Ten cells, each of 2 volts emf and 1 ohm internal resistance are connected in series. What is the current flowing through a resistance of 10 ohms connected across this combination of cells?
1 A
Calculating Current in a Series Combination of Cells
This problem involves calculating the total current flowing through an external resistor when multiple identical cells are connected in series. Understanding how EMFs and internal resistances combine in a series circuit is key to solving this problem.
Understanding Cells in Series
When cells are connected in series, their individual electromotive forces (EMFs) add up to give the total EMF of the combination, provided they are connected with the correct polarity (positive terminal of one cell connected to the negative terminal of the next). Similarly, their internal resistances also add up in a series combination.
Step-by-Step Calculation
1. Calculate the Total EMF of the Cells
We have 10 identical cells connected in series. Each cell has an EMF of 2 volts.
Total EMF ($E_{total}$) = Number of cells $\times$ EMF of each cell
$$E_{total} = 10 \times 2 \, \text{V} = 20 \, \text{V}$$
2. Calculate the Total Internal Resistance of the Cells
Each cell has an internal resistance of 1 ohm, and they are in series.
Total internal resistance ($r_{total}$) = Number of cells $\times$ Internal resistance of each cell
$$r_{total} = 10 \times 1 \, \Omega = 10 \, \Omega$$
3. Calculate the Total Resistance in the Circuit
The circuit consists of the external resistance connected across the series combination of cells. The total resistance of the circuit is the sum of the external resistance and the total internal resistance.
External resistance ($R$) = 10 ohms
Total resistance ($R_{circuit}$) = External resistance ($R$) + Total internal resistance ($r_{total}$)
$$R_{circuit} = R + r_{total} = 10 \, \Omega + 10 \, \Omega = 20 \, \Omega$$
4. Calculate the Current Flowing Through the Circuit
According to Ohm's Law, the current ($I$) flowing through a circuit is given by the total voltage (EMF) divided by the total resistance.
Current ($I$) = Total EMF ($E_{total}$) / Total resistance ($R_{circuit}$)
$$I = \frac{E_{total}}{R_{circuit}} = \frac{20 \, \text{V}}{20 \, \Omega} = 1 \, \text{A}$$
The current flowing through the resistance of 10 ohms connected across this combination of cells is 1 A.
Verification with Options
The calculated current is 1 A. Let's check the given options:
- Option 1: 1 A
- Option 2: 100 mA ($100 \times 10^{-3}$ A = 0.1 A)
- Option 3: 1 mA ($1 \times 10^{-3}$ A = 0.001 A)
- Option 4: 10 mA ($10 \times 10^{-3}$ A = 0.01 A)
Our calculated value of 1 A matches Option 1.
Revision Table: Key Concepts
| Concept | Description | Formula (for Series) |
|---|---|---|
| EMF of a Cell | Electromotive Force, the potential difference across the terminals of a cell when no current is drawn. | N/A (Individual) |
| Internal Resistance ($r$) | Resistance offered by the electrolyte and electrodes of a cell to the flow of current within the cell. | N/A (Individual) |
| Cells in Series | Connecting cells end-to-end such that the positive terminal of one connects to the negative terminal of the next. | Total EMF ($E_{total}$) = $\sum E_i$ Total internal resistance ($r_{total}$) = $\sum r_i$ |
| Ohm's Law (for a circuit with internal resistance) | Relates the current, total EMF, and total resistance in a circuit. | $I = \frac{E_{total}}{R_{external} + r_{total}}$ |
Additional Information: Cells in Parallel
While this problem uses cells in series, it's useful to know about parallel connections as well.
- When identical cells are connected in parallel, the total EMF remains the same as the EMF of a single cell. This is useful for providing more current capacity.
- The total internal resistance decreases in a parallel combination. If 'n' identical cells with internal resistance 'r' are in parallel, the total internal resistance is $r/n$.
- Connecting cells in parallel is generally done to increase the current capacity or lifespan of the battery under a given load, not to increase the voltage.
Understanding both series and parallel combinations of cells is crucial for analyzing more complex circuits involving multiple power sources.
Important Questions from Combination of Resistors — Series and Parallel
A cell of negligible resistance and e.m.f 2 volt is connected to series combination of 2 ohm, 3 ohm and 5 ohm. The potential difference across the 3 ohm resistance is:
3 resistors of 3 ohm each connected in series. What is the mean values of resistors?
Three resistor, each equal to 3 Ω are connected so as to form a triangle. The equivalent resistance between any two vertices of the triangle is:
- Two resistors R 1 and R 2 arranged in parallel combination in an electrical closed circuit are made of the same material and of the same thickness. If the length of R 2 is twice the length of R 1, then the total resistance R satisfies
Consider the following part of an electric circuit:
The total electrical resistance in the given part of the electric circuit is
