The temperature of a metallic sphere of radius R is increased by a small amount Δ T. If the linear coefficient of thermal expansion of the metal is α, the approximate increase in the volume of the sphere is:
- A.2π R³αΔ T
- B.3R³αΔ T
- C.4π R³αΔ T✓
- D.6R³αΔ T
Correct Answer
(C) 4π R³αΔ T
Solution & Explanation
For a solid, the coefficient of volume (cubical) expansion γ relates to the linear coefficient α by: γ = 3α. Fractional volume change: ΔV/V = γ·ΔT = 3α·ΔT. Volume of a sphere: V = (4/3)π R³. So ΔV = V·(3α·ΔT) = (4/3)π R³ × 3α·ΔT. The factor 3 cancels the 1/3: ΔV = 4π R³ α ΔT. This matches option C, 4π R³ α ΔT. Trap: use γ = 3α (not just α) — forgetting the factor 3 would wrongly give (4/3)π R³ α ΔT and miss the answer.
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