Question

A 100-W bulb and 25-W bulbs are designed for the same voltage . They have filaments of the same length and material. The ratio of the diameter of the 100-W bulb to the that of the 25- W bulbs is:

• A. 4:1
• B. 2:1
• C. $\sqrt{}2:1$
• D. 1:2

Answer 1

The correct option is B 2:1

Given that,

Power of first bulb,$P_1=100W$

Power of second bulb, $P_2=25W$

Power is given by,

$P=\frac{V^2}{R}$

$R=\frac{V^2}{P}$

$\frac{R_1}{R_2}=\frac{\left(\frac{V^2}{P_1}\right)}{\left(\frac{V^2}{P_2}\right)}=\frac{P_2}{P_1}........\left(1\right)$

And the resistance in terms of resistivity is given by:

$\frac{R_1}{R_2}=\frac{\frac{\rho l}{\pi {\left(d_1/2\right)}^2}}{\frac{\rho l}{\pi {\left(d_2/2\right)}^2}}=\frac{d_2^2}{d_1^2}$

$so,$

$\frac{R_1}{R_2}=\frac{d_2^2}{d_1^2}$

$\frac{d_2}{d_1}=\sqrt{\frac{P_2}{P_1}}$

$\frac{d_2}{d_1}=\sqrt{\frac{25}{100}}$

$\frac{d_2}{d_1}=\frac{1}{2}$

The ratio of the diameter of the 100 W bulb to the that of the 25 W bulbs is 2:1