two heading coils of resistances `10Omega and 20 Omega` are connected in parallel and connected to a battery of emf 12 V and internal resistance `1Omega` Thele power consumed by the n are in the ratio

Options

A. (a)`1:4`
B. (b)`1:3`
C. (c)`2:1`
D. (d)`4:1`

Answer

To solve the problem of finding the power consumed by two resistors connected in parallel, we can follow these steps:

### Step 1: Understand the Circuit Configuration
We have two resistors, R1 = 10Ω and R2 = 20Ω, connected in parallel to a battery with an EMF of 12V and an internal resistance of 1Ω.

### Step 2: Calculate the Total Resistance in the Circuit
The total resistance (R_total) of the circuit can be calculated using the formula for resistors in parallel:
\[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Substituting the values:
\[ \frac{1}{R_{total}} = \frac{1}{10} + \frac{1}{20} \]
Finding a common denominator (20):
\[ \frac{1}{R_{total}} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \]
Thus,
\[ R_{total} = \frac{20}{3} \, \Omega \]

### Step 3: Calculate the Total Circuit Resistance Including Internal Resistance
Now we need to consider the internal resistance of the battery (1Ω):
\[ R_{circuit} = R_{total} + R_{internal} = \frac{20}{3} + 1 = \frac{20}{3} + \frac{3}{3} = \frac{23}{3} \, \Omega \]

### Step 4: Calculate the Total Current in the Circuit
Using Ohm's Law (V = IR), we can find the total current (I) flowing through the circuit:
\[ I = \frac{V}{R_{circuit}} = \frac{12}{\frac{23}{3}} = \frac{12 \times 3}{23} = \frac{36}{23} \, A \]

### Step 5: Calculate the Voltage Across the Resistors
The voltage across the resistors (V_R) can be calculated using the voltage drop across the internal resistance:
\[ V_{internal} = I \times R_{internal} = \frac{36}{23} \times 1 = \frac{36}{23} \, V \]
Thus, the voltage across the resistors is:
\[ V_R = V - V_{internal} = 12 - \frac{36}{23} = \frac{276 - 36}{23} = \frac{240}{23} \, V \]

### Step 6: Calculate Power Across Each Resistor
Using the formula for power (P = V^2/R), we can find the power consumed by each resistor.

**For R1 (10Ω):**
\[ P_1 = \frac{V_R^2}{R_1} = \frac{\left(\frac{240}{23}\right)^2}{10} = \frac{57600}{5290} \, W \]

**For R2 (20Ω):**
\[ P_2 = \frac{V_R^2}{R_2} = \frac{\left(\frac{240}{23}\right)^2}{20} = \frac{57600}{10580} \, W \]

### Step 7: Find the Ratio of Power Consumed
To find the ratio of the power consumed by the two resistors:
\[ \frac{P_1}{P_2} = \frac{R_2}{R_1} = \frac{20}{10} = 2:1 \]

### Conclusion
The power consumed by the two resistors is in the ratio of 2:1.