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Question: A cylindrical capacitor has two co – axial cylinders of length 20 cm and radii 1.5 cm and 1.6 cm. Th...

A cylindrical capacitor has two co – axial cylinders of length 20 cm and radii 1.5 cm and 1.6 cm. The outer cylinder is earthed and inner cylinder is given a charge of 4μC.4\mu C. The capacitance of the system is (neglect end effects)

A

2.8×10−8F2.8 \times 10^{- 8}F

B

4.2×10−14F4.2 \times 10^{- 14}F

C

1.7×10−10F1.7 \times 10^{- 10}F

D

3.4×10−12F3.4 \times 10^{- 12}F

Answer

1.7×10−10F1.7 \times 10^{- 10}F

Explanation

Solution

: Here, length L=20cm=20×10−2mL = 20cm = 20 \times 10^{- 2}m

Inner radius ri=1.5cm=1.5×10−2mr_{i} = 1.5cm = 1.5 \times 10^{- 2}m

outer radius ro=1.6cm=1.6×10−2mr_{o} = 1.6cm = 1.6 \times 10^{- 2}m

charge q=4μC=4×10−6Cq = 4\mu C = 4 \times 10^{- 6}C

Capacitance,

C=2πε0Llog⁡e(rori)=2πε0L2.3log⁡10(rori)C = \frac{2\pi\varepsilon_{0}L}{\log_{e}\left( \frac{r_{o}}{r_{i}} \right)} = \frac{2\pi\varepsilon_{0}L}{2.3\log_{10}\left( \frac{r_{o}}{r_{i}} \right)}

=2π×8.85×10−12×20×10−22.3log⁡10(1.6×10−21.5×10−2)= \frac{2\pi \times 8.85 \times 10^{- 12} \times 20 \times 10^{- 2}}{2.3\log_{10}\left( \frac{1.6 \times 10^{- 2}}{1.5 \times 10^{- 2}} \right)}

=1.7×10−10F= 1.7 \times 10^{- 10}F