Two lamps rated 25 W - 220 V and 100 W - 220 V are connected to 220 V supply. Calculate the powers consumed by the lamps.

Two lamps rated 25 W - 220 V and 100 W - 220 V are connected to 220 V supply. Calculate the powers consumed by the lamps.
Solution:

Given,
For lamp 1,
power consumption, \(P_1\) = 25 W
voltage, \(V_1\) = 220 V

For lamp 2,
power consumption, \(P_2\) = 100 W
voltage, \(V_2\) = 220 V
Note! Here, we need to take care of the fact that the above information is for the case when this lamps are connected individually with 220 V mains. We now find the resistance \(R_1\) and \(R_2\) of this lamps.
We have,
\[P=\frac{V^2}{R}\] Using this relation, \[\begin{align*} P_1&=\frac{V_1^2}{R_1}\\ R_1&=\frac{V_1^2}{P_1}\\ &=\frac{(220)^2}{25}\\ \therefore R_1&=1936 \hspace{0.1cm} \Omega\\ \end{align*}\] Similarly, \[\begin{align*} R_2&=\frac{V_2^2}{P_2}\\ &=\frac{(220)^2}{100}\\ \therefore R_2&=484 \hspace{0.1cm} \Omega\\ \end{align*}\]
Now, we replace the lamps by their respective resistances to find the power consumed by each lamps when connected in series with 220 V supply.

Since the resistance is in series, total resistance, \(R\) = \(R_1\) + \(R_2\) = 1936 + 484 = 2420 \(\Omega\)

Total current through the circuit is then, \(I\) = \(\frac{V}{R}\) = \(\frac{220}{2420}\) = \(\frac{1}{11}\) A


Power consumed by lamp 1 is, \[\begin{align*} &=I^2 R_1\\ &=\left(\frac{1}{11}\right)^2 \times 1936\\ &=16 \hspace{0.1cm} W\\ \end{align*}\] Power consumed by lamp 2 is, \[\begin{align*} &=I^2 R_2\\ &=\left(\frac{1}{11}\right)^2 \times 484\\ &=4 \hspace{0.1cm} W\\ \end{align*}\]
Hence, the power consumed by the respective lamps are 16 W and 4 W.