OCR A Physics A-level
Topic 4.3: Electrical circuits
Notes
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Series and parallel circuits
Kirchhoff’s second law
Kirchhoff’s second law states in any circuit the sum of the electromotive force is equal to the
sum of the potential difference in a closed loop. This is a result of the conservation of energy.
In a series circuit, the current at every point in the circuit will be the same. The e.m.f. will be
split so that the total p.d. across each component is equal to the e.m.f. As a result the total
resistance of a circuit containing resistors in series will be the equal to the sum of the resistance
of each component, Rt = R1 + R2 + …
In a parallel circuit, the current in each loop adds up to the total current. The p.d. across each
loop is the same. This results in the total resistance of the circuit being given by the formula
1 1 1
= + +⋯
𝑅𝑅𝑡𝑡 𝑅𝑅1 𝑅𝑅2
where R 1 and R 2 are the total resistance values for each loop within the circuit. Within each loop
in the circuit, the current is the same at each point, and the p.d. is split across the components in
the loop.
Internal resistance
Internal resistance and lost volts
A source of e.m.f has some internal resistance. This means that not all of the energy transferred
to the charge carriers is available to the circuit, as some is transferred to the internal resistance of
the cell. This results in a difference between the measured p.d. across the terminals of the power
supply, and the actual e.m.f. of the cell, which is referred to as the ‘lost volts’, and is equal to the
p.d. across the internal resistor.
The internal resistance can be modelled as the source of e.m.f. 𝜀𝜀 being in series with its internal
resistance r. For a circuit with this power source attached to a resistor of resistance R, we can use
the equation
𝜺𝜺 = 𝑰𝑰(𝑹𝑹 + 𝒓𝒓)
to model the circuit. IR is equal to V, the terminal p.d. of the cell, and Ir is equal to the lost volts.
In A-level exam questions, you may be told the internal resistance of the cell as either a value or
as an unknown value to determine. If the question does not mention internal resistance, you can
assume that it is negligible and can be ignored in calculations.
Techniques to determine internal resistance
To determine the internal resistance of a cell, the cell with internal resistance r is connected in
series to an ammeter and a variable resistor. A voltmeter is connected in parallel around the cell.
The resistance of the variable resistor is varied, and the V and I readings recorded. The equation
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𝜀𝜀 = 𝑉𝑉 + 𝐼𝐼𝐼𝐼 is rearranged to 𝑉𝑉 = 𝜀𝜀 − 𝐼𝐼𝐼𝐼. When a graph of terminal p.d. (V) against current (I) is
plotted, the y intercept of the graph will be the e.m.f. of the cell and the negative gradient will
be the internal resistance of the cell.
Potential dividers
Potential divider circuits
From Kirchhoff’s second law, we know that in a series circuit the current is constant, and the p.d.
splits in a ratio proportional to the resistance of each component.
𝑉𝑉1 𝑅𝑅1
=
𝑉𝑉2 𝑅𝑅2
where V 1 and R 1 are the p.d. and resistance for the first component, and V 2 and R 2 are the p.d.
and resistance for the second component. We can rearrange this to get
𝑉𝑉1 𝑉𝑉2
=
𝑅𝑅1 𝑅𝑅2
By considering the ratio of the resistance for one component with the total circuit resistance, we
can determine the potential difference across this component, using the formula
𝑅𝑅2
𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 = × 𝑉𝑉𝑖𝑖𝑖𝑖
𝑅𝑅1 + 𝑅𝑅2
Where V in is the e.m.f. of the circuit, and V out is the potential difference across R 2 .
Potential divider circuits are set up with a potential difference distributed across 2 resistors, with
one of the resistors being connected to another circuit which uses the V out potential difference.
One of the resistors is typically a sensor or variable resistor, so that the resistance can be
altered. When the resistance of the V out resistor is decreased in proportion to the total circuit
resistance, the p.d. across this resistor, V out , will also decrease.
Techniques to investigate potential divider circuits
A thermistor or LDR can be used to vary the output of a potential divider circuit (such as in
street lamps or air conditioning systems). When V out is across a thermistor, a greater p.d. will be
provided when the temperature decreases, because this increases the resistance of the thermistor.
These circuits can be investigated using water baths and lamps to simulate different
environments, with a voltmeter across the desired component used to record the changes in V out
as the environmental factors affecting the resistance of the component are varied.
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