Animated Solution for Physics - Electrostatics: A long, hollow conducting cylinder is kept co-axially inside another long, hollow conducting cylinder of larger radius. Both the cylinders are initially electrically neutral.

The problem of coaxial conducting cylinders is a classic exploration of Gauss's Law and the fundamental relationship between electric fields and electric potential. Let's dive deep into the physics of this setup and understand exactly when and why a potential difference appears between the two cylinders.

Imagine two long, hollow conducting cylinders placed coaxially, one inside the other. Let the inner cylinder have a radius $R_1$ and the outer cylinder have a radius $R_2$. Initially, both cylinders are perfectly electrically neutral.

The core question asks us to determine under what conditions a potential difference will appear between these two cylinders. To answer this, we must recall the fundamental relationship between electric potential and electric field. The potential difference $\Delta V$ between two points is defined as the negative line integral of the electric field $\vec{E}$ along a path connecting them:

This equation tells us a profound truth: a potential difference exists between the two cylinders if and only if there is a non-zero electric field in the region between them (i.e., for $R_1<r<R_2$).

Gauss's Law

To find out if an electric field exists in that intermediate region, we turn to Gauss's Law. Gauss's Law states that the electric flux through any closed surface is proportional to the total charge enclosed by that surface:

If we construct a cylindrical Gaussian surface of radius $r$ such that $R_1<r<R_2$, the electric field at distance $r$ depends entirely on the charge enclosed within this imaginary surface. If $q_{\text{enclosed}}=0$, then $\vec{E}=0$, and consequently, $\Delta V=0$. If $q_{\text{enclosed}} eq 0$, then an electric field exists, and a potential difference will appear.

Now, let's systematically evaluate each option provided in the question based on our Gaussian framework.

Option (a): Charge density given to the inner cylinder If we place a charge on the inner cylinder, our Gaussian surface (which has a radius $r>R_1$) will enclose this charge. Since $q_{\text{enclosed}} eq 0$, an electric field $\vec{E}$ is generated in the region between the cylinders. Because the electric field is non-zero, a potential difference will definitely appear between the two cylinders. This makes option (a) a correct statement.

Option (b): Charge density given to the outer cylinder What happens if we only charge the outer cylinder? Our Gaussian surface is located at $r<R_2$. All the charge resides on the outer cylinder, which is outside our Gaussian surface. Therefore, the enclosed charge $q_{\text{enclosed}}$ is exactly zero. By Gauss's Law, the electric field $\vec{E}$ in the region between the cylinders is zero. No electric field means no potential difference. Thus, option (b) is incorrect.

Option (c): Uniform line charge along the axis If a uniform line charge is placed along the central axis, our Gaussian surface will enclose a portion of this line charge. Once again, $q_{\text{enclosed}} eq 0$, which means an electric field $\vec{E}=\frac{\lambda }{2\pi {ϵ}_{0}r}\hat{r}$ exists between the cylinders. This field will create a potential difference. However, option (c) claims that no potential difference appears, making the statement false.

Option (d): Same charge density given to both cylinders If both cylinders are given the same surface charge density $\sigma$, the inner cylinder will hold a total charge proportional to its surface area. When we draw our Gaussian surface between the cylinders, it will enclose the charge of the inner cylinder. The charge on the outer cylinder does not affect the field inside it. Since $q_{\text{enclosed}}$ is non-zero (it equals the charge on the inner cylinder), an electric field exists, and a potential difference appears. Option (d) claims no potential difference appears, which is incorrect.

Through the lens of Gauss's Law, we see that the electric field in the annular region between two coaxial cylinders is dictated solely by the charge enclosed within that region. Placing a charge on the inner cylinder guarantees an enclosed charge, an electric field, and a resulting potential difference.

Therefore, the only correct statement is that a potential difference appears when a charge density is given to the inner cylinder.