Question

Two copper wires, one of length $1m$ and the other of length $9m$, are found to have the same resistance. Their diameters are in the ratio:

• A. $3:1$
• B. $1:9$
• C. $9:1$
• D. $1:3$

Answer 1

Answer: (D)

$1:3$

The electrical resistance of an electrical conductor is the opposition to the passage of an electric current through that conductor.
The resistance of a wire is inversely proportional to the area of cross section of the wire and directly proportional to the length of the wire. That is, the resistance of wire decreases as the diameter of the wire increases and length of the wire decreases and the resistance of the wire increases as the diameter of the wire decreases and the length of the wire increases.
The resistance of the conductor is given as $R=\rho \frac{l}{A}$.
In this case, the wire is $9$ times longer than the first wire. As per the above statement, the resistance increases as the length increases. To retain the same resistance, diameter of the wire can be increased as it is inversely proportional to the resistance.
As the area of cross section is square of the diameter, when the diameter of the wire is increased by $3$ times, the resistance remains the same.
Hence, Their diameters are in the ratio $1:3$.