Three rods of same dimensions are arranged as shown in figure they have thermal conductivities K 1, K 2 and K 3. The points P and Q are maintained at different temeratures for the heat to flow at the same rate along P R Q and P Q then which of the following option is correct?

Question

Three rods of same dimensions are arranged as shown in figure they have thermal conductivities K 1, K 2 and K 3. The points P and Q are maintained at different temeratures for the heat to flow at the same rate along P R Q and P Q then which of the following option is correct? (A) K3=(1/2)(K1+K2) (B) K3=K1+K2 (C) K3=(K1 K2/K1+K2) (D) K3=-2(K1+K2)

Tardigrade Admin · Accepted Answer

The given arrangement of rods can be redrawn as follows

It is given that

H_{1} = H_{2}

\Rightarrow \frac{K A ( θ _{1} - θ _{2} )}{2 l} = \frac{K _{3} A ( θ _{1} - θ _{2} )}{l}

\Rightarrow K_{3} = \frac{K}{2} = \frac{K _{1} K _{2}}{K _{1} + K _{2}}

Q. Three rods of same dimensions are arranged as shown in figure they have thermal conductivities $K_{1}, K_{2}$ and $K_{3}$ . The points $P$ and $Q$ are maintained at different temeratures for the heat to flow at the same rate along $PRQ$ and $PQ$ then which of the following option is correct?

$K_{3} = \frac{1}{2} (K_{1} + K_{2})$

$K_{3} = K_{1} + K_{2}$

$K_{3} = \frac{K _{1} K _{2}}{K _{1} + K _{2}}$

$K_{3} = - 2 (K_{1} + K_{2})$

The given arrangement of rods can be redrawn as follows

It is given that $H_{1} = H_{2}$
$\Rightarrow \frac{K A ( θ _{1} - θ _{2} )}{2 l} = \frac{K _{3} A ( θ _{1} - θ _{2} )}{l}$
$\Rightarrow K_{3} = \frac{K}{2} = \frac{K _{1} K _{2}}{K _{1} + K _{2}}$

Q. Three rods of same dimensions are arranged as shown in figure they have thermal conductivities K1​,K2​ and K3​. The points P and Q are maintained at different temeratures for the heat to flow at the same rate along PRQ and PQ then which of the following option is correct?

K3​=21​(K1​+K2​)

K3​=K1​+K2​

K3​=K1​+K2​K1​K2​​

K3​=−2(K1​+K2​)