The Real World
So far in in the National 5 and Higher Physics courses, we have assumed that all electrical components are perfect. This is obviously not the case!
A clear example of this can be seen by observing the Voltage across a battery before and during operation:-
In the above circuit diagrams, there are two different Voltages shown. When the switch is open, the voltmeter across the cell reads 1.5 V, but when the switch is closed and a current flows, the reading drops to 1.48 V.
Why does allowing a Current to flow reduce the Voltage of a cell?
In order to explain this, we must take into account what is happening within the cell.
How a Cell (Battery) Works
A cell consists of two electrodes separated by an electrolyte. An electrolyte is a chemical that reacts with the electrodes to produce electrical Energy. When a load is attached to the cell, the circuit is completed and electrons can flow through the cell.
The Energy the Electrons need to move is provided by the chemical reactions that occur at the electrodes:-
1. At the Anode (+ve electrode) - An oxidation reaction occurs, creating a new compound within the cell and releasing Electrons.
2. At the Cathode (-ve Electrode) - A reduction reaction occurs, creating a new compound within the cell, absorbing Electrons.
These two processes cause a Current to flow through the circuit.
Note - In a rechargeable cell, these reactions are reversible. by forcing a current to flow in the opposite direction, the newly formed compounds can be split up into the original chemicals again, ready to start the process over.
Lost Voltage
The function of a battery above relies on two chemical reactions to produce a flow of Electrons. These reactions are exothermic, meaning that heat Energy is given out by the cell as the Current flows. This energy is lost from the cell as waste heat.
The more Current that flows through the cell, the more Energy that is lost as heat.
As we have seen earlier, Voltage is defined as the Energy per Coulomb, so if Energy is lost within the cell, less Voltage is available for the external circuit.
This missing Energy per Coulomb is known as the Lost Voltage.
E.M.F. , T.P.D and Lost Volts
In the above diagram, when the cell was not connected, the Voltage across the cell was measured to be 1.50 V.
When the cell is not connected, no current flows, so no Energy is lost as heat. This means that the Voltage measured is the maximum Voltage the cell can provide, which is called the Electromotive Force (E.M.F.).
The E.M.F. value is the Voltage of an ideal cell, a cell within which no energy is lost internally. It is the E.M.F. value that we have used in all calculations within Physics up until this point.
When a Current flows, however, we can now see that some energy is lost due to the Internal Resistance within the cell, so only a smaller amount is available for the external components of the circuit. The remaining available Voltage is defined as the Terminal Potential Difference, and is actual value measured across the cell's terminals when a current is flowing.
Summary of Internal Resistance Terms :-
1. Electromotive Force - The Energy supplied per Coulomb by the cell. It is measured as the Voltage of the cell when no current flows and has the units Volts.
2. Internal Resistance - The Resistance due to the chemicals within the cell, a small amount of Energy per Coulomb will be lost moving the charges through this resistance.
3. Lost Volts - The Potential Difference "lost" inside the cell due to Internal Resistance.
4. Terminal Potential Difference - The value displayed on a Voltmeter connected to the terminals of the cell. It is equal to the value of the E.M.F. minus the Lost Volts.
Internal Resistance as a Circuit Diagram
The diagram below shows a visual representation of the Internal Resistance within the above circuit:-
In the above diagram, the red dotted box shows what is within the cell.
A real cell can be thought of as a source of E.M.F. in series with an internal resistor. This internal resistor is then simply treated as any other series resistor in a circuit (see below).
Internal Resistance Calculations
The Electromotive Force (E.M.F.) was defined above as the Voltage measured when no Current flows within the circuit and the Terminal Potential Difference (T.P.D.) was defined above as the Voltage measured when a Current does flow.
By equating the two values above the following formula can be shown:-
E = Vtpd + Vlost
Where :-
E = Electromotive Force
Vtpd = Terminal Potential Difference
Vlost = Lost Volts
If Ohm's Law is applied using the values of the Resistances within the circuit, it can be shown that:-
Vtpd = I x R
Vlost = I x r
By combining these with the above equation, it can be shown that :-
E = IR + Ir
Where:-
E = Electromotive Force (V)
I = Current within circuit (A)
R = Resistance of external circuit (Ω)
r = Internal Resistance of the cell (Ω)
Example 1 -
A Resistor circuit is set up as shown:-
For the above Circuit, calculate :-
1. The Internal Resistance of the cell.
2. The reading on the Voltmeter.
3. The Terminal Potential Difference.
The Internal Resistance:-
E = IR + Ir
r = (E/I) - R
r = (1.5/0.15) - 9
r = 1 Ω
The reading on the Voltmeter:-
V = I x R
V = 0.15 x 9
V = 1.35 V
The Terminal Potential Difference:-
As there is only one load component in this circuit, the reading on the Voltmeter is equal to the T.P.D.
Vtpd = 1.35 V
Note - The T.P.D. could also have been found by rearranging E = Vtpd + Ir .
Internal Resistance by Graph Analysis
The E.M.F. and Internal Resistance can also be found experimentally using the following circuit:-
By varying the Resistance of the load resistor, a set of Voltages and Current values can be found. Below is a sample data set:-
When this data is plotted, the following graph is obtained:-
The graph above can be used to identify calculate:-
1. The E.M.F. value.
2. The Internal Resistance.
3. The Short Circuit Current.
The E.M.F. Value :-
The Electromotive Force is defined as the Voltage across the cell when no Current flows. This means that the E.M.F. is equal to the Y intercept.
In the above graph, the Y intercept, and therefore the E.M.F. value is 4.85 V
The Internal Resistance:-
As the above graph is a Voltage against Current, by applying Ohm's Law it can be shown
that the Internal Resistance is equal to the magnitude of the gradient.
In the above graph, the magnitude of the gradient and therefore the Internal Resistance is 2.25 Ω.
The Short Circuit Current:-
There is a maximum value of Current that a cell can provide, known as the Short Circuit Current. This occurs when there is no external resistance in the circuit, for example if a very thick wire is attached to both ends of the cell. If there is no external Resistance, the T.P.D. will be zero. This means that the Short Circuit Current is equal to the X intercept.
In the above graph, the X intercept and therefore the Short Circuit Current value is 2.16 A.
Note - The Short Circuit Current can also be found using the formula E = 0 + Ir and rearranging to solve for I.
