Consider two rods of equal cross - sectionai area A, one of Aluminium and other of iron joined end to end as shown in figure -4.8. Length of the two rods and their thermal coductivities are `l_(1) , k_(1)`and `l_(2),k_(2)` respectively . If the ends of the rods are maintained at temperature `T_(1)` and `T_(2),(T_(1)gt T_(2))` find the temperature of the junction in steady state .

Options

A. `(K_(1)T_(1) + K_(2)T_(2))/(T_(1) + T_(2))`
B. `(K_(1) T_(1)d_(1) + K_(2) T_(2) d_(2))/(K_(1)d_(2) + K_(2) d_(1))`
C. `(K_(1)T_(1)d_(2) + K_(2)T_(2)d_(1))/(K_(1)d_(2) + K_(2)d_(1))`
D. `(K_(1)T_(10 + K_(2)T_(2)))/(K_(1) + K_(2))`

Answer

C

Explanation

For the first slab, heat current, `H_(1) =(K_(1) T_(1)-T)A)/(d_(1))`
For the second slab, Heat current, `H_(2) = (K_(2)(T-T_(2))A)/(d_(2))`
As the slabs are connected in series same amount of heat current flows through the slabs.
`:. H_(1) = H_(2) or (K_(1) (T_(1) -T)A)/(d_(1)) = (K_(2) (T- T_(2))A)/(d_(2))`
or `K_(1)T_(1)d_(2) - K_(1)Td_(2) = K_(2)T d_(1) - K_(2) T_(2) d_(1)`
or `T(K_(2)d_(1) + K_(1) d_(2)) = K_(1)T_(1) d_(2) + K_(2) T_(2) d_(1)`
or `T = (K_(1) T_(1) d_(2) + K_(2) T_(2) d_(1))/(K_(2)d_(1) + K_(1)d_(2))`