A thin copper rod of uniform cross section A square metres and of length L metres has a spherical metal sphere of radius r metre at tis one end symmetrically attached to the copper rod. The thermal conductivity of copper is K and the emissivity of the spherical surface of the sphere is `epsi`.The free end of the copper rod is maintained at the temperature T kelving by supplying thermal energy from a P watt source. Steady state conditions are allowed ot be established while the rod is properly insulated aginst heat loss from its lateral surface. Surroundings are at `0^@C` Stefan's constant`=sigma W//m^(2) K^(4)`.
After the steady state conditions are reached,the temperature of the spherical end of the rod, `T_S` is

Options

A. `T_S=T-(PL)/(KA)`
B. `T_S=0^@C`
C. `T_S=(PL)/(KA)`
D. `T_S=T-(P(L+r))/(KA)`

Answer

D

Explanation

The distance from the end A of the copper rod (where power is supplied) and centre O of the metal sphere is `(L+r)`. Hence , in the steady state condions,
`P=(KA(T-T_S))/(L+r)`
Which gives `T_S=T-(P(L+r))/(KA)`
(The heat is trainsmitted up to centre of the spherical end and the sphere loses energy by radiation out of its spherical surface.)