Question

A wire of uniform cross-section has a resistance of $R\Omega$. What would be the resistance of a similar wire, made of the same material, but twice as long and of twice the diameter?

• A. $\frac{3}{2}R$
• B. $\frac{5}{2}R$
• C. $\frac{1}{2}R$
• D. $\frac{1}{4}R$

Answer 1

The correct option is C $\frac{1}{2}R$

Wire of uniform cross-section has a resistance of $R$.
let the diameter be $d$.
Let the length of the wire be $l$.
Cross sectional Area be $A$:
$A=\pi \frac{d^2}{4}$
Now the resistance of a wire can be expressed as:
$R=\rho \frac{l}{A}$
Where resistivity is $\rho$.
Now a new wire is taken made of the same material, but twice as long and of twice the diameter.
Let final length be $l_1$, diameter be $d_1$ and cross section Area be $A_1$.

$A_1=\pi \frac{d_1^2}{4}=\pi \frac{4d^2}{4}=4A$
$R_1=\rho \frac{l_1}{A_1}$

Putting the value of $l_1=2l$ and $A_1=4A$
We have, $R_1=\rho \frac{2l}{4A}=\rho \frac{l}{2A}$
$R_1=\frac{R}{2}$