Question

A cylindrical rod has resistance $\mathbf{R}$. If we triple its length and diameter, what is its resistance in terms of R

Short answer

The value of the resistance is $\frac{R}{3}$.

Step-by-step solution

Definition of Resistance

The term resistivity is opposition offered by the particular electric conductor to the flow the charges. It depends on the type of material used, while the resistance will depend on the length and the area of the conductor.

The expression for the resistance is,

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

Here, $\mathbf{\rho }$is the resistivity, Ris the resistance, L is the length and A is the area of the cross section.

Calculation of resistance

Consider the two resistance $R_1$and $R_2$such that there ratio is:

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

Consider the area of the cylindrical cross section is $A={\mathrm{\pi r}}^2$.

$\frac{R_1}{R_2}=\frac{L}{3L}\times \frac{{{\mathrm{\pi r}}_2}^2}{{{\mathrm{\pi r}}_1}^2}$

Solve further as,

$$\begin{gathered} \frac{R_1}{R_2}=\frac{1}{3}\times \frac{{\left(3r\right)}^2}{r^2} \\ {R}_2=\frac{R}{3} \end{gathered}$$

Hence, the resistance is $\frac{R}{3}$.