Combination of Cells in Series and Parallel
The study of the combination of cells in series and parallel is essential in class 12 physics, particularly in the NCERT chapter 3 on current electricity. This article is useful for various boards, including RBSE and others.
Electric Cell
"The cell is a device that can convert chemical energy into electrical energy."
(or)
" A device that is used to maintain a steady current in an electric circuit is called a cell or electrolytic cell. It has two electrodes one is positive (P) and other is negative (N)."
Types of Electric Cells
There are two types of cells
`\star` Primary cell
`\star` Secondary cell
EMF of a Cell or Electromotive Force
It is the maximum potential difference between two terminals of the circuit when the circuit is open.
EMF of the cell is
`W = \frac{W}{q}`
EMF of a cell is also known as electromotive force.
Internal Resistance of a Cell
The internal resistance (r) of a cell is defined as the resistance offered by the electrolyte (An electrolyte is a substance that produces ions when dissolved in water, enabling the conduction of electrical current) of the cell to the flow of current through it. It is denoted by r. Its SI unit is ohm.
Terminal Potential Difference
The maximum potential difference between two terminals of the circuit when the circuit is closed is known as the terminal voltage of the terminal potential difference (V) of the cell.
Relation between r, R, E and V
`r = (\frac {E - V}{V}) R`
Where,
r = Internal resistance,
R = External resistance,
E = EMF of cell
V = Terminal voltage of cell.
Combination of Cells
There are the following types of combinations of cells.
✯ Series Combination
✯ Parallel Combination
Series Combination of cells
When the negative terminal of one cell is connected to the positive terminal of the next cell, the cells are said to be connected in series. In this combination, endpoints are of opposite poles so that these points can connect external resistance.
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| Series Combination of Cells Diagram |
Consider two cells of emf `\epsilon_1` and `\epsilon_2` and internal resistance `r_1` and `r_2` are connected between the endpoints of external resistance R.
From Ohm's law the potential difference across resistance, R is
`V = V_A - V_C`
`R I = V_A - V_C`
The potential difference between A and B
`V_{AC} = (V_A - V_B) + (V_B - V_C)`
`V_{AC} = (\epsilon_1 - I r_1) + (\epsilon_2 - I r_2)` .....eqn (1)
`V_{AC} = \epsilon_1 - I r_1 + \epsilon_2 - I r_2`
`V_{AC} = \epsilon_1 - I r_1 + \epsilon_2 - I r_2`
`V_{AC} = \epsilon_1 + \epsilon_2 - I ( r_1 + r_2)` .....eqn (2)
If the equivalent e.m.f. of combination is `\epsilon_{eq}` and the equivalent resistance is `r_{eq}` then.
`V_{AC} = \epsilon_{eq} + I r_{eq}` .....eqn (3)
From eqn. 1 and eqn. 2
`\epsilon_{eq} = \epsilon_1 + \epsilon_2` .....eqn (4)
`r_{eq} = r_1 + r_2` .....eqn (5)
From eqn 1
`V_{AC} = (\epsilon_1 - I r_1) + (\epsilon_2 - I r_2)`
`R I = (\epsilon_1 - I r_1) + (\epsilon_2 - I r_2)`
`R I = \epsilon_1 - I r_1 + \epsilon_2 - I r_2`
`R I + I r_1 + I r_2 = \epsilon_1 + \epsilon_2 `
`R I + I r_1 + I r_2 = \epsilon_1 + \epsilon_2 `
`I (R + r_1 + r_2) = \epsilon_1 + \epsilon_2 `
`I = \frac {\epsilon_1 + \epsilon_2}{R + r_1 + r_2} `
`I = \frac {\epsilon_{eq}}{R + r_{eq}} `
Conclusion
✯ This formula can be used for n-cells
`I = \frac {\epsilon_1 + \epsilon_2 + .......+ \epsilon_n}{R + r_1 + r_2 + .........+ r_n} `
`I = \frac {\epsilon_{eq}}{R + r_{eq}} `
Thus, The equivalent emf of a series combination of n cells is equal to the sum of their individual emf.
`\epsilon_{eq} = \epsilon_1 + \epsilon_2 + ....... + \epsilon_n `
Same as Internal resistance
`r_{eq} = r_1 + r_2 + .......... + r_n`
✯ If n cells are connected having equal emfs `(\epsilon_1 = \epsilon_2 = .....)` and equal internal resistance `(r_1 = r_2 ....... )` then
`\epsilon_{eq} = \epsilon + \epsilon + ....... = \epsilon ` `{\because \epsilon_1 = \epsilon_2 = ..... = \epsilon}`
`\epsilon_{eq} = n \epsilon` and
`r_{eq} = r + r + .......... + r` `{r_1 = r_2 = .......... = r}`
`r_{eq} = n r`
Thus,
`I = \frac {\epsilon_{eq}}{R + r_{eq}} `
`I = \frac {n \epsilon}{R + n r} `
Parallel Combination of cells
If the positive terminals of all cells are connected to one point and all their negative terminals to another point, the cells are said to be connected in parallel.
.webp "Parallel Combination of Cells") |
| Parallel Combination of Cells |
For first cell `V = \epsilon_1 - I_1 r_1`
` I_1 = \frac {\epsilon_1 - V}{r_1}`
For second cell `V = \epsilon_2 - I_2 r_2`
` I_1 = \frac {\epsilon_2 - V}{r_2}`
At point A
`I = I_1 + I_2`
` I = \frac {\epsilon_1 - V}{r_1} + \frac{\epsilon_2 - V }{r_2}`
` I = \frac {\epsilon_1}{r_1} - \frac{V}{r_1} + \frac{\epsilon_2}{r_2} - \frac{V }{r_2}`
` I = \frac {\epsilon_1}{r_1} + \frac{\epsilon_2}{r_2} - \frac{V}{r_1} - \frac{V }{r_2}`
` I = \frac {\epsilon_1}{r_1} + \frac{\epsilon_2}{r_2} - V(\frac{1}{r_1} + \frac{1 }{r_2})`
` I = (\frac {\epsilon_1 r_1 + \epsilon_2 r_2}{r_1 r_2}) - V(\frac{r_1 + r_2}{r_1 r_2} )` ....... eqn (1)
` V(\frac {r_1 + r_2}{r_1 r_2}) = \frac {\epsilon_1 r_1 + \epsilon_2 r_2}{r_1 r_2} - I `
` V = \frac{r_1 r_2}{r_1 + r_2}( \frac {\epsilon_1 r_1 + \epsilon_2 r_2}{r_1 r_2} - I) `
` V = ( \frac {\epsilon_1 r_1 + \epsilon_2 r_2}{r_1 + r_2}) - I (\frac{r_1 r_2}{r_1 + r_2}) ` ...... eqn (2)
If the equivalent emf is `\epsilon_{eq}` and equivalent internal resistance is `r_{eq}` then
`V = \epsilon_{eq} - I r_{eq}`
Important Facts
Combination of Cells in Series
`\star` When cells are connected in series, their voltages add up. If you have n cells, each with an emf of `\epsilin`, the total emf is given by `\epsilon_{eq} = n \epsilon`.
`\star` The current flowing through each cell in a series combination is the same.
`\star` The total internal resistance of cells in series is the sum of their individual internal resistances. If each cell has an internal resistance r, then `r_{eq} = n r`.
`\star` Application of series combinations is that series combinations are used when higher voltage is required, such as in flashlight batteries and other high-voltage devices.
Combination of Cells in Parallel
`\star` The total current supplied by the parallel combination is the sum of the currents supplied by each cell.
`\star` Parallel combinations are used when higher current capacity is needed, such as in power banks and other devices requiring a stable voltage with increased capacity.
Current Electricity MCQs
Q. What will be the grouping of the cells when the current in the circuit is ne/(R + nr)?
Q. Calculate the number of dry dells, each of emf 2V and internal resistance 1V that is joined in series with a resistance of 30 ohms so that a current of 0.8A passes through it.
Q. There are 4 resistors, each with a resistance of 4 ohms. First, they are connected in series with a cell that has an internal resistance of 2 ohms. After that, they are connected in parallel to the same cell. Find the ratio of the currents in both two cases.
Q. 36 cells, each of emf 4V are connected in series combination and kept in a box. The combination shows an emf of 88 V on the outside. Calculate the number of cells reversed.
Related Question
1. What is parallel and series combination of cells?
2. How do you derive an expression for a parallel combination of cells?
3. Explain the various methods of combination of dells.
4. Derive an expression combination of cells for equivalent emf and equivalent internal resistance in series.
5. How cells are combined in parallel. Derive the expression current flowing in the external circuit. When is this combination useful?
6. How are cells combined in series? Derive the expression for the current flowing through the external circuit. When is this combination useful?
7. Define internal resistance and describe its impact on the performance of a cell.
8. What are the practical applications of using a combination of cells in series and parallel in electrical devices?
9. What is the function of a cell in an electric circuit?
10. Differentiate between a primary cell and a secondary cell.
11. What is the formula for equivalent emf of cells in parallel?
12. What is the formula for equivalent emf of cells in series?
13. Explain the benefits of using cells in series and parallel arrangements.
14. How does the combination of cell in parallel affect the overall voltage output compared to a single cell?
15. What is electromotive force of a cell?
16. Define emf and internal resistance of a cell.
17. What is electrolyte medium?
18. Write one difference between potential difference and emf.
19. What is potential difference in electricity?
Chapter 3: CURRENT ELECTRICITY
PHYSICS NOTES
- Electric Current Definition and Electric Currents in Conductors
- Ohm’s Law: Definition, Formula, Limitations, Derivation, Diagram, and Deduction of Ohm's Law
- Drift of Electrons and the Origin of Resistivity
- Resistivity, Resistivity of Various Materials, Temperature Dependence of Resistivity
- Mobility Formula in terms of Relaxation Time
- Electrical Energy, Power
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