Cells and batteries (both sources of electricity) have resistance, this resistance is called internal resistance.
Internal resistance is caused by the materials that make up the battery and this resistance increases over time. However, for simplicity in calculations it is often assumed to remain constant.
Internal resistance is usually denoted by a lowercase r.
Electromotive Force
Due to the internal resistance of a battery the electromotive force of a battery is less than the potential difference that that would be measured by a voltmeter placed across its terminals.
The below circuit shows a cell with a resistor, r (connected in series). This resistor represents the internal resistance of the cell. A second, external resistor (R) is also connected in series.
In a series circuit the total resistance can be calculated by adding together the resistances of the individual components. So for this circuit the total resistance would be:
Using Ohm’s Law (V=IR) the value of the electromotive force can be calculated using the following equation:
Where ε is the electromotive force measured in volts (V), I is the current measured in amps (A) and R is the resistance measured in Ohm’s (Ω).
IMPORTANT NOTE: This equation can only be used for series circuits.
This equation can also be written as:
Here IR is equal to the potential difference across the terminals and Ir is equal to the potential difference that is lost due to the internal resistance. The energy is usually lost as heat energy.
Due to the conservation of energy the electromotive force is always equal to the sum of the terminal potential difference and the potential difference that is lost.
Efficiency and Internal Resistance
As the potential difference difference measured across a battery is lower than the electromotive force this means that the internal resistance impacts the efficiency of the battery.
The greater the difference between the potential difference and electromotive force the less efficient the battery is.
The efficiency can be calculated using the following equation:
Where p.d. is the potential difference across the terminals and e.m.f. is the electromotive force of the battery. These are both measured in volts (V).
Worked Examples
Example 1
The internal resistance of a battery is 1.5Ω. The current supplied to an external resistor is 2A. If the external resistor has a resistance of 10Ω calculate the e.m.f. of the battery.
Example 2
A battery has an e.m.f. of 10V and supplies a current of 0.5A to an external resistor with a resistance of 12Ω. Calculate the internal resistance of the battery.
Example 3
A voltmeter measure the potential difference across a battery as 18V. The internal resistance of the battery is 5Ω and it supplies a current of 1A to an external resistor. The external resistor has a resistance of 15Ω.Calculate the efficiency of the battery.
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