A cylindrical metalic rod in thermal contact with two reservoirs of heat at its ends conducts an amo

Question

A cylindrical metalic rod in thermal contact with two reservoirs of heat at its ends conducts an amount of heat Q, in time t. The metallic rod is melted and the material is formed into rod of half the radius of the origional rod. What is the amount of the heatconducted by the new rod, when placed in thermal contact with the two reservoirs in time t ?

Accepted Answer

The amount of heat flows in time t through a cylindrical metallic rod of length L and uniform area of cross-section A(= πR²) with ends maintained at temperatures T₁ and T₂ (T₁ > T₂) is given by
Q =

.....(i)
where K is the thermal conductivity of the material of the rod.
Area of cross-section of new rod A′ = π

.....(ii)
As the volume of the rod remains unchanged
∴ AL = A′L′
where is the length the new rod
or
L′ = L

= 4L ......(iii)
Now, the amount of heat flows in same time t in the new rod with its ends maintained att the same temperature T₁ and T₂ is given by
Q′ =

......(vi)
Substituting the values of A′ and L′ from equations (ii) and (iii) in the above equation, we get
Q′ =

(using (i))

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