Two bulbs are marked 220 V, 100 W and 220 V, 50W respectively. They are connected in series to 220 V mains. Find the ratio of heats generated in them.

Answer

To solve the problem of finding the ratio of heats generated in two bulbs connected in series, we can follow these steps:

### Step 1: Determine the Resistance of Each Bulb
The resistance \( R \) of each bulb can be calculated using the formula:
\[ R = \frac{V^2}{P} \]
where \( V \) is the voltage rating of the bulb and \( P \) is the power rating.

**For Bulb 1 (100 W):**
\[ R_1 = \frac{220^2}{100} = \frac{48400}{100} = 484 \, \Omega \]

**For Bulb 2 (50 W):**
\[ R_2 = \frac{220^2}{50} = \frac{48400}{50} = 968 \, \Omega \]

### Step 2: Calculate the Total Resistance in Series
Since the bulbs are connected in series, the total resistance \( R_{total} \) is the sum of the individual resistances:
\[ R_{total} = R_1 + R_2 = 484 + 968 = 1452 \, \Omega \]

### Step 3: Calculate the Current Flowing Through the Circuit
Using Ohm's law, the current \( I \) flowing through the circuit can be calculated using:
\[ I = \frac{V_{total}}{R_{total}} = \frac{220}{1452} \]

### Step 4: Calculate the Power for Each Bulb
The power \( P \) dissipated in each bulb can be calculated using:
\[ P = I^2 R \]

**For Bulb 1:**
\[ P_1 = I^2 R_1 = \left(\frac{220}{1452}\right)^2 \times 484 \]

**For Bulb 2:**
\[ P_2 = I^2 R_2 = \left(\frac{220}{1452}\right)^2 \times 968 \]

### Step 5: Calculate the Ratio of Heats Generated
The heat generated in each bulb is proportional to the power dissipated in each bulb when they are connected for the same time duration. Thus, the ratio of heats \( H_1 : H_2 \) is given by:
\[ \frac{H_1}{H_2} = \frac{P_1}{P_2} = \frac{R_1}{R_2} \]

Substituting the values of \( R_1 \) and \( R_2 \):
\[ \frac{H_1}{H_2} = \frac{484}{968} = \frac{1}{2} \]

### Final Answer
The ratio of heats generated in the two bulbs is:
\[ H_1 : H_2 = 1 : 2 \]