Three rods of material ‘x’ and three rods of material y are connected as shown in figure. All the rods are of identical length and cross-section.

Q: Three rods of material ‘x’ and three rods of material y are connected as shown in figure. All the rods are of identical length and cross-section. If the end A is maintained at 60°C and the junction E at 10°C, find effective Thermal Resistance. Between A and E, Given length of each rod = l, area of cross-section = A, conductivity of x = K and conductivity of y = 2K.

(a) $\displaystyle \frac{4l}{3 K A}$

(b) $\displaystyle \frac{7l}{6 K A}$

(c) $\displaystyle \frac{4 K A}{3 l}$

(d) $\displaystyle \frac{7 K A}{3 l }$

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Ans: (b)
Sol: $\displaystyle Thermal \, Resistance \propto \frac{1}{Thermal \, conductivity}$

Thermal Resistance of BC is RBC = RCE = 2R

And , Thermal Resistance of BC is RBD = RDE = R (say)

$latex \displaystyle R = \frac{l}{2KA}$

As , Bridge is balanced ,  hence no heat current will flow in the rod CD

RBCE = 2R +2R = 4R

RBDE = R+R =2R

Now , RBCE & RBDE are in parallel

$\displaystyle R’ = \frac{4R\times 2R}{4R+2R} = \frac{4R}{3}$

AS , RAB & R’ are in series

Req = RAB + R’= 4R/3 + R = 7R/3