Question

Find the ratio of resistances of two copper rods X and Y of lengths 30 cm and 50 cm respectively and having a radii $3cm$ and $2cm$ respectively.
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• A. $5:3$
• B. $3:5$
• C. $4:15$
• D. $15:4$

Answer 1

The correct option is C $4:15$

Resistance (R) of a conductor depends on
(i) it's length (L)
(ii) it's area of cross-section (A) and
(iii) the resistivity of the material $\left(\rho \right)$
$R=\frac{\rho L}{A}$
$R_X=\frac{{\rho }_X{L}_X}{A_X}=\frac{{\rho }_{X}30}{\pi \times 3\times 3}$
$R_Y=\frac{{\rho }_Y{L}_Y}{A_Y}=\frac{{\rho }_{Y}50}{\pi \times 2\times 2}$
Since both the rods are made of copper only, therefore,
${\rho }_X={\rho }_Y$

$⟹$$\frac{R_X}{R_Y}=\frac{30\times \pi \times 2\times 2}{50\times \pi \times 3\times 3}$
$⟹\frac{R_X}{R_Y}=\frac{4}{15}$

Hence the ratio of resistances of wires X and Y will be $4:15$.