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NCERT Class XII Chapter
Electrostatic Potential and Capacitance
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Question : 30 of 37
Marks: +1, -0
A spherical capacitor has an inner sphere of radius 12 cm and outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 μC. The space between the concentric spheres is filled with liquid of dielectric constant 32.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere.
(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.
Solution:  
r1r_1 = 13 cm , r2r_2 = 12 cm , K = 32 , Q = 2.5 µC
(a) Capacitance of capacitor is
C = 4πε0kr1r2r1−r2\frac{4\pi\varepsilon_0 k r_1 r_2}{r_1 - r_2} = 1×329×109\frac{1 \times 32}{9 \times 10^9} × 13×02×12×10−2(13−12)×10−2\frac{13 \times 0^{2} \times 12 \times 10^{-2}}{(13-12) \times 10^{-2}} or C = 5.5 × 10−910^{-9} F
(b) Electric potential of inner sphere is VBV_B = VBB+VBAV_{BB} + V_{BA}
= 14πε0k[+Qr2−Qr1]\frac{1}{4\pi\varepsilon_0 k} \left[ +\frac{Q}{r_2} - \frac{Q}{r_1} \right] = Q4πε0k[r1−r2r1r2]\frac{Q}{4\pi\varepsilon_0 k} \left[ \frac{r_1 - r_2}{r_1 r_2} \right]
= 9×10932\frac{9 \times 10^9}{32} 2.5 × 10−610^{-6} [13−1213×12]\left[ \frac{13-12}{13 \times 12} \right] × 10−210−4\frac{10^{-2}}{10^{-4}} = 4.5 × 10210^2 V
(c) Capacitance of isolated sphere of radius 12 cm is
C0C_0 = 4πε0r24\pi\varepsilon_0 r_2 = 19×109\frac{1}{9 \times 10^9} × 12 × 10−210^{-2} or C0C_0 = 1.3 × 10−1110^{-11} F
Here C > C0C_0 , because a single conductor A can be charged to a electric potential till it reaches the breakdown value of surroundings. But when another earthed metallic conductor B is brought near it, negative charge induced on it decreases the electric potential on A, hence more charge can not be stored on A.
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