Three rods each of length l and cross sectional area A joined in series between two heat reservoirs as shown in the figure. Their conductivities are 2K, K and `(K)/(2)`, respectively. Assuming that the conductors are insulated from surroundings, the temperatures `T_(1)` and `T_(2)` of the junctions in steadly state condition are respectively.

Options

A. `(600)/(7).^(@)C, (400)/(7).^(@)C`
B. `(600)/(7).^(@)C, (700)/(4).^(@)C`
C. `(500)/(6).^(@)C, (600)/(5).^(@)C`
D. `(600)/(4).^(@)C, (400)/(7).^(@)C`

Answer

A

Explanation

In a series combination of heat conductors the rate of heat flow remains constant.
According to the question, the figure given below shows the conduction of heat through series combinations of rods.

where, `T_(1), T_(2)` = temperature of junctions,
I = heat current `= (d Q)/(d t)`
As, `(d Q_(1))/(dt) = (d Q_(2))/(dt) = (d Q_(3))/(dt) = (T_(i) - T_(f))/(k_(eq))`
So, `(k_(1)A_(1)(T_(i) - T_(1)))/(l_(1)) = (k_(2)A_(2)(T_(1) - T_(2)))/(l_(2)) = (k_(3)A_(3)(T_(2)-T_(f)))/(l_(3))`
`because A_(1) = A_(2) = A_(3)`
and `l_(1) = l_(2) = l_(3)`
`rArr 2k(100 - T_(1)) = k(T_(1) - T_(2)) = 0.5 (T_(2) - 0)` ...(i)
Equivalent coefficient of thermal conductivity,
`(1)/(k_(eq)) = (1)/(2k) + (1)/(k) + (2)/(k)`
`rArr k_(eq) = (2k)/(7)`
`because` Heat current, `(d Q)/(d t) = (d Q_(1))/(d t)`
`rArr (k_(eq) A(T_(i) - T_(f)))/(l) = (k_(1)A_(1)(T_(i) - t_(t)))/(l_(1))`
here, `l = l_(1) + l_(2) + l_(3) = l_(1) + l_(1) + l_(1) = 3l_(1)` and `A = A_(1) + A_(2) + A_(3) = A_(1) + A_(1) + A_(1) = 3A_(1)`
So, `(2k)/(7) xx (3A_(1))/(3l_(1)) (100 - 0) = (2k A_(1)(100 - T_(1)))/(l_(1))`
or `(1)/(7) xx 100 = 100 - T_(1)`
`rArr T_(1) = 100 - (100)/(7)`
`rArr T_(1) = (600^(@)C)/(7)`
Similarly,
From Eq. (i) we get
`2k(100 - T_(1)) = 0.5 k T_(2)`
`rArr 200 - 2 xx (600)/(7) = 0.5 T_(2)`
`rArr T_(2) = (400)/(7).^(@)C`
Hence, the correct option is (a).