The `n` rows each containing `m` cells in series are joined parallel. Maximum current is taken from this combination across an external resistance of `3 Omega` resistance. If the total number of cells used are 24 and internal resistance of each cell is `0.5 Omega` then

Options

A. `m = 8, n = 3`
B. `m = 6, n = 4`
C. `m = 12, n = 2`
D. `m = -2, n = 12`

Explanation

To solve the problem step by step, let's break down the information given and apply the relevant formulas.

### Step 1: Understand the configuration of cells
We have `n` rows and `m` cells in each row, which means the total number of cells is given by:
\[ n \times m = 24 \]

### Step 2: Identify the internal resistance
The internal resistance of each cell is given as \( r = 0.5 \, \Omega \).

### Step 3: Calculate the total internal resistance
When cells are arranged in parallel, the effective internal resistance \( R_{internal} \) can be calculated using the formula:
\[ R_{internal} = \frac{r}{n} \]
Where \( r \) is the internal resistance of each cell and \( n \) is the number of rows.

### Step 4: Set up the equation for maximum current
According to the problem, maximum current is taken across an external resistance \( R_{external} = 3 \, \Omega \). The condition for maximum current in mixed grouping of cells is:
\[ R_{external} = R_{internal} \]
This leads to the equation:
\[ 3 = \frac{0.5m}{n} \]

### Step 5: Substitute \( m \) from the total number of cells equation
From the total number of cells equation \( n \times m = 24 \), we can express \( m \) in terms of \( n \):
\[ m = \frac{24}{n} \]

### Step 6: Substitute \( m \) into the maximum current equation
Substituting \( m \) into the equation \( 3 = \frac{0.5m}{n} \):
\[ 3 = \frac{0.5 \times \frac{24}{n}}{n} \]
\[ 3 = \frac{12}{n^2} \]

### Step 7: Solve for \( n \)
Rearranging gives:
\[ 3n^2 = 12 \]
\[ n^2 = 4 \]
\[ n = 2 \]

### Step 8: Find \( m \)
Now substituting \( n = 2 \) back into the equation for \( m \):
\[ m = \frac{24}{n} = \frac{24}{2} = 12 \]

### Step 9: Verify the values
We have \( n = 2 \) and \( m = 12 \). Let's verify:
- Total cells: \( n \times m = 2 \times 12 = 24 \) (Correct)
- Internal resistance: \( R_{internal} = \frac{0.5 \times 12}{2} = 3 \, \Omega \) (Matches \( R_{external} \))

### Final Answer
The values of \( n \) and \( m \) are:
- \( n = 2 \)
- \( m = 12 \)