D.C. circuits

Syllabus points

• Recall and use the circuit symbols and draw circuit diagrams.
• State and apply Kirchhoff's first and second laws.
• Derive the formulae for resistors in series and in parallel.
• Understand the effects of the internal resistance of a source of e.m.f.
• Explain and use the potential divider as a source of variable p.d.

Kirchhoff's laws

First law: the sum of the currents into a junction equals the sum of the currents out. This is conservation of charge.

Second law: around any closed loop, the sum of the e.m.f.s equals the sum of the p.d.s. This is conservation of energy.

Naming the conserved quantity is usually worth a mark, and the two are easy to swap under pressure. Charge for the junction rule; energy for the loop rule.

Combining resistors

In series: the same current passes through each, and the p.d.s add.

R = R₁ + R₂ + R₃

In parallel: the same p.d. is across each, and the currents add.

1/R = 1/R₁ + 1/R₂ + 1/R₃

Two checks worth running on any answer:

• A series combination is always larger than the largest single resistor.
• A parallel combination is always smaller than the smallest single resistor.

If your answer breaks either rule, it is wrong, and this catches the frequent slip of forgetting to invert at the end of the parallel calculation.

For two resistors in parallel, R = R₁R₂ / (R₁ + R₂) is quicker. For n identical resistors of resistance R in parallel the combination is R/n.

Worked example. What is the resistance of four identical resistors R connected in parallel, compared with the same four in series?

Parallel: R/4. Series: 4R. The ratio is 1:16.

Internal resistance

A real cell has resistance of its own, and the energy it dissipates internally is not available to the circuit.

E = I(R + r) = V + Ir

where E is the e.m.f., r the internal resistance, R the external resistance and V the terminal potential difference.

So V = E − Ir. The terminal p.d. is always less than the e.m.f. when current flows, and the difference Ir is called the lost volts.

Three consequences that questions rely on:

• With no current drawn, V = E. A voltmeter of very high resistance across a cell therefore reads the e.m.f.
• The larger the current, the lower the terminal p.d. This is why car headlights dim when the starter motor turns.
• On a short circuit, R = 0, so the current is E/r, the maximum the cell can deliver.

Worked example. A cell of e.m.f. 1.5 V and internal resistance 0.50 Ω is connected to a 2.5 Ω resistor. Find the current and the terminal p.d.

I = E / (R + r) = 1.5 / 3.0 = 0.50 A.

V = E − Ir = 1.5 − 0.50 × 0.50 = 1.25 V.

Plotting V against I gives a straight line of gradient −r with intercept E, and reading r off as a positive gradient is a common error.

The maximum power is delivered to the external resistance when R = r, which is worth knowing even though the derivation is beyond AS.

Potential dividers

Two resistors in series across a supply divide the p.d. in the ratio of their resistances:

V_out = V_in × R₂ / (R₁ + R₂)

The output is taken across R₂. Getting the wrong resistor on the top of the fraction gives the complement of the right answer, so identify which resistor the output is measured across before writing anything.

Worked example. A 12 V supply is across a 4.0 kΩ and an 8.0 kΩ resistor in series. What is the p.d. across the 8.0 kΩ?

V = 12 × 8.0 / 12.0 = 8.0 V.

The larger resistance takes the larger share, which is the sanity check to run.

Sensing circuits

Replace one resistor with a thermistor or an LDR and the output responds to the environment.

The reasoning for these questions is always the same three steps, and it is worth doing them explicitly:

1. Decide what happens to the resistance of the sensor.
2. Decide what happens to its share of the total resistance.
3. Decide what happens to V_out.

For example, a thermistor as the upper resistor with the output across the fixed lower one. As the temperature rises, the thermistor's resistance falls, so it takes a smaller share of the supply, so the p.d. across the fixed resistor rises.

Swap them over and the conclusion reverses. So a question showing four circuits and asking in which one V rises with temperature is asking you to run those three steps on each, and the answer turns on which component the output is across.

A potentiometer is a potential divider with a sliding contact, giving a continuously variable output from zero to the full supply p.d.

Ammeters and voltmeters

• An ammeter is connected in series and should have zero resistance, so that it does not reduce the current it is measuring.
• A voltmeter is connected in parallel and should have infinite resistance, so that it draws no current from the branch it is across.

A real voltmeter of finite resistance connected across one resistor of a potential divider draws current and lowers the reading below the value calculated. Questions ask why a measured value is lower than the predicted one, and this is the answer.

Common mistakes

• Attributing Kirchhoff's first law to conservation of energy and the second to charge.
• Forgetting to invert 1/R at the end of a parallel calculation.
• Giving a parallel combination larger than one of its resistors.
• Saying the terminal p.d. equals the e.m.f. while current flows.
• Reading the internal resistance as a positive gradient from a V against I graph.
• Putting the wrong resistance on top in the potential divider equation.
• Saying an ammeter should have a high resistance.
• Ignoring the current a real voltmeter draws when explaining a low reading.