A cylindrical capacitor has two co-axial cylinders of length $15\; cm$ and radii $1.5 \;cm$ and $1.4\; cm .$ 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).
A cylindrical capacitor has two co-axial cylinders of length $15\; cm$ and radii $1.5 \;cm$ and $1.4\; cm .$ 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).
Length of a co-axial cylinder, $l=15 \,cm =0.15\, m$
Radius of outer cylinder, $r_{1}=1.5 \,cm =0.015\, m$
Radius of inner cylinder, $r_{2}=1.4 \,cm =0.014 \,m$
Charge on the inner cylinder, $q=3.5\, \mu \,C=3.5 \times 10^{-6} \,C$
Capacitance of a co-axial cylinder of radii $r_{1}$ and $r_{2}$ is given by the relation
$C=\frac{2 \pi \epsilon_{0} l}{\log _{e r_{2}}^{r_{1}}}$
Where, $\varepsilon_{0}=$ Permittivity of free space $=8.85 \times 10^{-12}\, N ^{-1} \,m ^{-2} \,C ^{2}$
$\therefore C =\frac{2 \pi \times 8.85 \times 10^{-12} \times 0.15}{2.3026 \log _{10}\left(\frac{0.15}{0.14}\right)}$
$=\frac{2 \pi \times 8.85 \times 10^{-12} \times 0.15}{2.3026 \times 0.0299}$$=1.2 \times 10^{-10} \,F$
Potential difference of the inner cylinder is given by,
$V=\frac{q}{C}$
$=\frac{3.5 \times 10^{-6}}{1.2 \times 10^{-10}}=2.92 \times 10^{4}\, V$
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