Question

Two spheres of different materials, one with double the radius and one-fourth wall thickness of the other, are filled with ice. If the time taken for complete melting of ice in the larger sphere is 25 minutes and for the smaller one is 16 minutes, the ratio of thermal conductivities of the materials of larger spheres to that of smaller sphere is:

• A. \(4:5\)
• B. \(5:4\)
• C. \(25:8\)
• D. 8:25

Hint:

For heat conduction: t ∝ fracthicknessk × area Larger area and higher conductivity reduce melting time.

Answer 1

The Correct Option is A

Step 1: Rate of heat flow through a spherical shell: (dQ)/(dt) ∝ (kA)/(x)
Step 2: Time for melting t ∝ (x)/(kA) Given: R₁ = 2R₂, x₁ = (x₂)/(4)
Step 3: Ratio of times: (t₁)/(t₂) = (x₁ k₂ A₂)/(x₂ k₁ A₁)
Step 4: Using A ∝ R²: (25)/(16) = ((x₂)/(4) · k₂ · R₂²)/(x₂ · k₁ · (2R₂)²) = (k₂)/(16k₁)
Step 5: (k₁)/(k₂) = (4)/(5)