Two wires of the same material and length have radii in the ratio of $2 : 3$ . Compare their resistance.

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Solution

$R = \frac{δ l}{A}, δ =$ same for both wire as they are of same material
So, $R_{1} = \frac{δ l}{A_{1}}$ $R_{2} = \frac{δ l}{A_{2}}$
Now, $A_{1} = π r^{2}$
$A_{2} = π r_{2}^{2}$
$\frac{r_{1}}{r_{2}} = \frac{2}{3} ∴ r_{1} = \frac{2}{3} r_{2}$
So, $π r^{2} = π {(\frac{2}{3} r_{2})}_{2}^{2}$
$⟹ A_{1} = π r_{2}^{2} (\frac{4}{9})$
$A_{1} = \frac{4}{9} A_{2}$
$∴ \frac{R_{1}}{R_{2}} = \frac{δ l}{A_{1}} \times \frac{A_{2}}{δ l} \frac{A_{2}}{A_{1}}$
$\frac{R_{1}}{R_{2}} = \frac{A_{2}}{\frac{4}{9} A_{2}} = \frac{9}{4}$

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