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Question: A cylindrical capacitor has two coaxial cylinders of length \[15cm\] and radii \[1.5cm\] and \[1.4cm...

A cylindrical capacitor has two coaxial cylinders of length 15cm15cm and radii 1.5cm1.5cm and 1.4cm1.4cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5μC3.5\mu C. Determine the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends).

Explanation

Solution

Cylindrical capacitor is like a coaxial cable. The capacitance is stated as a capacitance per unit length. A coaxial capacitor is made with two concentric metallic cylinders. In between two cylinders, there are two media with different relative permittivities.

Formula used:
To calculate the capacitance of the cylinder we can use
C=2π∈0llog⁡er1r2C = \dfrac{{2\pi { \in _0}l}}{{{{\log }_e}\dfrac{{{r_1}}}{{{r_2}}}}}

Complete step by step solution:
Let, the length of the co-axial cylinder is l=15cm=0.15ml = 15cm = 0.15m
The radius of the outer cylinder is r1=1.5cm=0.015m{r_1} = 1.5cm = 0.015m. The radius of the inner cylinder r2=1.4cm=0.014m{r_2} = 1.4cm = 0.014m
Inner cylinder gives charge q=3.5μC=3.5×10−6Cq = 3.5\mu C = 3.5 \times {10^{ - 6}}C
The capacitance of the cylinder is C=2π∈0llog⁡er1r2C = \dfrac{{2\pi { \in _0}l}}{{{{\log }_e}\dfrac{{{r_1}}}{{{r_2}}}}}
Where, ∈0{ \in _0}is the permittivity of free space. ∈0=8.85×10−12N−1m−2C2{ \in _0} = 8.85 \times {10^{ - 12}}{N^{ - 1}}{m^{ - 2}}{C^2}
∴C=2π×8.85×10−12×0.152.3026log⁡e(0.0150.014)F\therefore C = \dfrac{{2\pi \times 8.85 \times {{10}^{ - 12}} \times 0.15}}{{2.3026{{\log }_e}\left( {\dfrac{{0.015}}{{0.014}}} \right)}}F
C=2π×8.85×10−12×0.152.3026×0.0299FC = \dfrac{{2\pi \times 8.85 \times {{10}^{ - 12}} \times 0.15}}{{2.3026 \times 0.0299}}F
C=1.2×10−10FC = 1.2 \times {10^{ - 10}}F
The potential of the inner cylinder is V=qCV = \dfrac{q}{C}
\therefore V = \dfrac{{3.5 \times {{10}^{ - 6}}}}{{1.2 \times {{10}^{^{ - 6}}}}}volt$$$$ = 2.92 \times {10^4}volt

Note:
The unit of the capacitor is Farad or FF
The capacitor is used to store a large amount of charge in a small space. In electronics motors, electronic juicers, flour mills, and other electronic instruments cylindrical capacitors can be used. The cylindrical capacitor includes a hollow or a solid cylindrical conductor surrounded by the concentric hollow spherical cylinder. The potential differences vary in each capacitor. To get the desired capacitance in many electrical circuits, a group of cylindrical capacitors is needed. There are two modes in capacitors, capacitors in series and capacitors in parallel.