Question

A 100-watt bulb working on 200 volts and a 200-watt bulb working on 100 volts have:-
A. Resistances in the ratio of 4:1
B. Maximum current ratings in the ratio of 1:4
C. Resistances in the ratio of 2:1
D. Maximum current ratings in the ratio of 1:2

Answer 1

Hint: We will compute the values of the resistance and the current of both the bulbs, as, in the options, the ratios of the resistance and the current values of both the bulbs is given. Then, we will divide these values to find their ratios. Thus, we will obtain the ratio of the values of the resistance and the current.
Formulae used:
\[\begin{align}
  & R=\dfrac{{{V}^{2}}}{P} \\
 & I=\dfrac{P}{V} \\
\end{align}\]

Complete step-by-step solution
The formula that relates the power, the voltage, and the resistance of the bulb is given as follows.
\[R=\dfrac{{{V}^{2}}}{P}\]
Where R is the resistance, V is the voltage and P is the power.
The formula that relates the power, the voltage, and the resistance of the bulb is given as follows.
\[I=\dfrac{P}{V}\]
Where I is the current, V is the voltage and P is the power.
Firstly, we will compute the resistance values of both the bulbs.
From the data, we have the data as follows.
The power of bulb 1 is 100 watt.
The voltage using which the bulb glows is 200 volt.
Bulb 1: The resistance value of the first bulb is,
\[\begin{align}
  & R=\dfrac{{{V}^{2}}}{P} \\
 &\Rightarrow {{R}_{1}}=\dfrac{{{200}^{2}}}{100} \\
 &\Rightarrow {{R}_{1}}=400\Omega \\
\end{align}\]
From the data, we have the data as follows.
The power of bulb 2 is 200 watt.
The voltage using which the bulb glows is 100 volt.
Bulb 2: The resistance value of the second bulb is,
\[\begin{align}
  & R=\dfrac{{{V}^{2}}}{P} \\
 &\Rightarrow {{R}_{2}}=\dfrac{{{100}^{2}}}{200} \\
 &\Rightarrow {{R}_{2}}=50\Omega \\
\end{align}\]
The ratio of the resistance values of the bulbs is calculated as follows.
\[\begin{align}
  & \dfrac{{{R}_{1}}}{{{R}_{2}}}=\dfrac{400}{50} \\
 &\Rightarrow \dfrac{{{R}_{1}}}{{{R}_{2}}}=\dfrac{8}{1} \\
\end{align}\]
Therefore, the ratio of the resistance values of the bulbs is 8:1.
Now, we will compute the current values of both the bulbs.
From the data, we have the data as follows.
The power of bulb 1 is 100 watt.
The voltage using which the bulb glows is 200 volt.
Bulb 1: The current value of the first bulb is,
\[\begin{align}
  & I=\dfrac{P}{V} \\
 & {{I}_{1}}=\dfrac{100}{200} \\
 &\Rightarrow {{I}_{1}}=\dfrac{1}{2}A \\
\end{align}\]
From the data, we have the data as follows.
The power of bulb 2 is 200 watt.
The voltage using which the bulb glows is 100 volt.
Bulb 2: The current value of the second bulb is,
\[\begin{align}
  & I=\dfrac{P}{V} \\
 &\Rightarrow {{I}_{2}}=\dfrac{200}{100} \\
 &\Rightarrow {{I}_{2}}=\dfrac{2}{1}A \\
\end{align}\]
The ratio of the current values of the bulbs is calculated as follows.
\[\begin{align}
  & \dfrac{{{I}_{1}}}{{{I}_{2}}}=\dfrac{1}{2}\times \dfrac{1}{2} \\
 &\Rightarrow \dfrac{{{I}_{1}}}{{{I}_{2}}}=\dfrac{1}{4} \\
\end{align}\]
Therefore, the ratio of the current values of the bulbs is 1:4.
\[ \therefore \]The maximum current ratings in the ratio of 1:4.
As, the maximum current ratings in the ratio of 1:4, thus, option (B) is correct.

Note: The things to be on your finger-tips for further information on solving these types of problems are: The units of the physical parameters should be known. Even the formula for computing the values of the power, current, voltage, and resistance should be known.