Problem 45 - Arihant AIEEE Physics

Question

If a conductor encloses a charge, then in equilibrium: (a) its inner surface will have an opposite charge equal in magnitude to the charge enclosed (b) its inner surface has no charge (c) its inner surface will have same nature charge equal in the magnitude to the charge enclosed (d) its inner surface will have opposite nature but not equal in the magnitude of the charge enclosed

Step-by-Step Solution

Answer

The answer is (a): inner surface will have an opposite charge equal in magnitude to the charge enclosed.

Step 1: Understanding the Problem

The problem revolves around a conductor enclosing a charge and determining what happens to the charge distribution on the conductor's inner surface when the system is in electrostatic equilibrium.

Step 2: Understanding Charge Distribution in Conductors

In electrostatic equilibrium, the electric field inside a conductor is zero. To ensure this, any excess charge resides on the surface, and the inner surface will adjust to counteract any enclosed charge.

Step 3: Applying Gauss's Law

Using Gauss's Law, we know that for any closed surface, the electric flux is proportional to the enclosed charge. Within a conductor, the charge enclosed within the inner surface must influence the charge distribution such that the interior electric field remains zero.

Step 4: Determining Charge on Inner Surface

Given a charge enclosed by the conductor, the inner surface will develop a charge equal in magnitude but opposite in sign to neutralize the effect of the enclosed charge. This counters any electric field due to the enclosed charge, satisfying electrostatic equilibrium conditions.

Step 5: Selection of Correct Option

The correct description of the charge on the inner surface, given these conditions, is that it will be of opposite charge, equal in magnitude to the charge enclosed. Thus, option (a) is correct.

Key Concepts

Charge Distribution

In a conductor, charge distribution is crucial to maintaining electrostatic equilibrium. When charges are introduced to a conductor, they will move until they reach a stable configuration where the electric field inside is zero. This movement occurs because like charges repel each other, and excess charges will settle on the conductor's surface.
For the inside surface of a hollow conductor, if there is a charge enclosed, the charge distribution will adjust to maintain an equilibrium. The inner surface will develop a charge opposite in sign and equal in magnitude to the enclosed charge.
This ensures that the electric field inside the conductor is completely balanced, preventing any further movement of charges.

Gauss's Law

Gauss's Law is an essential tool in understanding electric fields and charge distributions in conductors. It states that the electric flux through a closed surface is directly proportional to the enclosed electric charge. Mathematically, it is expressed as: \[ \Phi = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\varepsilon_0} \] where:

• \( \Phi \) is the electric flux,
• \( \vec{E} \) is the electric field,
• \( d\vec{A} \) is a differential area on the closed surface,
• \( Q_{\text{enclosed}} \) is the total enclosed charge,
• \( \varepsilon_0 \) is the permittivity of free space.

When applied to a conductor, Gauss's Law helps illustrate why the electric field inside must be zero: if it were not, there would be a non-zero flux indicative of unbalanced charges or induced force. Thus, any charge enclosed by a conductor is countered by the inner surface charge, nullifying electric fields inside.

Conductor Properties

Conductors have specific properties that define their behavior in electrostatic conditions.

• **Zero Electric Field:** One key property is that the electric field within a conductor in electrostatic equilibrium is zero. This happens because free charges within the conductor move until they have no net force acting on them.
• **Surface Charge Distribution:** All excess charge in a conductor redistributes itself to the surface. This charge distribution ensures no interior electric field, which aligns with the principle that the field is zero inside a conductor during electrostatic equilibrium.
• **Equilibrium with Enclosed Charges:** If the conductor encloses a charge, the inner surface will develop charges that oppose and equal the enclosed charge in magnitude but are opposite in nature. This ensures that disruptions to the internal electric field are neutralized, maintaining equilibrium.

Conductors work efficiently as shields to external electric fields due to these properties.

Electric Field

The electric field is a fundamental concept for understanding electrostatic interactions. An electric field is a vector field around a charged object that represents the force exerted on other charges in its vicinity. In the context of conductors:

• **Zero Inside Conductors:** The electric field inside a conductor in electrostatic equilibrium is zero. Free charges move within the conductor until they are uniformly distributed over the surface with no net electric field inside.
• **Equilibrium Maintenance:** The electric field at any point on the conductor's surface is perpendicular to the surface. This configuration is essential for electrostatic equilibrium. If it were not perpendicular, charges would continue to move and not be in equilibrium.
• **Magnified Outside Effects:** While the interior electric field is nullified, the surface charge of conductors can affect the electric fields external to the conductor, influencing surrounding charges.

Understanding the behavior of electric fields around conductors is vital for applications in shielding and electrostatic protection.