Question

What length of the wire of specific resistance \[48 \times {10^{ - 8}}\Omega - m\] is needed to make a resistance of \[4.2\Omega \] (diameter of wire =0.4mm)
A. 4.1 m
B. 3.1 m
C. 2.1 m
D. 1.1 m

Answer 1

Hint: The resistivity is the property of the material so if different resistors are made of the same material then the resistivity of all the resistors will be equal. The resistance is proportional to the length and inversely proportional to the area of cross-section.

Formula used:
\[R = \dfrac{{\rho l}}{A}\]
where R is the resistance of the wire of length l and cross-sectional area A, \[\rho \] is the resistivity of the material of the wire.

Complete step by step solution:
The specific resistance (resistivity) of the wire is given as \[48 \times {10^{ - 8}}\Omega - m\].
\[\rho = 48 \times {10^{ - 8}}\Omega - m\]
The resistance of the wire is given as \[4.2\Omega \].
\[R = 4.2\Omega \]
The diameter of the wire is given as 0.4 mm. Then the diameter of the wire in the standard unit will be \[0.4 \times {10^{ - 3}}m\].
\[d = 0.4 \times {10^{ - 3}}m\]
The area of cross-section of the wire will be,
\[A = \dfrac{{\pi {d^2}}}{4}\]

Putting the value of the diameter of the wire, we get the area of cross-section of the wire as,
\[A = \dfrac{{\pi \times {{\left( {0.4 \times {{10}^{ - 3}}} \right)}^2}}}{4}{m^2} \\ \]
\[\Rightarrow A = 1.26 \times {10^{ - 7}}{m^2}\]
Now, using the formula of the resistance,
\[R = \dfrac{{\rho L}}{A} \\ \]
\[\Rightarrow L = \dfrac{{RA}}{\rho }\]
Putting the values, we get the required length of the wire as,
\[L = \dfrac{{4.2 \times 1.26 \times {{10}^{ - 7}}}}{{48 \times {{10}^{ - 8}}}}m \\ \]
\[\therefore L = 1.10\,m\]
Hence, to have the resistance of the wire as 4.2 Ohms, we need the length of wire to be 1.10 m and diameter of the wire is 0.4 mm.

Therefore, the correct option is D.

Note: We should be careful about the shape of the cross-section of the wire while finding the area and the unit of the given measurements should be used in standard units. If the measurements are given in other equivalent units then we need to convert it into the standard unit.