Question

A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger radius. Both the cylinders are initially electrically neutral. (A) A potential difference appears between the two cylinders when a charge density is given to the inner cylinder (B) A potential difference appears between the two cylinders when a charge density is given to the outer cylinder (C) No potential difference appears between the two cylinders when a uniform line charge is kept along the axis of the cylinders (D) No potential difference appears between the two cylinders when same charge density is given to both the cylinders

Short answer

(A) and (B) are true, because giving charge density to either the inner or the outer cylinder will create an electric field between them, leading to a potential difference. (C) is true because the line charge's effect is shielded by the conducting material. (D) is false because even with the same charge density, the cylinders' varied radii lead to a potential difference.

Step-by-step solution

Conceptual Analysis - Charging the Inner Cylinder

When a charge density is given to the inner cylinder, due to the charge distribution, an electric field will be established between the inner and outer cylinders. By definition, a potential difference occurs whenever there is an electric field in the region.

Implication for the Outer Cylinder

Similarly, if a charge density is provided to the outer cylinder while the inner remains neutral, an electric field develops, again causing a potential difference between the two cylinders.

Evaluating the Effect of a Line Charge

A uniform line charge along the axis affects both cylinders equally, assuming the cylinders are perfectly conducting and the line charge is equidistant from both. No potential difference arises because the effect of the line charge will be shielded and no net electric field will exist in the conductor material itself.

Assessing Equal Charge Density

If both cylinders are given the same charge density, they will still have a potential difference between them, as the amount of charge (and therefore the potential) will differ due to their different radii, which affects the capacitance and stored charge.

Key concepts

Electric Charge Distribution

Understanding the electric charge distribution is crucial when studying conductors and the electric fields they influence. In our exercise, a charge density given to the hollow conducting cylinder results in charges distributing uniformly on its surface. This is because conductors allow free movement of charges, which repel each other due to being of like sign, thereby spreading out as much as possible.

When the inner cylinder receives a charge density, it creates an electric field directed outward toward the outer cylinder. This field exists in the space between the two cylinders, demonstrating how charge distribution affects the electric field in the environment. Conversely, if the outer cylinder is charged instead, the charges redistribute themselves over its outer surface, producing an electric field directed inward towards the inner cylinder.

Potential Difference

Potential difference, also known as voltage, is a measure of work done to move a charge from one point to another. In conductive systems like our cylinders, the potential difference appears when there's an established electric field between them.

If only the inner cylinder is charged, the potential difference arises due to the electric field that stretches from the inner to the outer cylinder. Similarly, charging the outer cylinder creates a potential difference, this time with the direction of the field inversion. The potential difference, in either case, can be thought of as the energy landscape that charges would experience moving between the two cylinders.

Electric Field in Conductors

The behavior of electric fields within conductors is a fascinating topic. Inside a conductor at electrostatic equilibrium, the electric field is zero because excess charges reside on the surface. In the scenario where a line charge is placed along the axis between two cylinders, the charges in the conducting materials rearrange themselves to negate the external electric field within the conductive material itself.

This shielding effect leads to no potential difference being established between the two cylinders for a uniform line charge scenario. The charges perfectly balance the external field, a principle often referred to as electrostatic shielding, and is why sensitive electronic components are sometimes encased in conductive materials.

Line Charge Effect

When addressing the line charge effect, we're considering how a linear distribution of charge influences the surrounding area. A uniform line charge creates an electric field radiating outwards from the line itself. However, in the unique case of our coaxial conductive cylinders, the cylinders act as barriers that respond to the line charge by rearranging their free electrons.

This rearrangement neutralizes the electric field within the conductor, again presenting the concept of shielding. Consequently, inside the conductor and between the two cylinders, the line charge's effect is essentially nullified. The result is that no potential difference arises between the cylinders, illustrating how geometry and distribution can dictate the influence a line charge has on its environment.