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Cathode Ray Experiment

A rod CD of t...

Question

A rod CD of thermal resistance 5.0 K/W is joined at the middle of an identical rod AB as shown in figure The ends A, B and D are maintained at 100 $^{\circ}$ C, 0 $^{\circ}$ C and 25 $^{\circ}$ C respectively. Find the heat current in CD.

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Solution

The correct option is C
4W

The thermal resistance of AC is equal to that of CB and is equal to 2.5 K/W. Suppose, the temperature at C is $θ$ . The heat currents through AC, CB AND CD are

$\frac{Δ Q_{1}}{Δ t} = \frac{100^{\circ} C - θ}{2.5 K / W}$

$\frac{Δ Q_{2}}{Δ t} = \frac{θ - 0^{\circ} C}{2.5 K / W}$

and $\frac{Δ Q_{3}}{Δ t} = \frac{θ - 25^{\circ} C}{5.0 K / W}$

We also have

$\frac{Δ Q_{1}}{Δ t} = \frac{Δ Q_{2}}{Δ t} + \frac{Δ Q_{3}}{Δ t}$

or, $\frac{100^{\circ} C - θ}{2.5} = \frac{θ - 0^{\circ} C}{2.5} + \frac{θ - 25^{\circ} C}{5}$

or, $225^{\circ} C = 5 θ$

or, $θ = 45^{\circ} C$

Thus, $\frac{Δ Q_{3}}{Δ t} = \frac{45^{\circ} C - 25^{\circ} C}{5.0 K / W} = \frac{20 K}{5.0 K / W}$

=4.0W.

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