if the cell had been connected in parallel (instead of in series) which of the graphs would have shown the relationship between total current I and n?

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the relationship between total current \( I \) and the number of identical cells \( n \) when connected in parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
When \( n \) identical cells are connected in parallel, each cell has an electromotive force (emf) \( e \) and an internal resistance \( r \).

2. **Effective Resistance Calculation**:
In a parallel connection, the effective resistance \( R_{\text{eff}} \) of the cells can be calculated using the formula:
\[ R_{\text{eff}} = \frac{r}{n} \]
This is because the total resistance in parallel is the reciprocal of the sum of the reciprocals of individual resistances.

3. **Current Calculation**:
The total current \( I \) flowing from the battery can be calculated using Ohm's Law:
\[ I = \frac{E_{\text{eff}}}{R_{\text{eff}}} \]
In this case, since the emf of the cells in parallel remains the same, \( E_{\text{eff}} = e \). Substituting the effective resistance:
\[ I = \frac{e}{R_{\text{eff}}} = \frac{e}{\frac{r}{n}} = \frac{n \cdot e}{r} \]

4. **Relationship Between Current and Number of Cells**:
From the equation \( I = \frac{n \cdot e}{r} \), we can see that the current \( I \) is directly proportional to the number of cells \( n \) (since \( e \) and \( r \) are constants). Thus:
\[ I \propto n \]

5. **Graphical Representation**:
The relationship \( I \propto n \) indicates that if we plot \( I \) on the y-axis and \( n \) on the x-axis, the graph will be a straight line passing through the origin, indicating a linear relationship.

6. **Conclusion**:
Therefore, the graph that represents the relationship between total current \( I \) and the number of cells \( n \) when connected in parallel is a straight line with a positive slope. From the given options, the first option represents this relationship.