Question

Four wires of same materials have the following dimensions:
Wire $1$: Length $l$, Cross section area $A$
Wire $2$: Length $l$, Cross section area $2A$
Wire $3$: Length $0.5l$, Cross section area $1.5A$
Wire $4$: Length $2l$, Cross section area $0.5A$

Which wire has the highest resistance?

• A. Wire $1$
• B. Wire $2$
• C. Wire $3$
• D. Wire $4$

Answer 1

The correct option is D Wire $4$

Resistance of a conductor is given by $R=\rho \frac{L}{a}$ where, $\rho$ is the resistance, $a$ is the cross sectional area and $L$ is the length of the conductor.
For wire $1$, $a=A$ and $L=l$
So, $R_1=\rho \frac{l}{A}$
For wire $2$, $a=2A$ and $L=l$
So, $R_2=\rho \frac{l}{2A}$
For wire $3$, $a=1.5A$ and $L=0.5l$
So, $R_3=\rho \frac{0.5l}{1.5A}=\rho \frac{0.33l}{A}$
For wire $4$, $a=0.5A$ and $L=2l$
So, $R_4=\rho \frac{2l}{0.5A}=\rho \frac{4l}{A}$

$⟹R_4>R_1>R_2>R_3$

Hence, for the same value of $\rho$, wire $4$ has the maximum resistance.