To get maximum current through a resistance of `2.5Omega,` one can use m rows of cells, each row having n cells. The internal resistance of each cell is `0.5 Omega` what are the values of n and m, if the total number of cells is 45.

Options

A. 3,15
B. 5,6
C. 9,5
D. 15,3

Explanation

To solve the problem, we need to find the values of \( n \) (the number of cells in each row) and \( m \) (the number of rows) such that the total number of cells is 45, and we achieve maximum current through a resistance of \( 2.5 \, \Omega \) with each cell having an internal resistance of \( 0.5 \, \Omega \).

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- Each row has \( n \) cells.
- The total number of cells is given as \( 45 \), so we can write the equation:
\[ n \times m = 45 \]

2. **Calculating EMF and Internal Resistance**:
- The EMF of each row is \( nE \) (where \( E \) is the EMF of a single cell).
- The internal resistance of each row is \( 0.5n \).
- If \( m \) rows are connected in parallel, the total internal resistance \( R_{internal} \) becomes:
\[ R_{internal} = \frac{0.5n}{m} \]

3. **Total Resistance in the Circuit**:
- The total resistance \( R_{total} \) in the circuit is the sum of the external resistance \( R_L \) and the internal resistance:
\[ R_{total} = R_L + R_{internal} = 2.5 + \frac{0.5n}{m} \]

4. **Current Expression**:
- The current \( I \) through the circuit can be expressed as:
\[ I = \frac{E_{total}}{R_{total}} = \frac{nE}{2.5 + \frac{0.5n}{m}} \]

5. **Substituting for \( m \)**:
- From the equation \( n \times m = 45 \), we can express \( m \) as:
\[ m = \frac{45}{n} \]
- Substitute \( m \) into the current equation:
\[ I = \frac{nE}{2.5 + \frac{0.5n}{\frac{45}{n}}} = \frac{nE}{2.5 + \frac{0.5n^2}{45}} = \frac{nE}{2.5 + \frac{n^2}{90}} \]

6. **Maximizing Current**:
- To find the value of \( n \) that maximizes the current, we differentiate \( I \) with respect to \( n \) and set the derivative to zero:
\[ \frac{dI}{dn} = 0 \]
- After differentiating and simplifying, we arrive at:
\[ 2.5 = \frac{n^2}{90} \]
- Rearranging gives:
\[ n^2 = 2.5 \times 90 = 225 \]
- Taking the square root:
\[ n = 15 \]

7. **Finding \( m \)**:
- Substitute \( n \) back into the equation for \( m \):
\[ m = \frac{45}{n} = \frac{45}{15} = 3 \]

### Final Answer:
Thus, the values of \( n \) and \( m \) are:
\[ n = 15, \quad m = 3 \]