A cell of emf \( E \) and internal resistance \( r \) is connected to an external variable resistance \( R \). Plot a graph showing the variation of terminal voltage \( V \) of the cell as a function of current \( I \), supplied by the cell. Explain how the emf of the cell and its internal resistance can be found from it.
Show Hint
To determine the emf and internal resistance of a cell, measure the terminal voltage at different currents and plot \( V \) versus \( I \). The slope and the intercept of the graph give the internal resistance and emf, respectively.
The terminal voltage \( V \) of a cell is related to its emf \( E \) and internal resistance \( r \) by the following equation:
\[
V = E - I r
\]
where:
- \( V \) is the terminal voltage,
- \( E \) is the emf of the cell,
- \( r \) is the internal resistance,
- \( I \) is the current supplied by the cell.
As the external resistance \( R \) is varied, the current \( I \) supplied by the cell changes, and consequently, the terminal voltage \( V \) also changes.
Thus, the terminal voltage is a linear function of the current, where the slope of the graph represents the negative value of the internal resistance \( -r \), and the intercept on the voltage axis corresponds to the emf \( E \).
Graph:
The graph of \( V \) versus \( I \) will be a straight line with the following characteristics:
1. The slope of the line is \( -r \), which is the internal resistance of the cell.
2. The \( y \)-intercept of the graph gives the emf \( E \), since when the current \( I = 0 \), the terminal voltage \( V = E \).
How to Find \( E \) and \( r \) from the Graph:
1. Emf (\( E \)): The emf of the cell can be found by looking at the \( y \)-intercept of the graph. When \( I = 0 \), the terminal voltage \( V \) is equal to the emf \( E \). So, the value of \( V \) at \( I = 0 \) gives the emf of the cell.
2. Internal Resistance (\( r \)): The slope of the graph represents \( -r \), the negative of the internal resistance. So, the magnitude of the slope gives the value of the internal resistance \( r \).
For example, from the graph above, the slope of the line is \( -2 \), meaning the internal resistance \( r = 2 \, \Omega \), and the intercept on the voltage axis is \( 6 \, \text{V} \), meaning the emf \( E = 6 \, \text{V} \).
