Parallel Combination of Cells: Effective EMF & Internal Resistance

TITLE: Parallel Combination of Cells: Effective EMF & Internal Resistance DESCRIPTION: Learn how to calculate the effective EMF and internal resistance when multiple cells are connected in parallel, a key concept in current electricity for JEE.

Concept Overview

This question tests the understanding of how to combine multiple cells connected in parallel to find an equivalent single cell. Specifically, it focuses on deriving the formulas for the effective electromotive force (EMF) and the effective internal resistance of such a parallel combination. This is crucial for analyzing complex circuits involving multiple power sources.

Step 1: Consider two cells connected in parallel. Let's consider two cells with EMFs $E_1$ and $E_2$, and internal resistances $r_1$ and $r_2$, respectively. They are connected in parallel, and this combination is then connected to an external resistor $R$.

Step 2: Analyze the potential difference across the parallel combination. When cells are connected in parallel, the potential difference across their terminals is the same. Let this potential difference be $V$. For the first cell, the terminal voltage is given by $V = E_1 - I_1 r_1$, where $I_1$ is the current flowing through the first cell. Similarly, for the second cell, $V = E_2 - I_2 r_2$.

Step 3: Express currents in terms of EMF, internal resistance, and terminal voltage. From the equations in Step 2, we can express the currents $I_1$ and $I_2$ as:

Step 4: Apply Kirchhoff's Current Law at the junction. Let $I$ be the total current flowing out of the parallel combination. According to Kirchhoff's Current Law, the sum of currents entering a junction equals the sum of currents leaving it. In this case, the total current $I$ is the sum of the currents from each cell: $I = I_1 + I_2$.

Step 5: Substitute the expressions for $I_1$ and $I_2$ into the total current equation. Substitute the expressions from Step 3 into the equation from Step 4:

Step 6: Rearrange the equation to solve for $V$. Let's rearrange the equation to group terms with $V$:

Now, isolate $V$:

Step 7: Compare with the formula for a single equivalent cell. An equivalent single cell with EMF $E_{eq}$ and internal resistance $r_{eq}$ connected to an external resistor $R$ would have a terminal voltage $V = E_{eq} - I r_{eq}$. Comparing this with the equation derived in Step 6, we can identify the effective EMF and effective internal resistance.

Step 8: Generalize for $n$ cells in parallel. For $n$ cells connected in parallel with EMFs $E_1, E_2, \dots, E_n$ and internal resistances $r_1, r_2, \dots, r_n$, the effective EMF and internal resistance are given by:

This can be rewritten as:

The formula for $r_{eq}$ is analogous to resistors connected in parallel.

Key Takeaways:

• When cells are connected in parallel, the potential difference across each cell is the same.
• The effective internal resistance of cells in parallel is calculated similarly to resistors in parallel: the reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances.
• The effective EMF is a weighted average of individual EMFs, weighted by the inverse of their internal resistances.
• If all cells have the same EMF ($E$) and internal resistance ($r$), then $E_{eq} = E$ and $r_{eq} = r/n$, where $n$ is the number of cells.

Answer: The effective EMF ($E_{eq}$) and internal resistance ($r_{eq}$) for $n$ cells connected in parallel are given by: