The problem involves two rods, A and B, made of different materials, welded together. The thermal conductivities of the two rods are given as K₁ and K₂. We need to find the thermal conductivity of the composite rod.
To solve this, we use the concept of thermal resistance in series. When two materials are joined end-to-end, like these rods A and B, the total thermal resistance (R_total) is the sum of the individual resistances of each material.
Thermal resistance (R) is inversely proportional to thermal conductivity (K). The relationship can be expressed as:
R = L / (K * A)
where:
L is the length of the rod
K is the thermal conductivity
A is the cross-sectional area of the rod
For the two rods welded together:
For rod A: R₁ = L₁ / (K₁ * A)
For rod B: R₂ = L₂ / (K₂ * A)
The total resistance (R_total) is the sum of the individual resistances:
R_total = R₁ + R₂ = L₁ / (K₁ * A) + L₂ / (K₂ * A)
Now, the total thermal conductivity (K_total) is the reciprocal of the total resistance. To simplify:
K_total = 1 / R_total
Substituting the formula for R_total:
K_total = 1 / (L₁ / (K₁ * A) + L₂ / (K₂ * A))
Simplifying further:
K_total = (K₁ * K₂ * A) / (L₁ * K₂ + L₂ * K₁)
So, the thermal conductivity of the composite rod depends on the lengths and thermal conductivities of the individual rods.
However, without specific lengths (L₁ and L₂) provided, we cannot compute the exact value of K_total. Based on the multiple-choice options, the correct general expression for the thermal conductivity of the composite rod is typically found by considering an average of the two thermal conductivities, weighted by the length of each rod.
Given that the question does not provide specific lengths, the most likely answer is (a) (K₁ + K₂) / 2, which represents an average.
Answer: (a) (K₁ + K₂) / 2
