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, $$\begin{array}{rl}{P}_1& =\frac{V_1^2}{R_1}\\ R_1& =\frac{V_1^2}{P_1}\\ & =\frac{(220{)}^2}{25}\\ \therefore R_1& =1936\mathrm{\Omega }\end{array}$$ Similarly, $$\begin{array}{rl}{R}_2& =\frac{V_2^2}{P_2}\\ & =\frac{(220{)}^2}{100}\\ \therefore R_2& =484\mathrm{\Omega }\end{array}$$
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 $\mathrm{\Omega }$

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


Power consumed by lamp 1 is, $$\begin{array}{rl}& =I^2{R}_1\\ & ={\left(\frac{1}{11}\right)}^2\times 1936\\ & =16W\end{array}$$ Power consumed by lamp 2 is, $$\begin{array}{rl}& =I^2{R}_2\\ & ={\left(\frac{1}{11}\right)}^2\times 484\\ & =4W\end{array}$$
Hence, the power consumed by the respective lamps are 16 W and 4 W.