The figure shows a conducting sphere 'A' of radius 'a' which is surrounded by a neutral conducting spherical shell B of radius 'b' (a). Initially switches S 1 , S 2

Question

The figure shows a conducting sphere 'A' of radius 'a' which is surrounded by a neutral conducting spherical shell B of radius 'b' (>a). Initially switches S 1 ​ , S 2 ​ and S 3 ​ are open and sphere 'A' carries a charge Q. First the switch ' S 1 ​ ' is closed to connect the shell B with the ground and then opened. Now the switch ' S 2 ​ ' is closed so that the sphere ' A ' is grounded and then S 2 ​ is opened. Finally, the switch ' S 3 ​ ' is closed to connect the spheres together. Find the heat (in joule) which is produced after closing the switch S 3 ​ . [Consider b = 4 cm , a = 2 cm and Q = 80 μ C ]

TestHub · Accepted Answer

When outer surface is grounded charge '-Q' resides on the inner surface of sphere 'B' Now sphere A is connected to earth potential on its surface becomes zero. Let the charge on the surface A becomes q

$\frac{kq}{a} - \frac{kQ}{b} = 0 \Rightarrow q = \frac{a}{b} Q$

Consider the figure. In this position energy stored

$E_{1} = \frac{1}{8 π ε _{0} a} [\frac{a}{b} Q]^{2} + \frac{Q ^{2}}{8 π ε _{0} b} + \frac{1}{4 π ε _{0} b} [\frac{a}{b} Q]$

when ' $S_{3}$ ' is closed, total charge will appear on the outer surface of shell ' B '. In this position energy stored

$E_{2} = \frac{1}{8 π ε _{0} b} (\frac{a}{b} - 1)^{2} Q^{2} Heat produced = E_{1} - E_{2} = \frac{Q ^{2} a ( b - a )}{8 π ε _{0} b ^{3}} = 180]$

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