Question

Nichrome wire of length ‘l’ radius ‘r’ has a resistance of 10Ω. How would the resistance of the wire change when (i) only the diameter is doubled? (ii) only the length of the wire is doubled?

Answer 1

Hint: Specific resistance formula applies to solve this problem. Specific resistance is equal to the resistance of an object. It depends upon the nature of the material, length, and area of cross-section.

Complete step by step solution:
Given data,
Length of the wire = $l$
Resistance of the wire $ = 10\Omega $
Radius of the wire =$ r$
We know specific resistance formula,
${\text{R = }}\rho \dfrac{l}{{\text{A}}}$
(Where $R$ is resistance, $l$ is the length; $A$ is the area of cross-section, and $\rho$ is the resistivity)

(i) When the diameter doubles, the radius of the wire also doubles.
Therefore area = $\pi {{\text{r}}^2} = \pi {(2{\text{r)}}^2}$
$ \Rightarrow 4\pi {r^2} = 4A$
Let the new resistance be R1,
$\therefore {{\text{R}}_1} = \rho \dfrac{l}{{4{\text{A}}}} = \dfrac{{\text{R}}}{4}$
$ \Rightarrow {{\text{R}}_1} = \dfrac{{10}}{4} = 2.5\Omega $

(ii) Only when the length of the wire is doubled.
We apply specific resistance formula,
${\text{R = }}\rho \dfrac{l}{{\text{A}}}$
Let the new resistance $R_2$
$\therefore {{\text{R}}_2} = \rho \dfrac{{2l}}{{\text{A}}}$
$ \Rightarrow {{\text{R}}_2} = 2\rho \dfrac{l}{{\text{A}}} = 2{\text{R}}$
(Because length of the wire is doubled)
$ \Rightarrow {{\text{R}}_2} = 2 \times 10 = 20\Omega $

$\therefore$ The resistance of the wire will be $2.5\Omega $ when only the diameter of the wire is doubled. The resistance of the wire will be $20\Omega $ when only the length of the diameter is doubled.

Additional information:
Nichrome has high resistivity, low conductivity, and a small temperature coefficient of resistivity. Another advantage of Nichrome is that it does not deteriorate mechanically or chemically even at high temperatures. It is used for making elements of an electric heater.

Note:
The specific resistance of the conductor depends upon the temperature of the conductor also. If the temperature of a conductor increases, its resistivity also increases. If the temperature of the conductor decreases, its resistivity also decreases. It does not depend upon the dimensions of the objects. It depends upon the nature of the material of the conductor.