Question 820: Electrostatics & Capacitance

Question

A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger radius. Both the cylinder are initially electrically neutral.

Accepted Answer

Let the radii of the inner and outer cylinders be $r_1$ and $r_2$, respectively. The problem has cylindrical symmetry and Gauss's law can be applied to find the electric field $E$.
In the case (A), surface charge density $\sigma$ is given to the inner cylinder. Take a cylinder of radius $r\left(r_1<ight. r
$$ E(2 \pi r l)=\frac{q_{\mathrm{enc}}}{\epsilon_0}=\frac{\sigma\left(2 \pi r_1 light)}{\epsilon_0}, \quad 	ext { i.e., } \quad E=\frac{\sigma r_1}{\epsilon_0 r} . $$
Potential difference between the two cylinders is given by
$$ \Delta V=-\int_{r_1}^{r_2} E \mathrm{~d} r=-\frac{\sigma r_1}{\epsilon_0} \ln \left(\frac{r_2}{r_1}ight) $$
In the case (B), charge density $\sigma$ is given to the outer cylinder. Apply Gauss's law to get $E=0$ (because $q_{\mathrm{enc}}=0$ ) and hence $\Delta V=0$.
In the case (C), uniform linear charge density $\lambda$ is kept along the common axis of cylinders. Apply Gauss's law as in case (A) to get the electric field
$$ E(2 \pi r l)=\lambda l / \epsilon_0, \quad 	ext { i.e., } \quad E=\lambda /\left(2 \pi \epsilon_0 right) . $$
Potential difference between the two cylinders is
$$ \Delta V=-\int_{r_1}^{r_2} E \mathrm{~d} r=-\frac{\lambda}{2 \pi \epsilon_0} \ln \left(\frac{r_2}{r_1}ight) $$
In the case (D) charge density $\sigma$ is given to both the cylinders. Using the argument similar to the case (A), we get, $\Delta V=-\frac{\sigma r_1}{\epsilon_0} \ln \left(\frac{r_2}{r_1}ight)$.

Answer

A potential difference appears between the two cylinders when a charge density is given to the inner cylinder

Answer

A potential difference appears between the two cylinders when a charge density is given to the outer cylinder

Answer

No potential difference appears between the two cylinders when a uniform line charge is kept along the axis of the cylinders

Answer

No potential difference appears between the two cylinders when same charge density is given to both the cylinders

Electrostatics & Capacitance

A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger radius. Both the cylinder are initially electrically neutral.

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