Question

The resistance of a certain wire is 20 Ohms. Another wire of the same material has a diameter 1/3 as great and twice the length. What is the resistance of the second wire?

Answer 1

We know, that the resistance of a wire is given by

$R=\rho\frac{L}{A},$

where $\rho$ is the resistivity of the material,

is the length of the resistance, and

is the area of its cross section.

It is given in the question that both wires are made of the same material. Therefore, we can conclude that the resistivities are equal for both of them.

We can thus say,

$R_1=\rho\frac{L_1}{A_1}$

and

$R_2=\rho\frac{L_2}{A_2}$

Dividing the two equations, we get

$\frac{R_1}{R_2}=\frac{L_1}{L_2}\cdot \frac{A_2}{A_1}$

Area of cross section is given by

$A=\frac{\pi d^2}{4}$

Thus, the equation derived above simplifies to

$\frac{R_1}{R_2}=\frac{L_1}{L_2}\cdot \left(\frac{d_2}{d_1}\right) ^{2}$

Putting $L_2=2L_1$ and $d_2=\frac{d_1} {3}$ ​​​​​, we can solve for $R_2$ ​​​​​​.

$\frac{20\:\Omega}{R_2}=\frac{L_1}{2L_1}\cdot \left(\frac{\frac{d_1}{3}}{d_1}\right) ^{2}$

$\Rightarrow R_2=360\:\Omega.$​​​​​​