Problem 144 - Master Resource Book in JEE Main Physics

Question

A solid spherical conductor of radius \(R\) has a spherical cavity of radius \(a(a

Step-by-Step Solution

Answer

The correct answer is (b) \(-Q, +Q, 0\).

Step 1: Understanding the Problem

We have a solid spherical conductor with a spherical cavity inside it. There is a charge, \(+Q\), positioned at the center of the cavity. We need to determine the charges induced on the inner surface of the cavity and at any point within the conductor where \(a < r < R\).

Step 1: Induced Charge on the Inner Surface

Due to electrostatic induction, when a charge \(+Q\) is placed at the center of the cavity, it will induce a charge on the inner surface of the conductor. By Gauss's law and symmetry, the inner surface of the cavity will have an induced charge of \(-Q\) to cancel the electric field inside the conducting material.

Step 2: Charge at the Center

The charge at the center is simply the charge \(+Q\) that is given in the problem. The induced charge on the inner surface does not affect the charge at the center.

Step 3: Charge at a Position r(a<r<R)

Inside a conductor in electrostatic equilibrium, the electric field is zero. Consequently, no excess charge resides in the bulk of the conductor. Hence, at any point within the solid part of the conductor where \(a < r < R\), the charge is \(0\).

Step 4: Analyzing the Options

Given the results from the previous steps, we are looking for the option where the charge on the inner surface is \(-Q\), the charge at the center is \(+Q\), and the charge at any \(r\) within the conductor is \(0\). This matches option (b) \((-Q, +Q, 0)\).

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism, providing a relationship between electric charges and electric fields. It states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, this is expressed as:\[\Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0}.\]Here, \(\Phi_E\) represents the electric flux, \(\mathbf{E}\) is the electric field, and \(Q_{enc}\) is the charge enclosed within the surface. \(\varepsilon_0\) is the permittivity of free space.

Gauss's Law is particularly useful when dealing with symmetrical charge distributions, as it allows for calculating the electric field without having to use complex integration. In the problem of a spherical conductor with a cavity, Gauss's Law helps determine the distribution of charges by considering a Gaussian surface within the conductor.

• The symmetry of a spherical conductor allows for simplifying the calculation of induced charges on surfaces.
• It explains why charges are distributed on the surface in such a way that the electric field inside the conductor is zero.
• Applying Gauss's Law here helps us to conclude that the induced charge on the inner surface must be \(-Q\) to maintain balance.

Conductors in Electrostatics

In electrostatics, conductors play a significant role due to their unique ability to allow free movement of charges. When a conductor reaches electrostatic equilibrium, the electric field within it is zero. This is because any free charges inside the conductor will rearrange themselves until there is no net force acting on them.

Conductors possess the following key properties in electrostatics:

• Charge Redistribution: When a charge is introduced near a conductor, its free electrons redistribute themselves to counteract the external electric field, ensuring that the internal field is nullified.
• Surface Charges: In equilibrium, any excess charge resides solely on the conductor's surface.
• Potential Uniformity: The entire conductor is at a constant electric potential. This ensures no current flows within it as there's no potential difference.

In the scenario described, the charge \(+Q\) at the cavity center induces a \(-Q\) charge on the cavity's inner surface, allowing the electrostatic potential throughout the conductor to remain constant.

Electric Field Inside Conductors

A fundamental characteristic of conductors in electrostatic situations is that the electric field inside them is zero. This is due to the behavior of charges within conductors, which move freely until they reach an equilibrium where the net electric force on them is zero. As a result, we observe several important consequences:

• Inner Field is Null: With a zero electric field inside, conductors ensure that no electric force acts on any charge within the bulk.
• Induced Surface Charges: The charges induced on the conductor's inner surface counterbalance any external fields, as seen with \(+Q\) inducing \(-Q\) on the cavity surface.
• Field Exclusion from Interior: Since no electric field exists internally, any point within the conductor beyond the cavity, \(a < r < R\), contains no additional charge or field influence, confirming a charge of zero and field nullity.

Understanding the behavior of electric fields inside conductors leads to practical applications where shielding and charge containment are required, serving as the basis for concepts like Faraday cages.