Question

Four wires are made from the same material. Which wire has the lowest resistance?
A. Length of wire/cm$-20$, Diameter of wire/mm$-0.20$
B. Length of wire/cm$-20$, Diameter of wire/mm$-0.40$
C. Length of wire/cm$-40$, Diameter of wire/mm$-0.20$
D. Length of wire/cm$-40$, Diameter of wire/mm$-0.40$

Answer 1

Hint: To solve this question we will use the formula of resistance that gives a relation between the area of the wire, the length of the wire and the corresponding resistance. According to the formula resistance is directly proportional to the length of the wire and is inversely proportional to the area of the diameter.

Complete answer:
Let us first write down the formula for resistance of a wire:
$R=\rho \dfrac{l}{A}$
Here, $\rho $ is the resistivity of the wire, $l$ is the length of the wire and $A$ is the cross-sectional area of the wire.
We know that the cross sectional of a wire in terms of the diameter of the wire is:
$A=\pi \dfrac{{{d}^{2}}}{4}$
If we substitute the value of cross-sectional are in the equation of resistance, then we would get:
$R=\dfrac{4\rho l}{\pi {{d}^{2}}}$
On observing the equation, we can conclude that:
$R\propto l$
And,
$R\propto \dfrac{1}{{{d}^{2}}}$
This means that the wire with lowest resistance must have the lowest length of the wire and the highest magnitude of diameter. If we check the options, we can see that the correct choice must be the wire which has length of wire/cm$-20$ and Diameter of wire/mm$-0.40$.

So, the correct answer is “Option B”.

Note: Resistance of a wire is the property by which it tries to oppose the flow of electric current through the wire when a voltage difference is applied to it. If the wires are made up of the same material then they will have the same value of resistivity. The factors that decide which wire would have a higher or lower resistivity are length and the area of the wires.