A spherical capacitor has an inner sphere of radius $12 \;cm$ and an outer sphere of radius $13\; cm .$ The outer sphere is earthed and the inner sphere is given a charge of $2.5\; \mu \,C .$ The space between the concentric spheres is filled with a liquid of dielectric constant $32$
$(a)$ Determine the capacitance of the capacitor.
$(b)$ What is the potential of the inner sphere?
$(c)$ Compare the capacitance of this capacitor with that of an isolated sphere of radius $12 \;cm .$ Explain why the latter is much smaller.
A spherical capacitor has an inner sphere of radius $12 \;cm$ and an outer sphere of radius $13\; cm .$ The outer sphere is earthed and the inner sphere is given a charge of $2.5\; \mu \,C .$ The space between the concentric spheres is filled with a liquid of dielectric constant $32$
$(a)$ Determine the capacitance of the capacitor.
$(b)$ What is the potential of the inner sphere?
$(c)$ Compare the capacitance of this capacitor with that of an isolated sphere of radius $12 \;cm .$ Explain why the latter is much smaller.
Radius of the inner sphere, $r_{2}=12 \,cm =0.12\, m$
Radius of the outer sphere, $r_{1}=13 \,cm =0.13 m$
Charge on the inner sphere, $q=2.5\, \mu\, C=2.5 \times 10^{-6}\, C$
Dielectric constant of a liquid, $\epsilon_{r}=32$
$(a)$ Capacitance of the capacitor is given by the relation,
$C=\frac{4 \pi \epsilon_{0} \epsilon_{r} r_{1} r_{2}}{r_{1}-r_{2}}$ Where,
$\epsilon_{0}=$ Permittivity of free space $=8.85 \times 10^{-12} \,C ^{2} \,N ^{-1} \,m ^{-2}$
$\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \,N\, m ^{2}\, C ^{-2}$
$\therefore C=\frac{32 \times 0.12 \times 0.13}{9 \times 10^{9} \times(0.13-0.12)}$
$=5.5 \times 10^{-9}\, F$
Hence, the capacitance of the capacitor is approximately $5.5 \times 10^{-9} \,F$
$(b)$ Potential of the inner sphere is given by,
$V=\frac{q}{C}$
$=\frac{2.5 \times 10^{-6}}{5.5 \times 10^{-9}}=4.5 \times 10^{2} \,V$
Hence, the potential of the inner sphere is $4.5 \times 10^{2} \,V$
$(c)$ Radius of an isolated sphere, $r=12 \times 10^{-2} \,m$
Capacitance of the sphere is given by the relation, $C^{\prime}=4 \pi \in_{0} r$
$=4 \pi \times 8.85 \times 10^{-12} \times 12 \times 10^{-12}$
$=1.33 \times 10^{-11} \,F$
The capacitance of the isolated sphere is less in comparison to the concentric spheres. This is because the outer sphere of the concentric spheres is earthed. Hence, the potential difference is less and the capacitance is more than the isolated sphere.
Similar Questions
An uncharged parallel plate capacitor having a dielectric of constant $K$ is connected to a similar air-cored parallel capacitor charged to a potential $V$. The two share the charge and the common potential is $V'$. The dielectric constant $K$ is
An uncharged parallel plate capacitor having a dielectric of constant $K$ is connected to a similar air-cored parallel capacitor charged to a potential $V$. The two share the charge and the common potential is $V'$. The dielectric constant $K$ is
In the figure a capacitor is filled with dielectrics. The resultant capacitance is
In the figure a capacitor is filled with dielectrics. The resultant capacitance is
The area of the plates of a parallel plate capacitor is $A$ and the gap between them is $d$. The gap is filled with a non-homogeneous dielectric whose dielectric constant varies with the distance $‘y’$ from one plate as : $K = \lambda \ sec(\pi y/2d)$, where $\lambda $ is a dimensionless constant. The capacitance of this capacitor is
The area of the plates of a parallel plate capacitor is $A$ and the gap between them is $d$. The gap is filled with a non-homogeneous dielectric whose dielectric constant varies with the distance $‘y’$ from one plate as : $K = \lambda \ sec(\pi y/2d)$, where $\lambda $ is a dimensionless constant. The capacitance of this capacitor is
Two parallel plate capacitors $C_1$ and $C_2$ each having capacitance of $10 \mu F$ are individually charged by a $100\,V$ $D.C.$ source. Capacitor $C _1$ is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor $C _2$ is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor $C_1$ is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be $.........V.$ (Assuming Dielectric constant $=10$ )
Two parallel plate capacitors $C_1$ and $C_2$ each having capacitance of $10 \mu F$ are individually charged by a $100\,V$ $D.C.$ source. Capacitor $C _1$ is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor $C _2$ is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor $C_1$ is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be $.........V.$ (Assuming Dielectric constant $=10$ )
- [JEE MAIN 2023]
Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $\theta$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is $1.5 \mathrm{~g} / \mathrm{cc}$, the dielectric constant of water will be
(Take density of water $=1 \mathrm{~g} / \mathrm{cc}$ )
Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $\theta$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is $1.5 \mathrm{~g} / \mathrm{cc}$, the dielectric constant of water will be
(Take density of water $=1 \mathrm{~g} / \mathrm{cc}$ )
- [JEE MAIN 2024]