Two batteries with emf 12 V and 13V are connected in parallel across a load resistor of `10Omega` . The internal resistances of the two batteries are `1Omega and 2Omega` respectively . The voltage across the load lies between

Options

A. 11.7 V and 11.8 V
B. 11.6 V and 11.7 V
C. 11.5 V and 11.6 V
D. 11.4 V and 11.5 V

Answer

To solve the problem of finding the voltage across the load resistor when two batteries are connected in parallel, we will follow these steps:

### Step 1: Identify the Given Values
- EMF of Battery 1 (E1) = 12 V
- Internal Resistance of Battery 1 (R1) = 1 Ω
- EMF of Battery 2 (E2) = 13 V
- Internal Resistance of Battery 2 (R2) = 2 Ω
- Load Resistance (R_load) = 10 Ω

### Step 2: Calculate the Equivalent Internal Resistance
When batteries are connected in parallel, the equivalent internal resistance (R_eq) can be calculated using the formula:
\[ R_{eq} = \frac{R_1 \cdot R_2}{R_1 + R_2} \]
Substituting the values:
\[ R_{eq} = \frac{1 \cdot 2}{1 + 2} = \frac{2}{3} \, \Omega \]

### Step 3: Calculate the Equivalent EMF
The equivalent EMF (E_eq) for batteries connected in parallel with different EMFs can be calculated using the formula:
\[ E_{eq} = \frac{E_1}{R_1} + \frac{E_2}{R_2} \]
We need to find the total current flowing through the circuit, which is given by:
\[ I = \frac{E_{eq}}{R_{eq} + R_{load}} \]
To find E_eq, we first calculate the contributions of each battery:
\[ E_{eq} = \frac{12}{1} + \frac{13}{2} \cdot \frac{2}{3} \]
Calculating this gives:
\[ E_{eq} = 12 + 6.5 = 18.5 \, V \]

### Step 4: Calculate the Total Resistance in the Circuit
The total resistance in the circuit (R_total) is the sum of the equivalent internal resistance and the load resistance:
\[ R_{total} = R_{eq} + R_{load} = \frac{2}{3} + 10 = \frac{2}{3} + \frac{30}{3} = \frac{32}{3} \, \Omega \]

### Step 5: Calculate the Current in the Circuit
Using Ohm's Law, we can find the current (I) flowing through the circuit:
\[ I = \frac{E_{eq}}{R_{total}} = \frac{18.5}{\frac{32}{3}} = \frac{18.5 \cdot 3}{32} = \frac{55.5}{32} \approx 1.734375 \, A \]

### Step 6: Calculate the Voltage Across the Load Resistor
The voltage (V_load) across the load resistor can be calculated using Ohm's Law:
\[ V_{load} = I \cdot R_{load} = 1.734375 \cdot 10 \approx 17.34375 \, V \]

### Step 7: Determine the Range of Voltage Across the Load
Since the voltage across the load lies between the voltages provided by the two batteries, we can conclude that:
- The voltage across the load will be between the lower voltage (from the weaker battery) and the higher voltage (from the stronger battery).

### Final Answer
The voltage across the load lies between approximately **11.57 V** and **12.33 V**.