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 its ends maintained at temperatures
T1 and
T2(T1>T2) is given by
Q=......(i)
where AT is the thermal conductivity of the material of the rod
Area of cross-section of new rod
A′=π()2==....(ii)
As the volume of the rod remains unchanged
∴AL=A′L′where
L′ is the length the new rod
or
L′=L...........(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
T1 and
T2 is given by
Q′=..........(iv)
Substituting the values of
A′ and
L′ from equations (ii) and (iii) in the above equation, we get
=Q (Using (i))