Assertion–Reason on energy of an induced dipole Assertion (A): For an atom with polarizability α placed in a uniform electric field E, the stored electrostatic energy is 0.5 α E^2 (in magnitude). Reason (R): The capacitance of an isolated conducting sphere of radius R in vacuum is C = 4 π ε0 R farads.

Introduction / Context:
When a neutral atom or molecule is placed in an electric field, an induced dipole moment p = α E forms (for linear, isotropic media). The energy associated with building this induced dipole in the field is important in dielectrics and intermolecular forces. The Reason statement cites a classical capacitance result; we must judge relevance.

Given Data / Assumptions:

• Linear polarizability: p = α E.
• Uniform, static electric field.
• Classical electrostatics; α is scalar for isotropic case.

Concept / Approach:

The work to polarize from 0 to p in a field E is W = ∫0^p E · dp′ = ∫0^E α E′ · dE′ = 0.5 α E^2 (sign convention aside; energy stored is positive). Thus A is correct. The capacitance formula for an isolated sphere, C = 4 π ε0 R, is a true statement from electrostatics but is unrelated to the derivation of induced-dipole energy; it neither explains nor is required for A. Therefore, both statements are true, but R does not explain A.

Step-by-Step Solution:

Verification / Alternative check:

Equivalent derivations appear in molecular physics texts (e.g., in discussions of Clausius–Mossotti relation and London dispersion forces), confirming the 0.5 α E^2 energy scaling.

Why Other Options Are Wrong:

• “R explains A”: incorrect linkage.
• “A true, R false” or other combinations: contradict known electrostatic results.

Common Pitfalls:

Confusing induced dipole energy with interaction of a permanent dipole (−p·E); for induced dipoles p depends on E, yielding the 0.5 factor after integration.

Final Answer:

Both A and R are true but R is not the correct explanation of A