Question

Given below are two statements : One is labelled as Assertion \(A\) and the other is labelled as Reason R 
Assertion A : Two metallic spheres are charged to the same potential One of them is hollow and another is solid, and both have the same radii Solid sphere will have lower charge than the hollow one. 
Reason R : Capacitance of metallic spheres depend on the radii of spheres.
In the light of the above statements, choose the correct answer from the options given below.

• A. Both \(A\) and \(R\) are true and \(R\)is the correct explanation of \(A\)
• B. Both \(A\) and \(R\) are true but \(R\) is not the correct explanation of \(A\)
• C. A is true but \(R\) is false
• D. A is false but \(R\) is true

Hint:

For questions involving assertions and reasons, analyze each statement indepen dently. Recall the relevant formulas and concepts to determine the truth or false hood of each statement.

Answer 1

The Correct Option is D

To answer this question, let's analyze the given assertion and reason.

Assertion (A): Two metallic spheres are charged to the same potential. One of them is hollow and another is solid, and both have the same radii. The solid sphere will have a lower charge than the hollow one.

Reason (R): Capacitance of metallic spheres depends on the radii of spheres.

First, let's understand the assertion. For spherical conductors, the potential \(V\) is given by:

\(V = \frac{Q}{C}\)

where \(Q\) is the charge and \(C\) is the capacitance. The capacitance of a sphere is expressed as:

\(C = 4\pi \varepsilon_0 R\)

where \(R\) is the radius of the sphere and \(\varepsilon_0\) is the permittivity of free space.

For two spheres with the same radii at the same potential, the capacitance is the same because it depends only on the radius. Since both are charged to the same potential and have the same capacitance, the charges \(Q\) are equal. This contradicts the assertion that the solid sphere will have a lower charge. Thus, the assertion is false.

Now, let's look at the reason. The reason states that the capacitance of metallic spheres depends on their radii. This statement is true because, from the formula for capacitance, it is clear that capacitance indeed depends on the radius of the sphere.

Conclusion: The correct option is "A is false but \(R\) is true" because the assertion incorrectly suggests different charges for spheres at the same potential and radius, while the reason correctly states the radius-dependent nature of capacitance.