Question

how does the resistance of a wire change if area of cross section is tripled and reduced to one fourth

Answer 1

Dear student,

The resistance of wire is given by $R=\frac{\rho L}{A}$
Where L is length of wire,
Where A is area of cross section,
Where ρ is the resistivity

If the area of cross section is tripled: The resistance is inversely proportional to area of cross section.Hence when area of cross section increases the resistance decrease.therefore amount of current flowing through the circuit increases.
$$\begin{gathered} R\text{'}=\frac{\rho L}{3A} \\ \frac{R\text{'}}{R}=\frac{1}{3} \\ R\text{'}=\frac{1}{3}R \end{gathered}$$

The resistance when area is tripled will be one third of original resistance.

When the area is reduced to one fourth of original,

$$\begin{gathered} R\text{'}=\frac{\rho L}{{\displaystyle \frac{A}{4}}} \\ R\text{'}=\frac{4\rho L}{A} \\ \frac{R\text{'}}{R}=4 \\ R\text{'}=4R \end{gathered}$$

The resistance is increased four times the original.

Regards.