How many lamps each of `50W`and `100W`can be connected in parallel across a `120V` battery of internal resistance `10 Omega` so the each glows to full power ?

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the problem of how many lamps of 50W and 100W can be connected in parallel across a 120V battery with an internal resistance of 10Ω so that each glows at full power, we can follow these steps:

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

For the 50W lamp:
\[ R_{50W} = \frac{120^2}{50} = \frac{14400}{50} = 288 \, \Omega \]

For the 100W lamp:
\[ R_{100W} = \frac{120^2}{100} = \frac{14400}{100} = 144 \, \Omega \]

### Step 2: Calculate the current required by each lamp
The current \( I \) through each lamp can be calculated using the formula:
\[ I = \frac{P}{V} \]

For the 50W lamp:
\[ I_{50W} = \frac{50}{120} = \frac{5}{12} \, \text{A} \]

For the 100W lamp:
\[ I_{100W} = \frac{100}{120} = \frac{5}{6} \, \text{A} \]

### Step 3: Determine the total current required for \( N \) lamps
If \( N \) lamps are connected in parallel, the total current required for the 50W lamps is:
\[ I_{total, 50W} = N \times I_{50W} = N \times \frac{5}{12} \, \text{A} \]

For the 100W lamps:
\[ I_{total, 100W} = N \times I_{100W} = N \times \frac{5}{6} \, \text{A} \]

### Step 4: Calculate the total resistance of the circuit
The total resistance \( R_{total} \) of the circuit when \( N \) lamps are connected in parallel is given by:
\[ R_{total} = R_{internal} + R_{lamp} \]
Where \( R_{internal} \) is the internal resistance of the battery (10Ω) and \( R_{lamp} \) is the equivalent resistance of \( N \) lamps in parallel.

For \( N \) lamps of 50W:
\[ R_{lamp, 50W} = \frac{R_{50W}}{N} = \frac{288}{N} \]
Thus,
\[ R_{total, 50W} = 10 + \frac{288}{N} \]

For \( N \) lamps of 100W:
\[ R_{lamp, 100W} = \frac{R_{100W}}{N} = \frac{144}{N} \]
Thus,
\[ R_{total, 100W} = 10 + \frac{144}{N} \]

### Step 5: Apply Ohm's law to find the maximum \( N \)
Using Ohm's law, we can set up the equation for the total current from the battery:
\[ I = \frac{V}{R_{total}} \]
For the 50W lamps:
\[ \frac{N \times \frac{5}{12}}{1} = \frac{120}{10 + \frac{288}{N}} \]
Cross-multiplying gives:
\[ N \times \frac{5}{12} \left(10 + \frac{288}{N}\right) = 120 \]
Solving this equation will yield the maximum number of 50W lamps.

For the 100W lamps:
\[ \frac{N \times \frac{5}{6}}{1} = \frac{120}{10 + \frac{144}{N}} \]
Cross-multiplying gives:
\[ N \times \frac{5}{6} \left(10 + \frac{144}{N}\right) = 120 \]
Solving this equation will yield the maximum number of 100W lamps.

### Final Calculation
Solving the equations for both types of lamps will give us the maximum number of lamps that can be connected in parallel while glowing at full power.