Question

In India electricity is supplied for domestic use at 220 V. It is supplied at 110 V in the USA. If the resistance of a 60W bulb for use in India is R. The resistance of a 60W bulb for use in the USA will be:-
A). $2R$
B). ${\displaystyle \frac{R}{2}}$
C). ${\displaystyle \frac{R}{4}}$
D). $R$

Answer 1

Hint: Use the formula for power in terms of voltage and resistance. Then calculate the resistance of the bulb in India. Using the same formula, calculate the resistance of the bulb in the USA. Then, obtain the resistance of the bulb in the USA in terms of the resistance of the bulb in India.

Complete step-by-step solution:
Given: Voltage in India = 220V
             Voltage in USA = 110V
            Power= 60 Watt
Power is expressed by,
$Power={\displaystyle \frac{{(Voltage)}^2}{Resistance}}$
Rearranging the above equation we get,
$Resistance={\displaystyle \frac{{(Voltage)}^2}{Power}}$
$\Rightarrow R={\displaystyle \frac{V^2}{P}}$ ...(1)
Substituting value in the equation. (1)
Therefore, in India,
$R={\displaystyle \frac{{220}^2}{60}}$
$\Rightarrow R={\displaystyle \frac{48400}{60}}$
$\Rightarrow R={\displaystyle \frac{2420}{3}}$ ...(1)
Similarly, in the USA,
$R^{{}^{\prime }}={\displaystyle \frac{{110}^2}{60}}$
$\Rightarrow R^{{}^{\prime }}={\displaystyle \frac{12100}{60}}$
$\Rightarrow R^{{}^{\prime }}={\displaystyle \frac{605}{3}}$ ...(2)
Substituting equation. (2) in equation. (1) we get,
$R=4R^{{}^{\prime }}$
$\Rightarrow R^{{}^{\prime }}={\displaystyle \frac{R}{4}}$
Thus, the resistance of a 60W bulb for use in USA will be ${\displaystyle \frac{R}{4}}$
Hence, the correct answer is option C i.e. ${\displaystyle \frac{R}{4}}$.

Note: You can solve this problem using an alternate method. You can take the ratio of the square of voltage to resistance for both the India and USA and then equate them as the power of the bulb is the same. Simplify them and obtain a relation between the resistances of both the countries. And then write the resistance of the bulb in the USA in terms of the resistance of the bulb in India.