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A metal rod AB of uniform cross sectional area is conducting heat from end A to end B and temperature difference between ends A and B is $100^{o} C$ . Mid point of the rod is C. Coefficient of thermal conductivity of the rod linearly increases from $K_{C}$ at end A to 2 $K_{C}$ at end B. Then temperature difference between A and C is

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equal to $50^{o} C$

less than $50^{o} C$

more than $50^{o} C$

may be less than $50^{o} C$

answer is B.

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Detailed Solution

Since coefficient of thermal conductivity of the rod gradually increases from end A to end B and thermal resistance $R = \frac{l}{K A}$ , it is evident that thermal resistance of the part AC is less than that of the part CB is $R_{A C} < R_{C B}$ .

Now rate of heat conduction $Q = \frac{Δ θ_{A C}}{R_{A C}} = \frac{Δ θ_{A B}}{R_{A C} + R_{B C}}$

$∴ Δ θ_{A C} = \frac{Δ θ_{A B}}{1 + \frac{R_{B C}}{R_{A C}}} < \frac{Δ θ_{A B}}{2}$

$∴ Δ θ_{A C} < 50^{o} C$

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