Question

An aluminium wire \[\left( {{\alpha }_{Al}}=4\times {{10}^{-3}}/{}^{o}C \right)\] resistance \[{{R}_{1}}\] and carbon wire \[\left( {{\alpha }_{C}}=0.5\times {{10}^{-3}}/{}^{o}C \right)\] resistance \[{{R}_{2}}\] are connected in series to have a resultant resistance of 18 ohm at all temperatures. The value of \[{{R}_{1}}\] and \[{{R}_{2}}\] in ohms
A. 2,16
B. 12,6
C. 13,5
D. 14,4

Answer 1

Hint: In this question we are asked to calculate the individual resistance of two resistors connected in series. It is said that the total resistance is constant at varying temperatures. The equation of temperature coefficient gives us the relation between the resistance and change in temperature. Therefore, we will be using this to calculate individual resistance of given resistors.

Formula Used:
\[R={{R}_{o}}\alpha \Delta T\]
Where,
R is the total resistance at T=T
\[{{R}_{o}}\] is the resistance at T = 0
\[\alpha \] is the temperature coefficient
\[\Delta T\] is the difference in temperature (\[T-{{T}_{o}}\])

Complete answer:
For two resistors connected in series, we know that, Total resistance is given by,
\[R={{R}_{1}}+{{R}_{2}}\] ……………. (1)
We also know that, for variation in temperature, the total resistance is given by,
\[R=({{R}_{1}})[1+{{\alpha }_{Al}}\Delta T]+({{R}_{2}})[1+{{\alpha }_{C}}\Delta T]\]
On solving
We get,
\[R={{R}_{1}}+{{R}_{2}}+[{{R}_{1}}{{\alpha }_{Al}}\Delta T+{{R}_{2}}{{\alpha }_{C}}\Delta T]\]
It is given that total resistance is given as 18
Therefore,
\[R=18+[{{R}_{1}}{{\alpha }_{Al}}\Delta T+{{R}_{2}}{{\alpha }_{C}}\Delta T]\]
Now, it is said that the total resistance is constant at all the varied temperature i.e. \[R=18\]
Therefore, we can say that
\[[{{R}_{1}}{{\alpha }_{Al}}\Delta T+{{R}_{2}}{{\alpha }_{C}}\Delta T]=0\]
After substituting the given values
We get,
\[4\times {{10}^{-3}}{{R}_{1}}-0.5\times {{10}^{-3}}{{R}_{2}}=0\]
On solving,
\[{{R}_{2}}=8{{R}_{1}}\] …………………….. (2)
Substituting (2) in (1)
We get,
\[{{R}_{1}}+8{{R}_{1}}=18\]
On solving,
\[{{R}_{1}}=2\]
Substituting above value in (2)
We get,
\[{{R}_{2}}=16\]
Therefore, \[{{R}_{1}}=2\] and \[{{R}_{2}}=16\]

Therefore, the correct answer is option A.

Note:
Temperature coefficient describes the relative change of resistance with given temperature. A positive temperature coefficient means that its resistance increases with increasing temperature. Resistance is the property of the material to avoid the flow of current. For many metals as temperature increases the resistance of material also increases.