3.0 A current passing through Two batteries of different emf and internal resistances connected in series with each other and with an external load resistor. The current reversed, the current becomes 1.0 A. the ratio of the emf of the two batteries is

Options

A. 2.5
B. 2
C. 1.5
D. 1

Answer

To solve the problem step by step, we will analyze the circuit with the two batteries and derive the relationship between their EMFs based on the given currents.

### Step 1: Understand the Circuit Configuration
We have two batteries connected in series with an external load resistor. The first battery has an EMF of \( E_1 \) and an internal resistance of \( r_1 \), and the second battery has an EMF of \( E_2 \) and an internal resistance of \( r_2 \).

### Step 2: Write the Current Equation for the Initial Configuration
When the batteries are aligned in the same direction, the total EMF is \( E_1 + E_2 \). The total resistance in the circuit is \( R + r_1 + r_2 \), where \( R \) is the external load resistance. The current \( I_1 \) in this configuration is given by:
\[ I_1 = \frac{E_1 + E_2}{R + r_1 + r_2} \]
Given that \( I_1 = 3 \, \text{A} \), we can write:
\[ 3 = \frac{E_1 + E_2}{R + r_1 + r_2} \quad \text{(1)} \]

### Step 3: Write the Current Equation for the Reversed Configuration
When the current is reversed, the batteries are connected in opposite directions. Assuming \( E_1 > E_2 \), the total EMF in this case is \( E_1 - E_2 \). The current \( I_2 \) is given by:
\[ I_2 = \frac{E_1 - E_2}{R + r_1 + r_2} \]
Given that \( I_2 = 1 \, \text{A} \), we can write:
\[ 1 = \frac{E_1 - E_2}{R + r_1 + r_2} \quad \text{(2)} \]

### Step 4: Divide the Two Equations
Now, we will divide equation (1) by equation (2):
\[ \frac{I_1}{I_2} = \frac{E_1 + E_2}{E_1 - E_2} \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ \frac{3}{1} = \frac{E_1 + E_2}{E_1 - E_2} \]
This simplifies to:
\[ 3 = \frac{E_1 + E_2}{E_1 - E_2} \]

### Step 5: Cross-Multiply and Rearrange
Cross-multiplying gives:
\[ 3(E_1 - E_2) = E_1 + E_2 \]
Expanding and rearranging:
\[ 3E_1 - 3E_2 = E_1 + E_2 \]
\[ 3E_1 - E_1 = 3E_2 + E_2 \]
\[ 2E_1 = 4E_2 \]
Dividing both sides by 2:
\[ E_1 = 2E_2 \]

### Step 6: Determine the Ratio of the EMFs
From the equation \( E_1 = 2E_2 \), we can express the ratio of the EMFs:
\[ \frac{E_1}{E_2} = \frac{2}{1} \]

### Final Answer
The ratio of the EMF of the two batteries is \( 2:1 \).

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