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

Step 1: Analyze Assertion A

The potential of a conducting sphere (solid or hollow) is given by: \[ V = \frac{KQ}{R} \] where \(V\) is the potential, \(K\) is a constant, \(Q\) is the charge, and \(R\) is the radius.

If two spheres have the same potential (\(V_1 = V_2\)) and the same radius (\(R_1 = R_2\)), then: \[ \frac{KQ_1}{R_1} = \frac{KQ_2}{R_2} \] Since \(R_1 = R_2\), it follows that \(Q_1 = Q_2\). Therefore, the assertion that the solid sphere will have a lower charge is false. The charges must be equal.

Step 2: Analyze Reason R

The capacitance of a spherical conductor is given by: \[ C = 4\pi \epsilon_0 R \] where \(C\) is the capacitance, \(\epsilon_0\) is the permittivity of free space, and \(R\) is the radius. This shows that the capacitance of a metallic sphere depends only on its radius. Therefore, the reason is true.

Conclusion: Assertion A is false, but Reason R is true. Therefore, the correct option is (4).