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Physics-

General

Easy

Question

A composite metal bar of uniform section is made up of length $25 blank c m$ of copper, $10 blank c m$ of nickel and $15 blank c m$ of aluminium. Each part being in perfect thermal contact with the adjoining part. The copper end of the composite rod is maintained at $100 ℃$ and the aluminium end at $0 ℃$ . The whole rod is covered with belt so that no heat loss occurs at the sides. If $K subscript C u end subscript equals 2 K subscript A l end subscript$ and $K subscript A l end subscript equals 3 K subscript N i end subscript$ , then what will be the temperatures of $C u minus N i$ and $N i minus A l$ junctions respectively

$23.33 ℃$ and $78.8 ℃$
$83.33 ℃$ and $20 ℃$
$50 ℃$ and $30 ℃$
$30 ℃$ and $50 ℃$

The correct answer is: $83.33 ℃$ and $20 ℃$

If suppose $K subscript N i end subscript equals K rightwards double arrow K subscript A l end subscript equals 3 K$ and $K subscript C u end subscript equals 6 K$
Since all metal bars are connected in series
So $open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C u end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript A l end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript N i end subscript$
$a n d blank fraction numerator 3 over denominator K subscript e q end subscript end fraction equals fraction numerator 1 over denominator K subscript C u end subscript end fraction plus fraction numerator 1 over denominator K subscript A l end subscript end fraction plus fraction numerator 1 over denominator K subscript N i end subscript end fraction equals fraction numerator 1 over denominator 6 K end fraction plus fraction numerator 1 over denominator 3 K end fraction plus fraction numerator 1 over denominator K end fraction equals fraction numerator 9 over denominator 6 K end fraction$
$rightwards double arrow K subscript e q end subscript equals 2 K$

Hence, it $open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C u end subscript$
$rightwards double arrow fraction numerator K subscript e q end subscript A left parenthesis 100 minus 0 right parenthesis over denominator l subscript C o m b i n a t i o n end subscript end fraction equals fraction numerator K subscript C u end subscript A left parenthesis 100 minus theta subscript 1 end subscript right parenthesis over denominator l subscript C u end subscript end fraction$
$rightwards double arrow blank fraction numerator 2 K blank A left parenthesis 100 minus 0 right parenthesis over denominator left parenthesis 25 plus 10 plus 15 right parenthesis end fraction equals fraction numerator 6 K blank A left parenthesis 100 minus theta subscript 1 end subscript right parenthesis over denominator 25 end fraction rightwards double arrow theta subscript 1 end subscript equals 83.33 ℃$
Similar if $open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript A l end subscript$
$rightwards double arrow fraction numerator 2 K blank A left parenthesis 100 minus 0 right parenthesis over denominator 50 end fraction equals fraction numerator 3 K blank A left parenthesis theta subscript 2 end subscript minus 0 right parenthesis over denominator 15 end fraction rightwards double arrow theta subscript 2 end subscript equals 20 ℃$

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