b
The amount of heat flows in time \(t\) through a cylindrical metallic rod of length \(L\) and uniform area of \(cross-section\,A(=\pi\,R^2)\) with its ends maintained at temperatures \(T_1\) and \(T_2\) \((T_1>T_2)\) is given by

\(Q = \frac{{KA\left( {{T_1} - {T_2}} \right)t}}{L}\)                          \(,,,(i)\)

Where \(K\) is the thermal conductivity of the material of the rod.

Area of \(cross-section\) of new rod

\(A' = \pi {\left( {\frac{R}{2}} \right)^2} = \frac{{\pi {R^2}}}{4} = \frac{A}{4}\)                 \(...(ii)\)

As the volume of the rod remains unchanged

\(\therefore AL = A'L'\)

Where \(L'\) is the length the new rod

\(or\,\,\,\,L' = L\frac{A}{{A'}}\)                           \(,,,(iii)\)

\( = 4L\)                                                     \((Using (ii))\)

Now, the amount of heat flows in same time \(t\) in the new rod with its ends maintained at the same temperatures \(T_1\) and \(T_2\) is given by

\(Q' = \frac{{KA'\left( {{T_1} - {T_2}} \right)t}}{{L'}}\)                   \(...(iv)\)

Substituting the values of \(A'\) and \(L'\) from equations \((ii)\) and \((iii)\) in the above equation, we get

\(Q' = \frac{{K\left( {A/4} \right)\left( {{T_1} - {T_2}} \right)t}}{4L}\)

\( = \frac{1}{{16}}\frac{{KA\left( {{T_1} - {T_2}} \right)t}}{L} = \frac{1}{{16}}Q\)