Two heating coils of resistance `10 Omega and 20 Omega` are connected in parallel and connected to a battery of emf 12V and internal resistance `1 Omega`. The power consumed by them is in the ratio

Options

A. `1 :4`
B. `1:3`
C. `2:1`
D. `4:1`

Explanation

To find the ratio of power consumed by two heating coils connected in parallel, we can follow these steps:

### Step 1: Identify the resistances and the battery configuration
We have two heating coils with resistances:
- \( R_1 = 10 \, \Omega \)
- \( R_2 = 20 \, \Omega \)

They are connected in parallel to a battery with:
- EMF \( E = 12 \, V \)
- Internal resistance \( r = 1 \, \Omega \)

### Step 2: Calculate the total resistance in the circuit
The total resistance \( R_t \) of the parallel combination of \( R_1 \) and \( R_2 \) can be calculated using the formula for resistors in parallel:
\[ \frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} \]
Substituting the values:
\[ \frac{1}{R_t} = \frac{1}{10} + \frac{1}{20} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \]
Thus,
\[ R_t = \frac{20}{3} \, \Omega \]

### Step 3: Calculate the total resistance in the circuit
The total resistance in the circuit, including the internal resistance of the battery, is:
\[ R_{total} = R_t + r = \frac{20}{3} + 1 = \frac{20}{3} + \frac{3}{3} = \frac{23}{3} \, \Omega \]

### Step 4: Calculate the total current flowing through the circuit
Using Ohm's law, the total current \( I \) flowing through the circuit can be calculated as:
\[ I = \frac{E}{R_{total}} = \frac{12}{\frac{23}{3}} = \frac{12 \times 3}{23} = \frac{36}{23} \, A \]

### Step 5: Calculate the voltage across each coil
The voltage across the parallel combination of the coils can be calculated using:
\[ V = I \times R_t = \frac{36}{23} \times \frac{20}{3} = \frac{720}{69} \, V \]

### Step 6: Calculate the power consumed by each coil
The power consumed by each coil can be calculated using the formula:
\[ P = \frac{V^2}{R} \]
For coil 1:
\[ P_1 = \frac{V^2}{R_1} = \frac{V^2}{10} \]
For coil 2:
\[ P_2 = \frac{V^2}{R_2} = \frac{V^2}{20} \]

### Step 7: Find the ratio of power consumed
The ratio of power consumed by the two coils is given by:
\[ \frac{P_1}{P_2} = \frac{\frac{V^2}{R_1}}{\frac{V^2}{R_2}} = \frac{R_2}{R_1} \]
Substituting the values:
\[ \frac{P_1}{P_2} = \frac{20}{10} = 2 \]
Thus, the ratio of power consumed by the two coils is:
\[ P_1 : P_2 = 2 : 1 \]

### Final Answer
The power consumed by the coils is in the ratio of \( 2 : 1 \).

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