Problem 75 - Master Resource Book in JEE Main Physics

Question

The electrostatic potential on the surface of a charged conducting sphere is \(100 \mathrm{~V}\). Two statements are made in this regard. \(S_{1}\) : At any point inside the sphere, electric intensity is zero. \(S_{2}:\) At any point inside the sphere, the electrostatic potential is \(100 \mathrm{~V}\). Which of the following is a correct statements. [NCERT Exemplar] (a) \(S_{1}\) is true but \(S_{2}\) is false. (b) Both \(S_{1}\) and \(S_{2}\) are false (c) \(S_{1}\) is true, \(S_{2}\) is also true and \(S_{1}\) is the cause of \(S_{2}\) (d) \(S_{1}\) is true, \(S_{2}\) is also true but the statements are independent.

Step-by-Step Solution

Answer

Option (c) \(S_1\) is true, \(S_2\) is also true and \(S_1\) is the cause of \(S_2\).

Step 1: Evaluate statement S1

For a charged conducting sphere, the electric field inside the conductor is zero due to the distribution of charges on the surface of the conductor. This means that within the conductor, the electric intensity is zero. Therefore, statement \( S_1 \) is true.

Step 2: Evaluate statement S2

Since there is no electric field inside the charged conducting sphere, the potential remains constant throughout the conductor. The potential at every point inside the conductor must be equal to the potential on the surface of the sphere. Given that the potential on the surface is \(100 \text{ V}\), the potential at any point inside the sphere is also \(100 \text{ V}\). Hence, statement \( S_2 \) is true.

Step 3: Determine the Relationship Between S1 and S2

Since the electric intensity is zero inside the sphere (as per \( S_1 \)), this leads to a constant potential everywhere inside the sphere, validating \( S_2 \). Therefore, \( S_1 \) is the reason \( S_2 \) is true.

Step 4: Select the Correct Option

Both \( S_1 \) and \( S_2 \) are true statements, and \( S_1 \) being true is the cause of \( S_2 \) being true. Therefore, the correct option is (c) \( S_1 \) is true, \( S_2 \) is also true and \( S_1 \) is the cause of \( S_2 \).

Key Concepts

Charged Conducting Sphere

A charged conducting sphere is a fascinating object in electrostatics. Imagine it as a sphere made of a conductive material, like metal, that has been charged with electricity. When a conductive material is charged, the excess charge resides entirely on its surface.
This is because the electrons repel each other, causing them to spread out evenly on the outer surface. This unique property causes some specific effects in terms of electric fields and potentials inside the sphere.
Inside the sphere, there are no electric fields, effectively creating an electrically hollow sphere. This is because the charges on the outer surface cancel out any electrical forces within. That's one of the reasons why it's useful for protecting sensitive equipment from outside electric fields, such as in a Faraday cage.

Electric Field Intensity

Electric field intensity represents the force that a charge experiences in an electric field. In simple terms, it's the push or pull exerted by the electric field on a charged particle.
Inside a charged conducting sphere, the electric field intensity is zero. But why is that?
Because for conductors in electrostatic equilibrium, the electric field inside the material is zero. Charges arrange themselves on the conductor's surface to ensure there’s no net electric field inside.

• This arrangement is crucial in maintaining equilibrium.
• The absence of electric field inside means there's no "push" on any additional charge that might be inside the sphere.
• This forms the foundation for understanding why the potential inside is uniform.

Electrostatic Potential

Electrostatic potential is a measure of the work done in bringing a charge from a point at infinity to a point in an electric field, without any acceleration.
For a charged conducting sphere, understanding potential can be quite intuitive. Knowing that the electric field inside is zero, the potential inside the sphere remains constant. Thus, the potential at any point inside the sphere is equal to the potential on its surface.

• If the surface potential is given as 100 V, so is the potential throughout the interior.
• This constant potential results from the non-existent electric field inside the sphere.
• Think of it like a plateau: it's at the same height (potential) everywhere, thanks to the flat (zero intensity) field.

NCERT Exemplar Problems

NCERT Exemplar Problems are designed to deepen students' understanding of various topics. They often present realistic scenarios that encourage analytical thinking.
In the context of our exercise about a charged conducting sphere, NCERT problems help consolidate core electrostatic concepts.

• They guide you through applying the fundamental principles of electrostatics to solve and understand problems.
• By engaging with these problems, concepts like electric field intensity and electrostatic potential become clear.
• This exemplifies how theoretical knowledge can be applied in practical problems, making learning more engaging and comprehensive.