Question

Seven identical rods of material of thermal conductivity k are connected as shown in Fig. All the rods are of identical length l and cross sectional area A If the one end B is kept at `100^@C` and the other end is kept at `0^@C` what would be the temperatures of the junctions C, D and `E(theta_C`,`theta)D` and `theta_E`) in the steady state?
A. `theta_Cgttheta_Egttheta_D`
B. `theta_E=50^@C` and `theta_D=37.5^@C`
C. `theta_E=50^@C`,`theta_C=62.5^@C` and `theta_D=37.5^@C`
D. `theta_E=50^@C`,`theta_C=60^@C` and `theta_D=40^@C`

Answer 1

Correct Answer - A::C

This problem can be solved like electric current problem.
`R_1`,`R_2`,`R_3`,`R_4`,`R_5`,`R_6` and `R_7` be the rates of heat flow through AE,EB,AC,CD,CE,ED and DB, respectively.
Since `R_1=R_2` `theta_E=50^@C` .(i)
`R_5=R_6` `R_3=R_4+R_5=R_7` .(ii)
`R_4+R_6=R_7`
`(KA(theta_C-50))/(l)=(kA)/(l)(theta_C-50)+(kA)/(l)(theta_C-theta_D)=(kA)/(l)theta_D`
`theta_C+theta_D=100`
`2theta_C-2theta_D=50impliestheta_C=62.5^@C`
`theta_D=37.5^@C`
`theta_Cgttheta_Egttheta_D`