Twenty four cells each of emf 1.5 V and internal resistance 0.5 ohms are to be connected to a 3 ohm resistance. For maximum current through this resistance the nuber of rows and number of columns that you connect these cells is.

Options

A. 12 cells in series 2 ros in parallel
B. 8 cells in series 3 rows in parallel
C. 4 cells in series 6 rows in parallel
D. 6 cells in series 4 rows in parallel

Explanation

To solve the problem of connecting 24 cells each with an emf of 1.5 V and an internal resistance of 0.5 ohms to a 3-ohm external resistance for maximum current, we will follow these steps:

### Step 1: Understand the Configuration
We need to determine how to arrange the 24 cells in terms of rows (series) and columns (parallel) to maximize the current through the external resistance.

### Step 2: Define Variables
Let:
- N = number of cells in series (rows)
- M = number of parallel paths (columns)

### Step 3: Calculate Effective Resistance
The effective resistance (R_eff) when N cells are connected in series and M paths are in parallel is given by:
\[ R_{\text{eff}} = \frac{N \cdot r}{M} \]
where \( r \) is the internal resistance of each cell (0.5 ohms).

### Step 4: Set Up the Equation for Maximum Current
For maximum current, the external resistance (R) should equal the effective resistance:
\[ R = R_{\text{eff}} \]
Substituting the values:
\[ 3 = \frac{N \cdot 0.5}{M} \]

### Step 5: Rearranging the Equation
From the equation above, we can express N in terms of M:
\[ N = 6M \]

### Step 6: Total Number of Cells
Since there are a total of 24 cells, we have the equation:
\[ N \cdot M = 24 \]

### Step 7: Substitute N in the Total Cells Equation
Substituting \( N = 6M \) into the total cells equation:
\[ 6M \cdot M = 24 \]
This simplifies to:
\[ 6M^2 = 24 \]
\[ M^2 = 4 \]
\[ M = 2 \]

### Step 8: Calculate N
Now, substituting M back to find N:
\[ N = 6M = 6 \cdot 2 = 12 \]

### Conclusion
Thus, the number of cells connected in series (N) is 12, and the number of parallel paths (M) is 2. Therefore, the cells should be connected as follows:
- **Number of rows (series)**: 12
- **Number of columns (parallel)**: 2

### Final Answer
The number of rows and columns to connect the cells for maximum current through the 3-ohm resistance is:
- **Rows (N)**: 12
- **Columns (M)**: 2

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