Derivation of terminal voltage
Part (a): Terminal Voltage Across the Cell
Consider a cell with emf and internal resistance , connected in series with a variable resistor . The total resistance in the circuit is . The current in the circuit is given by Ohmβs law applied to the entire circuit:
The terminal voltage across the cell is the voltage across the resistor :
Alternatively, the terminal voltage can be found by considering the potential drop across the internal resistance:
Substitute :
This confirms our expression for the terminal voltage. % Analyze the behavior of terminal voltage
Now, analyze the behavior of as varies from 0 to a very large value:
- When :
The terminal voltage is zero because the cell is short-circuited, and the entire emf is dropped across the internal resistance.
- When :
- As :
The terminal voltage approaches the emf , as the current becomes very small, and the voltage drop across the internal resistance ( ) becomes negligible. % Describe the graphical variation for terminal voltage
The graph of versus starts at when , increases rapidly at first, then more gradually, and asymptotically approaches as becomes very large. The curve is a hyperbolic growth shape, reflecting the form of the equation . % Derivation of current
Part (b): Current Supplied by the Cell
The current supplied by the cell is the same as the current in the circuit: % Analyze the behavior of current
Analyze the behavior of as varies:
- When :
This is the maximum current, corresponding to a short circuit.
- When :
- As :
The current approaches zero as the total resistance becomes very large. % Describe the graphical variation for current
The graph of versus starts at when , decreases rapidly at first, then more slowly, and asymptotically approaches as becomes very large. The curve is a hyperbolic decay shape, reflecting the form of the equation .