Transmission of Heat | JEE Main previous year questions with Solutions

Q 1 :

Three conductors of same length having thermal conductivity $k_{1}, k_{2} and k_{3}$ are connected as shown in figure.

Area of cross sections of 1st and 2nd conductor are same and for 3rd conductor it is double of the 1st conductor. The temperatures are given in the figure. In steady state condition, the value of $θ$ is _______ °C.

(Given : $k_{1} = 60 {Js}^{- 1} m^{- 1} K^{— 1}$ , $k_{2} = 120 {Js}^{- 1} m^{- 1} K^{— 1}$ , $k_{3} = 135 {Js}^{- 1} m^{- 1} K^{— 1}$ ) [2025]

0%

(40)

$R_{1} = \frac{2 l}{k_{1} A}, R_{2} = \frac{2 l}{k_{2} A}, R_{3} = \frac{l}{k_{3} A}$

$(100 - θ) (\frac{1}{R_{1}} + \frac{1}{R_{2}}) = \frac{(θ - 0)}{R_{3}}$

$(100 - θ) \frac{(k_{1} + k_{2}) A}{2 l} = \frac{θ k_{3} A}{l}$

$(100 - θ) (k_{1} + k_{2}) = 2 θ k_{3}$

$θ = \frac{100 (k_{1} + k_{2})}{k_{1} + k_{2} + 2 k_{3}} = \frac{100 \times 180}{450} = 40$

Q 2 :

Two cylindrical rods A and B made of different materials, are joined in a straight line. The ratio of lengths, radii and thermal conductivities of these rods are: $\frac{L_{A}}{L_{B}} = \frac{1}{2}, \frac{r_{A}}{r_{B}} = 2$ and $\frac{K_{A}}{K_{B}} = \frac{1}{2}$ . The free ends of rods A and B are maintained at 400 K, 200 K, respectively. The temperature of rods interface is ________ K, when equilibrium is established. [2025]

0%

(360)

$R_{1} = \frac{l_{1}}{K_{1} A_{1}}, R_{2} = \frac{l_{2}}{K_{2} A_{2}}$

$\frac{d Q}{d t} = \frac{△ T}{R}$

${(\frac{d Q}{d t})}_{1} = {(\frac{d Q}{d t})}_{2}$

$\frac{400 - T}{R_{1}} = \frac{T - 200}{R_{2}}$

$\frac{400 - T}{T - 200} = \frac{R_{1}}{R_{2}} = (\frac{l_{1}}{l_{2}}) {(\frac{r_{2}}{r_{1}})}^{2} \times \frac{K_{2}}{K_{1}}$

$\frac{400 - T}{T - 200} = \frac{1}{2} \times {(\frac{1}{2})}^{2} \times 2 = (\frac{1}{4})$

1600 – 4T = T –200

5T = 1800 $\Rightarrow$ T = 360 K

Q 3 :

Two plates A and B have thermal conductivities ${84 W m}^{- 1} K^{- 1}$ and ${126 W m}^{- 1} K^{- 1}$ respectively. They have same surface area and same thickness. They are placed in contact along their surfaces. If the temperatures of the outer surfaces of A and B are kept at $100 ° C$ and $0 ° C$ respectively, then the temperature of the surface of contact in steady state is ________ ${}^{\circ}C$ . [2023]

0%

(40)

Let the temperature of the contact surface is T, then

$H_{A} = H_{B}$

$\frac{K_{A} A (T_{A} - T)}{L} = \frac{K_{B} A (T - T_{B})}{L}$

$84 (100 - T) = 126 (T - 0)$

$2 (100 - T) = 3 T$

$\Rightarrow T = 40 ° C$

Q 4 :

Rods $x$ and $y$ of equal dimensions but of different materials are joined as shown in the figure. Temperatures of end points $A$ and $F$ are maintained at $100 ° C$ and $40 ° C$ respectively. Given the thermal conductivity of rod $x$ is three times of that of rod $y$ , the temperature at junction points $B$ and $E$ are (close to): [2026]

89 °C and 73 °C respectively
80 °C and 60 °C respectively
80 °C and 70 °C respectively
60 °C and 45 °C respectively

1

2

3

4

(1)

$Let [R = \frac{ℓ}{3 K A}]$

$H = [\frac{100 - 40}{\frac{11 R}{2}}] . . . (1)$

$H = \frac{100 - T_{B}}{R} . . . (2)$

$H = \frac{T_{E} - 40}{3 R} . . . (3)$

Using (1) and (2)

$120 = 1100 - 11 T_{A}$

$T_{B} = 89 ° C$

Using (1) and (3)

$T_{E} = 73 ° C$

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