Question

A cylindrical capacitor has two coaxial cylinders of length \[15cm\] and radii \[1.5cm\] and \[1.4cm\]. The outer cylinder is earthed and the inner cylinder is given a charge of \[3.5\mu C\]. 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).

Answer 1

Hint: 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
\[C = \dfrac{{2\pi { \in _0}l}}{{{{\log }_e}\dfrac{{{r_1}}}{{{r_2}}}}}\]

Complete step by step solution:
Let, the length of the co-axial cylinder is \[l = 15cm = 0.15m\]
The radius of the outer cylinder is \[{r_1} = 1.5cm = 0.015m\]. The radius of the inner cylinder \[{r_2} = 1.4cm = 0.014m\]
Inner cylinder gives charge \[q = 3.5\mu C = 3.5 \times {10^{ - 6}}C\]
The capacitance of the cylinder is \[C = \dfrac{{2\pi { \in _0}l}}{{{{\log }_e}\dfrac{{{r_1}}}{{{r_2}}}}}\]
Where, \[{ \in _0}\]is the permittivity of free space. \[{ \in _0} = 8.85 \times {10^{ - 12}}{N^{ - 1}}{m^{ - 2}}{C^2}\]
\[\therefore C = \dfrac{{2\pi \times 8.85 \times {{10}^{ - 12}} \times 0.15}}{{2.3026{{\log }_e}\left( {\dfrac{{0.015}}{{0.014}}} \right)}}F\]
\[C = \dfrac{{2\pi \times 8.85 \times {{10}^{ - 12}} \times 0.15}}{{2.3026 \times 0.0299}}F\]
\[C = 1.2 \times {10^{ - 10}}F\]
The potential of the inner cylinder is \[V = \dfrac{q}{C}\]
\[\therefore V = \dfrac{{3.5 \times {{10}^{ - 6}}}}{{1.2 \times {{10}^{^{ - 6}}}}}volt\]\[ = 2.92 \times {10^4}volt\]

Note:
The unit of the capacitor is Farad or \[F\]
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.