Question

At what temperature would the resistance of a copper conductor be double its resistance at ${0^ \circ }C$? Given $\alpha $for copper is $3.9 \times {10^{ - 3}}^{^ \circ }{C^{ - 1}}$
A.${256.4^ \circ }C$
B.${512.8^ \circ }C$
C.${100^ \circ }C$
D.$256.4K$

Answer 1

Hint: Recall the concept of resistance and resistivity. Resistance is the property of a conductor in which the flow of current or the movement of electrons is offered a blockage or opposition. While resistivity is the amount of resistance offered per unit length to a conductor for an area of unit cross section. It depends on the material of the conductor.

Complete answer:
Step I:
To explain the variation of resistance of a conductor with the change in temperature, the formula of temperature coefficient of resistance is used. The formula is written as
$R = {R_0}(1 + \alpha t)$---(i)
Where $R$ is the resistance at temperature t
${R_0}$ is the resistance at ${0^ \circ }C$
$\alpha $ is the temperature coefficient of resistance and its value is given $3.9 \times {10^{ - 3}}$
Step II:
Given that the resistance of copper conductor is double its resistance at ${0^ \circ }C$
Therefore,
$ \Rightarrow R = 2{R_0}$
Or $\dfrac{R}{{{R_0}}} = 2$
Step III:
Substitute the value of ‘R’ from equation (i),
$ \Rightarrow \dfrac{{{R_0}(1 + \alpha t)}}{{{R_0}}} = 2$
$1 + \alpha t = 2$
$\alpha t = 2 - 1$
$\alpha t = 1$
Or $t = \dfrac{1}{\alpha }$
Step IV:
Substitute the value of $\alpha $in the above equation and solve for the value of t
$ \Rightarrow t = \dfrac{1}{{3.9 \times {{10}^{ - 3}}}}$
On solving, the value of temperature is
$ \Rightarrow t = {256.4^ \circ }C$
Step V:
Therefore, at temperature $ = {256.4^ \circ }C$ the resistance of copper will be double its resistance at ${0^ \circ }C$

Hence Option A is the right answer.

Note:
It is to be noted that though both resistance and resistivity are the physical properties of a conductor, they both are different terms. Resistance varies directly with the length and temperature and inversely with the cross sectional area. But resistivity depends only on the nature and the temperature of the conductor. It increases linearly with temperature.