Electric Circuit

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 Ohm’s 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 .

Step 1: List the given data.
- Cell A: , .
- Cell B: , .
- Cell C: , .
The cells are in parallel, so they share the same terminal voltage . We need to find the currents , , and . Step 2: Set up the current equations.
The current through each cell is: Since no external load is mentioned, apply Kirchhoff’s Current Law (KCL) at the junction (net current is zero): Step 3: Solve for .
Step 4: Calculate the currents.
- Cell A: - Cell B: - Cell C: Step 5: Interpret the results.
- : Negative, so Cell A is being charged.
- : Negative, so Cell B is being charged.
- : Positive, so Cell C supplies current. Step 6: Verify.
The sum is zero, confirming the solution. % Final Answer Final Answer:
The currents are: