Question

Two metallic blocks M₁ and M₂ of the same cross-section area are connected (as shown in the figure). If the thermal conductivity of M₂ is K then the thermal conductivity of M₁ will be : [Assume steady state heat conduction]

• A. 10K
• B. 8K
• C. 12.5K
• D. 2K

Answer 1

The Correct Option is B

To solve this problem, we need to consider the concept of thermal conductivity and steady state heat conduction. In a composite system, the rate of heat transfer through each segment is the same in the steady state.

The formula for heat transfer through a material slab in steady state is given by:

Q = \frac{K \cdot A \cdot (T_1 - T_2)}{L}

where:

• K = Thermal conductivity
• A = Cross-sectional area
• T_1 and T_2 = Temperature difference across the slab
• L = Length of the slab

Given:

• Temperature across M_1: 100°C to 80°C
• Temperature across M_2: 80°C to 0°C
• Length of M_1: 16 cm
• Length of M_2: 8 cm

In steady state, the heat transfer rates through M_1 and M_2 are equal:

\frac{K_1 \cdot A \cdot (100 - 80)}{16} = \frac{K_2 \cdot A \cdot (80 - 0)}{8}

We know K_2 = K.

Substituting the known values, we have:

\frac{K_1 \cdot A \cdot 20}{16} = \frac{K \cdot A \cdot 80}{8}

Cancel out A on both sides and simplify:

\frac{K_1 \cdot 20}{16} = \frac{K \cdot 80}{8}

K_1 \cdot 20 = K \cdot 80 \cdot 2

K_1 = \frac{160K}{20}

K_1 = 8K

Hence, the thermal conductivity of M_1 is 8K.