Introduction
Recent decades have seen a huge expansion in consumer electronics. While the functions of new devices differ widely, their power still largely comes from either batteries or mains power supplies.
In the simple circuits met so far, we have pictured power supplies as being able to provide as much current as the components require. In reality, this is not the case.
Recent decades have seen a huge expansion in consumer electronics. While the functions of new devices differ widely, their power still largely comes from either batteries or mains power supplies.
In the simple circuits met so far, we have pictured power supplies as being able to provide as much current as the components require. In reality, this is not the case.
EMF basics
The
energy
A system has energy when it has the capacity to do work. The scientific unit of energy is the joule.energy provided by any
power
The power of system is a measurement of the rate at which energy is transferred from one form to another. The scientific unit
of power is the watt.power supply, such as a battery, to each
coulomb
The coulomb is the unit of charge. One coulomb is approximately equivalent to the charge carried by 6.25 × 10^{18} electrons.coulomb of charge leaving a battery or
power
The power of system is a measurement of the rate at which energy is transferred from one form to another. The scientific unit
of power is the watt.power supply is called the
electromotive force
The electromotive force (often abbreviated to EMF) of a battery is a measurement of its capacity to provide energy to the
charges.electromotive force or
EMF
EMF is the common abbreviation for the electromotive force – a measurement of a battery's capacity to provide energy to the
charges.EMF. The electromotive force is measured in units of joules per
coulomb
The coulomb is the unit of charge. One coulomb is approximately equivalent to the charge carried by 6.25 × 10^{18} electrons.coulomb (JC^{−1}) or volts (V).Click on the figure below to interact with the model.
Turn on the circuit in Fig.1 by clicking on the switch. Then connect the voltmeters across the buzzer and lamp by dragging them into the spaces in the circuit. Note the values for the p.d.s across the two components.
You should have discovered one example of a general rule which states that the sum of the battery EMFs is always equal to the sum of the p.d.s across the components. This can be summarized in the equation below:
Confirm that the sum of the battery EMFs is always equal to the sum of the p.d.s across the components by changing the values of the battery voltages in Fig.1. To do this, click on the battery voltages and alter their values in the range 0–10 V.
Internal resistance
In Fig.1, the buzzer and the lamp were attached to the terminals of the batteries. They could just as easily have been connected
to the output terminals of a power supply. When any set of components is connected across the terminals of a battery or a
power supply like this, they are collectively called the external circuit. This is to distinguish such components from the internal circuit within the power supply. The
resistance
The opposition to the flow of current provided by a circuit is called resistance. Resistance is measured in units called Ohms.resistance of the
external circuit
All the components connected across the terminals of a battery or power supply are collectively known as the external circuit.external circuit is often referred to as the load resistance.In Fig.1 the load resistance is the only resistance in the circuit. However, this is not the case in the circuit in Fig.2 below.
Click on the figure below to interact with the model.
Again, move the voltmeters into place to measure the p.d. across the two components.
We can explain the 'lost volts' in Fig.2 by the fact that the LED and the lamp are not the only sources of resistance in the circuit. In fact, the battery also has a small internal resistance.
Any battery or power supply with an
internal resistance
All batteries or power supplies have internal resistance. This resistance has the effect of reducing the output p.d. as the
current supplied increases.internal resistance can be pictured as shown in Fig.3. Here, the internal resistance of 2 Ω is shown as a resistor in series with the power source.Internal resistance is a feature of all practical power supplies. Different designs of battery will have different internal resistances, and the manufacturers will often take steps to minimize the internal resistance.
The 12 V output from a car battery will have the same output
voltage
The voltage across a component is the electrical energy transferred by 1 coulomb of charge passing through the component.voltage as a 12 V domestic battery. However, the car battery needs to have a much lower internal resistance and the design needed
to achieve this results in the car battery being larger and heavier. The voltmeter in the Fig.4 below shows the 'lost volts', even though in practice this is a quantity that could not be measured experimentally.
Click on the figure below to interact with the model.
Set the variable resistor in the circuit of Fig.4 to 10 Ω. Close the switch and note the values of the p.d.s across the internal resistance and the load resistor.
The p.d. actually available at the battery terminals is called the terminal p.d. If the EMF of the battery is E, the
terminal p.d.
The terminal p.d. is the voltage available at the terminals of a battery or power supply. The terminal p.d. varies depending
on how much current is being drawn.terminal p.d. is V_{tpd} and the p.d. across the internal resistance is V_{lost}, we can say:The above equation holds, no matter what the value of the load resistor or the
current
The rate of flow of charge past any specific point in a circuit. The base unit of current is the Ampere.current drawn from the supply. We can verify this using Fig.4. Set the variable resistor to a value of 60 Ω.Check this again for another value of the variable resistor.
Effects of internal resistance
Fig.5 below shows two identical circuits allowing different numbers of lamps to be connected to the batteries. Circuit B
has internal resistance in the battery, whereas Circuit A shows the hypothetical case where there is no internal resistance.Click on the figure below to interact with the model.
Complete the following table for Circuit A.

Now complete the following table for Circuit B.

Fig.5 demonstrates clearly that, when a battery has internal resistance, increasing the current taken from the battery reduces the voltage available to the external circuit. This is why manufacturers of power supplies and batteries often try to reduce the internal resistance of their products.
Determining the maximum current
When a power supply or battery has internal resistance, increasing the current provided to an external circuit reduces the
p.d. across this circuit. As the current taken from the supply increases, the current through the internal resistance increases
and so more of the p.d. will be across the internal resistance. Consequently less p.d. will be available for the external
circuit.
We can see this idea mathematically by using the equation from above:where E is the EMF, V_{tpd} is the p.d. across the terminals of the battery, and V_{lost} is the 'lost' voltage.
Since the internal resistance r and the load resistor R are in series the same current, I, passes through both resistors. 
Therefore, 
As I increases, Ir increases and, since E is constant, V_{tpd} must decrease. 
So as the current I drawn from a battery increases, the p.d. available at its output (V_{tpd}) decreases. 
If the current becomes sufficiently large, the output p.d. will fall to zero. The value of current for which this happens is called I_{max}.
Click on the figure below to interact with the model.
The battery in Fig.6 above has an internal resistance (which is not shown on the circuit diagram). Use the variable resistor to alter the current drawn from the battery and observe how the current varies.
When the resistance across the battery is zero, we say that the battery is shortcircuited. Maximum current flows when the battery is shortcircuited.
Determining the EMF
We have seen what happens when the maximum current is obtained from a power supply. Alternatively, if no current is drawn
from the supply, the output voltage V_{tpd} and the EMF E of the battery are equal.We can again show this idea mathematically by reusing the equation from above:
If no current is drawn from the supply I = 0 so Ir = 0 
Therefore E = V_{tpd}. 
The EMF of the supply is equal to the output p.d. when no current is being supplied.
Click on the figure below to interact with the model.
The battery in Fig.7 above has an internal resistance (which is not shown on the circuit diagram). Use the variable resistor to alter the current drawn from the battery and observe how the p.d. across the bulb varies.
Click on the figure below to interact with the model.
Close the switch to see how the reading on the voltmeter in Fig.8 changes.
Determining the internal resistance
In Fig.9 a variable resistor is connected to a battery. We will use this circuit to determine the EMF, the maximum current,
and the internal resistance of the battery. Click on the figure below to interact with the model.
Adjust the variable resistor to alter the current passing through the circuit in Fig.9 above.
Set the variable resistor in Fig.9 to a range of different positions and enter values for the terminal p.d. V_{tpd}, and the current I into the table of Fig.10.
The line in the graph of Fig.10 is given by the equation E = V_{tpd} + Ir. We can use this equation to explain the shape of the line.
As I increases, Ir increases. 
Since E is constant, V_{tpd} must decrease. 
When the current increases and becomes equal to I_{max}, then the output p.d. will have fallen to zero. 
The current reaches I_{max} when the resistance across the output of the battery is zero. 
As we saw previously, this is the current that flows through a short circuit. 
On the graph, the value of I_{max} is given by the point at which the line cuts the xaxis. By extending the line in Fig.10 to where it cuts the xaxis, we can find the value for I_{max}.
We have already seen that the EMF is the largest voltage available from the battery, and that this occurs when I = 0. On the graph, this is illustrated by the point where the line cuts the graph's yaxis.
Now that we know the values of the EMF and the shortcircuit current I_{max} we can calculate the internal resistance of the battery. The steps below lead through the analysis of how to do this.
We know that 
We also know that when the current is equal to I_{max}, the output p.d. is zero, V_{tpd} = 0 
We can rearrange this to find an expression for the shortcircuit current. 
We can rearrange once more to find an expression for the internal resistance r. 
Alternatively we can use the data in a different way to find the internal resistance of the power source. The line in the graph of Fig.10 is given by the equation:
Click on the V_{tpd} term to rearrange the equation. It is now in the standard form 'y = mx + c' where V_{tpd} is the variable plotted on the yaxis and I the variable plotted on the xaxis. The intercept gives the EMF of the supply and the negative of the gradient indicates its internal resistance.
The graph above shows the graph of terminal p.d. plotted against current for a power supply.
Power in circuits
The experimental data generated from Fig.10 shows the different currents passing through a resistor for different values
of the p.d. across it. We can multiply these values to get the power dissipated in the resistor at different currents. Similarly
we can use
Ohm's Law
Ohm's Law states that at constant temperature the current in a conductor is directly proportional to the d.p. across the conductor.
The constant of proportionality is called the resistance of the sample.Ohm's Law to get the values of the resistance connected to the battery by simply dividing the p.d. readings by the current readings.A graph of power delivered versus load resistance shows that the power delivered to the external circuit is not constant. The power delivered rises to a maximum when the internal and external resistances are equal.
Summary
All the components connected across the terminals of a battery or power supply are collectively known as the external circuit. The total resistance of these components is known as the load resistance.
All batteries or power supplies have internal resistance. This resistance has the effect of reducing the output p.d. as the current supplied increases.
The output p.d. V_{tpd} for any current I can be found using the equation:
where E is the EMF of the supply and r is the internal resistance of the supply.
If the EMF of the power supply and the shortcircuit current I_{max} are known, the internal resistance of the supply can be determined using the equation below:
All the components connected across the terminals of a battery or power supply are collectively known as the external circuit. The total resistance of these components is known as the load resistance.
All batteries or power supplies have internal resistance. This resistance has the effect of reducing the output p.d. as the current supplied increases.
The output p.d. V_{tpd} for any current I can be found using the equation:
where E is the EMF of the supply and r is the internal resistance of the supply.
If the EMF of the power supply and the shortcircuit current I_{max} are known, the internal resistance of the supply can be determined using the equation below:
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