A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Figure). Show that the capacitance of a spherical capacitor is given by
$C=\frac{4 \pi \varepsilon_{0} r_{1} r_{2}}{r_{1}-r_{2}}$
where $r_{1}$ and $r_{2}$ are the radii of outer and inner spheres, respectively.
Radius of the outer shell $=r_{1},$
radius of the inner shell $=r_{2}$
The inner surface of the outer shell has charge $+Q$.
The outer surface of the inner shell has induced charge $-Q$
Potential difference between the two shells is given by,
$V=\frac{Q}{4 \pi \epsilon_{0} r_{2}}-\frac{Q}{4 \pi \epsilon_{0} r_{1}}$
Where, $\epsilon_{0}=$ Permittivity of free space
$V =\frac{Q}{4 \pi \epsilon_{0}}\left[\frac{1}{r_{2}}-\frac{1}{r_{1}}\right]$
$=\frac{Q\left(r_{1}-r_{2}\right)}{4 \pi \epsilon_{0} r_{1} r_{2}}$
Capacitance of the given system is given by,
$C=\frac{\text { Charge }(Q)}{\text { Potential difference }(V)}$
$=\frac{4 \pi \epsilon_{0} r_{1} r_{2}}{r_{1}-r_{2}}$
Hence, proved.
The capacity of a spherical conductor in $ MKS$ system is
Two similar conducting balls having charges $+q$ and $-q$ are placed at a separation $d$ from each other in air. The radius of each ball is $r$ and the separation between their centres is $d(d > > r)$. Calculate the capacitance of the two ball system
Consider the situation shown in the figure. The capacitor $A$ has a charge $q$ on it whereas $B$ is uncharged. The charge appearing on the capacitor $B$ a long time after the switch is closed is
Two identical charged spherical drops each of capacitance $C$ merge to form a single drop. The resultant capacitance is
A parallel-plate capacitor is connected to a resistanceless circuit with a battery until the capacitor is fully charged. The battery is then disconnected from the circuit and the plates of the capacitor are moved to half of their original separation using insulated gloves. Let $V_{new}$ be the potential difference across the capacitor plates when the plates have moved. Let $V_{old}$ be the potential difference across the capacitor plates when they were connected to the battery $\frac{V_{new}}{V_{old}}=$......