A) \[\frac{{{q}^{2}}}{\pi {{\varepsilon }_{0}}r}\]
B) \[\frac{{{q}^{2}}}{4\pi {{\varepsilon }_{0}}r}\]
C) \[\frac{{{q}^{2}}}{8\pi {{\varepsilon }_{0}}r}\]
D) \[\frac{{{q}^{2}}}{16\pi {{\varepsilon }_{0}}r}\]
Correct Answer: D
Solution :
| the capacitance of inner sphere is \[{{C}_{1}}=4\pi {{\varepsilon }_{0}}r\] |
| the capacitance of the outer shell is \[{{C}_{2}}=4\pi {{\varepsilon }_{0}}(2r)=8\pi {{\varepsilon }_{0}}r\] |
| Before connection, the total energy is |
| \[{{U}_{1}}=\frac{{{q}^{2}}}{2{{C}_{1}}}=\frac{{{q}^{2}}}{8\pi {{\varepsilon }_{0}}r}\] |
| After connection, the entire charge q of the inner sphere is transferred to the outer shell. Hence, energy after connection is |
| \[{{U}_{2}}=\frac{{{q}^{2}}}{2{{C}_{2}}}=\frac{{{q}^{2}}}{16\pi {{\varepsilon }_{0}}r}\]\[\therefore \]Heat generated = \[{{U}_{1}}-{{U}_{2}}\] |
| \[=\frac{{{q}^{2}}}{8\pi {{\varepsilon }_{0}}r}-\frac{{{q}^{2}}}{16\pi {{\varepsilon }_{0}}r}=\frac{{{q}^{2}}}{16\pi {{\varepsilon }_{0}}r}\] |
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