Four identical cells of emf `epsilon` and internal resistance r are to be connected in series. Suppose, if one of the cell is connected wrongly, then the equivalent emf and effective internal resistance of the combination is

Options

A. 2E and 4r
B. 4E and 4r
C. 2E and 2r
D. 4E and 2r

Answer

To solve the problem, we need to determine the equivalent EMF and effective internal resistance when four identical cells, each with an EMF of \( \epsilon \) and an internal resistance of \( r \), are connected in series, with one cell connected in the reverse direction.

### Step-by-Step Solution:

1. **Identify the Configuration**:
We have four cells connected in series. Three cells are connected in the correct orientation, while one cell is connected in the reverse direction.

2. **Calculate the Equivalent EMF**:
- The EMF of each of the three correctly connected cells is \( \epsilon \).
- The EMF of the incorrectly connected cell is \( -\epsilon \) (since it opposes the current).
- Therefore, the total equivalent EMF \( E_{eq} \) can be calculated as:
\[ E_{eq} = \epsilon + \epsilon + \epsilon - \epsilon = 3\epsilon - \epsilon = 2\epsilon \]

3. **Calculate the Effective Internal Resistance**:
- The internal resistance of each cell is \( r \).
- When cells are connected in series, their internal resistances add up.
- Thus, the total internal resistance \( R_{eq} \) is:
\[ R_{eq} = r + r + r + r = 4r \]

4. **Final Result**:
- The equivalent EMF of the combination is \( 2\epsilon \).
- The effective internal resistance of the combination is \( 4r \).

### Summary:
- **Equivalent EMF**: \( 2\epsilon \)
- **Effective Internal Resistance**: \( 4r \)