First slide

Conduction

Question

Two identical conducting rods AB and CD are connected to a circular conducting ring at two diametrically opposite points B and C. The radius of the ring is equal to the length of rods AB and CD. The area of cross-section, thermal conductivity of the rod and ring are equal. Points A and D are maintained at temperatures of 100⁰C and 0⁰C. Temperature at point C will be

Moderate

A

62⁰C
B

37⁰C
C

28⁰C
D

45⁰C

Solution

$R_{A B} = R_{C D} = R (say)$
Length of semi-circle $= π l$
$∴$ Resistance of semi-circle $= π R (∵ R \propto l)$
   $R_{B C} = \frac{π R}{2} (in parallel)$
Heat current,    $H = \frac{100 - 0}{R + R + \frac{π}{2} \cdot R} = \frac{28}{R} units$
Now,    $θ_{C} - 0^{0} = H R = \frac{28}{R} R$
$∴ θ_{C} = 28^{0} C$

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