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

When we have a long, hollow conducting cylinder placed inside another larger hollow conducting cylinder, both of which are initially electrically neutral, we can analyze the situation using the principles of electrostatics. Let’s break down what happens when we apply a voltage or charge to one of the cylinders, and how this affects the electric field and charge distribution.

Understanding the Setup

Imagine two cylindrical shells: the inner one has a smaller radius \( r_1 \) and the outer one has a larger radius \( r_2 \). Both are made of conductive material, which means they can redistribute their charges freely. Initially, both cylinders have no net charge, meaning the total positive charge equals the total negative charge.

Charging the Inner Cylinder

Now, let’s say we apply a positive charge \( +Q \) to the inner cylinder. Here’s what happens:

• Charge Redistribution: The positive charge on the inner cylinder will induce a negative charge on the inner surface of the outer cylinder. This occurs because the electric field created by the positive charge attracts electrons from the outer cylinder, causing them to accumulate on the inner surface.
• Outer Surface Charge: To maintain overall neutrality, the outer surface of the outer cylinder will acquire a positive charge of \( +Q \) (equal in magnitude but opposite in sign to the induced charge on the inner surface).

Electric Field Analysis

Next, let’s consider the electric fields in different regions:

• Inside the Inner Cylinder: The electric field inside a conductor in electrostatic equilibrium is zero. Therefore, inside the hollow part of the inner cylinder, the electric field \( E = 0 \).
• Between the Cylinders: In the region between the inner and outer cylinders (from \( r_1 \) to \( r_2 \)), the electric field can be calculated using Gauss's Law. For a cylindrical Gaussian surface of radius \( r \) (where \( r_1 < r < r_2 \)), the electric field \( E \) is given by:
• \( E \cdot (2\pi r L) = \frac{Q}{\epsilon_0} \)
• Thus, \( E = \frac{Q}{2\pi \epsilon_0 r L} \), where \( L \) is the length of the cylinders and \( \epsilon_0 \) is the permittivity of free space.
• Outside the Outer Cylinder: Beyond the outer cylinder (for \( r > r_2 \)), the electric field behaves as if all the charge were concentrated at the center. Therefore, the electric field \( E \) is given by:
• \( E = \frac{Q}{2\pi \epsilon_0 r^2} \)

Key Takeaways

This scenario illustrates several important concepts in electrostatics:

• The ability of conductors to redistribute charge in response to an external electric field.
• The concept of induced charge, where the presence of a charge on one conductor affects the charge distribution on another conductor.
• The application of Gauss's Law to determine electric fields in symmetric charge distributions.

In summary, when a charge is applied to the inner conducting cylinder, it induces a charge on the outer cylinder, creating distinct electric fields in different regions. This example beautifully demonstrates the principles of electrostatics and the behavior of conductors in electric fields.