Problem 6 - Master Resource Book in JEE Main Physics

Question

A hollow metallic sphere of radius \(10 \mathrm{~cm}\) is given a charge of \(3.2 \times 10^{-9} \mathrm{C}\). The electric intensity at a point \(4 \mathrm{~cm}\) from the centre is (a) \(9 \times 10^{-9} \mathrm{NC}^{-1}\) (b) \(288 \mathrm{NC}^{-1}\) (c) \(2.88 \mathrm{NC}^{-1}\) (d) zero

Step-by-Step Solution

Answer

The electric field intensity at the point is zero.

Step 1: Understand the Problem

We need to find the electric field intensity at a point inside a hollow sphere that carries a charge. The point is closer to the center than the shell's radius.

Step 2: Determine Inside Surface Field

A hollow metallic sphere exhibits a unique property: inside a conductor, the electric field is zero. This stems from the fact that charges only reside on the outer surface of a conductor.

Step 3: Evaluate the Point Location

The point in question, at 4 cm from the center, is inside the 10 cm radius of the hollow sphere, placing it 6 cm away from the shell. Hence, it is within the conductive material where the internal electric field is zero.

Step 4: Conclude Electric Field Value

Since the electric field inside a conductor is zero, and our point is inside (4 cm < 10 cm), the electric field intensity at that point is zero.

Key Concepts

Hollow Metallic Sphere

A hollow metallic sphere is a fascinating object in the study of electrostatics. Imagine it as a hollow ball made of conductive material, like a metal. Because it is a conductor, an important property is that any excess charge added to it will distribute itself on the outer surface. This is due to each charge repelling the others, ultimately achieving a state where they are distributed evenly on the outer shell.
Consequently, there are no charges present on the inner surface or inside the hollow part of the sphere. This is because the charges want to be as far apart from each other as possible, settling on the outer boundary of the sphere. This unique characteristic leads us to the conclusion that the hollow section inside the sphere has no electric field, which ties directly into our next topic.

Electric Field Intensity

Electric field intensity, simply put, is a measure of the force experienced by a charge placed within an electric field. The intensity is described by the vector quantity, pointing from positive to negative charges, indicating both the direction and the magnitude of the force.
In the case of a hollow sphere, particularly inside it, this intensity becomes an intriguing concept because, as we saw in the previous section, the electric field inside a hollow conductor, like our metallic sphere, is effectively zero.
This happens because:

• The electric charges are solely on the exterior surface of the sphere, leaving the inside devoid of any charges to create an electric field.
• A test charge placed anywhere inside the hollow part will experience no net force because there are no electric forces acting on it.

Understanding this helps us realize that inside a hollow metallic sphere, no electric field intensity exists.

Gauss's Law

Gauss's Law is a powerful tool in electromagnetism. It relates the electric field and the charge distribution through a mathematical framework. The law states that the total electric flux through a closed surface is proportional to the total charge enclosed by that surface. Mathematically, it's expressed as: \[\Phi_E = \oint E \cdot dA = \frac{Q_{enc}}{\varepsilon_0} \]where \(\Phi_E\) is the electric flux, \(E\) is the electric field, \(dA\) is a differential area on the closed surface, \(Q_{enc}\) is the enclosed charge, and \(\varepsilon_0\) is the permittivity of free space.
For our hollow metallic sphere, this law beautifully shows why the electric field inside is zero. Since there's no charge enclosed in a Gaussian surface drawn inside the hollow part, the electric flux through that surface is zero, leading to an electric field value of zero within.

• It is crucial for proving the absence of an electric field inside conductive materials.
• It helps in understanding charge distributions on conductors, ensuring charges only reside on the surface.

By applying Gauss's Law, we confirm the behavior of electric fields within and outside conductive materials, reiterating theoretical predictions.