \[ab=9\]                                       ..... (ii) From equations (i) and (ii) \[b-\frac{9}{b}=\frac{1}{1000}\] Þ 1000 b2 ? b ? 9000 = 0 Þ \[b=\frac{1\pm \sqrt{{{(-1)}^{2}}\,-\,4(1000)\,(-\,9000)}}{2\times 1000}\]  {Solving of quadratic equation} Þ \[b=\frac{1\pm \sqrt{36\times {{10}^{6}}}}{2000}\approx \,\frac{\sqrt{36\times {{10}^{6}}}}{2000}=3\,m.\]