Problem 79 - Master Resource Book in JEE Main Physics

Question

A hollow conducting sphere of radius, \(R\) has a charge \((+Q)\) on its surface. What is the electric potential within the sphere at a distance, \(r=R / 3\) from its centre? (a) \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r}\) (b) \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{2}}\) (c) \(\frac{1}{4 \pi E_{0}} \cdot \frac{Q}{R}\) (d) Zero

Step-by-Step Solution

Answer

The electric potential at \\(r = \frac{R}{3}\\) is \\( \frac{1}{4 \pi E_{0}} \cdot \frac{Q}{R} \\) (option c).

Step 1: Understanding the Sphere Properties

A hollow conducting sphere has the unique property that the electric potential inside the cavity, and on the surface of the sphere, is uniform and equal to the surface potential. Since there is no charge inside the cavity, the potential at any point in the cavity is constant and is determined by the charge on the outer surface of the sphere.

Step 2: Electric Potential Formula for a Conducting Sphere

The electric potential on the surface of a conducting sphere with charge \(Q\) and radius \(R\) is given by \( \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{R} \). This potential is the same throughout the interior of the hollow sphere.

Step 3: Apply Conditions Inside the Sphere

Inside a conducting sphere, the electric potential does not change with distance from the center. Therefore, even at \(r = \frac{R}{3}\), the potential remains the same as on the surface: \( \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{R} \).

Key Concepts

Hollow Conducting Sphere

A hollow conducting sphere is a fascinating object in the study of electricity because of its unique properties. When you have a sphere that conducts electricity, all the charge resides on its outer surface. This happens because like charges repel each other, and they try to get as far away from each other as possible, which means spreading out over the surface.

Inside this hollow sphere, the electric field is zero. Why? Because any electric field inside would push charges around, but since the charges have already settled on the surface, there's no movement, leading to a zero field. This is a specific behavior of conductors and is one of the concepts of electrostatics.

• No charge resides inside the hollow part of the sphere.
• The electric field inside the sphere's cavity is zero.
• The potential everywhere inside is constant and equal to the potential on the surface.

Electric Potential Inside a Conductor

Understanding the electric potential inside a conductor reveals an interesting principle: inside a hollow conducting sphere, the potential is constant. This means every point inside, no matter how close or far from the surface, shares the same potential value. This uniformity arises because the electric field, as we discussed, is zero inside the sphere. Given that electric potential changes occur due to differences in the electric field, a zero field means zero change.
To determine the electric potential inside a conducting sphere, you examine the potential on its surface. For a sphere with charge \(Q\) and radius \(R\), the surface potential is given by the formula \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{R}\). Since there’s no gradient (change) in the potential from the surface going inside, this same value applies throughout the sphere’s interior.

• Potential inside is equivalent to the potential on the surface.
• No change in potential indicates no electric field present.
• Uniform potential provides a safe environment from electric forces.

Charge Distribution on Conductors

Conductors have a unique way of managing electric charge. When charge is added to a conductor, it spreads out evenly across the surface. This distribution occurs because similar charges repel, and they continuously move until they are evenly dispersed on the outer surface of the conductor.
This means, for a hollow conducting sphere, the entire charge \(Q\) is found on its outer surface, not inside the hollow cavity. This behavior leads to some insightful properties:

• All excess charge resides only on the surface.
• There is no electric field inside the conductor due to this distribution.
• The surface charge distribution ensures the electric potential inside remains constant.

Furthermore, this distribution affects how the potential is calculated. The charges on the surface create a uniform potential inside the sphere, as discussed earlier, providing interesting applications such as in Faraday cages, where this principle helps shield delicate equipment from outside electric fields.