Electric Circuits

Table of Contents
1. Potential difference and electromotive force (emf)
2. Measuring voltage: the voltmeter
3. Electric current
4. Measuring current: the ammeter
5. Activity suggestions
6. Worked examples
7. Resistance
8. Resistors in series
9. Resistors in parallel
10. Practical experiments and observations
11. Common components and symbols
12. Instrument connection summary
13. Exercises
14. Summary
15. Units and physical quantities
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Potential difference and electromotive force (emf)

When an electric circuit is complete, charges move through the circuit only if a force drives them. This force must do work on the charges to make them move. In most simple circuits the force is provided by a battery or cell. A battery converts chemical potential energy into electrical energy and can do work on charges to drive them around a closed circuit.

Definition - Potential difference

The potential difference (voltage) between two points is the work done per unit charge to move a small positive test charge between those points. It is denoted by V and has the SI unit volt (V), where 1 V = 1 J·C^-1.

Electromotive force (emf) and terminal potential difference

The electromotive force (emf) of a battery is the maximum work per unit charge the battery can supply to move charge from one terminal to the other when no current is drawn (open circuit). The measured potential difference across the battery terminals when no current flows equals the emf.

When the battery is connected in a complete circuit and current flows, the potential difference across its terminals is generally smaller than the emf because some work is done inside the battery to move charges through its internal structure. The terminal potential difference V_terminal is related to the emf E and the battery's internal resistance r by

V_terminal = E - I·r

where I is the current leaving the battery. The difference E - V_terminal equals the work per unit charge done inside the battery (or the voltage drop across the internal resistance).

Measuring voltage: the voltmeter

A voltmeter measures the potential difference between two points in a circuit. A voltmeter is always connected in parallel with the component across which the potential difference is to be measured. Ideal voltmeters have very high resistance so they draw negligible current from the circuit, minimising disturbance of the circuit conditions.

Electric current

Definition - Electric current

Electric current I is the rate at which electric charge Q passes a fixed point in a circuit. It is measured in amperes (A), where 1 A = 1 C·s^-1. The relationship is

I = Q / t

Conventional current is defined as the flow of positive charge; by this convention it flows from the positive terminal of a battery, through the circuit, to the negative terminal. In metallic conductors the actual charge carriers are electrons, which move in the opposite direction to conventional current.

Flow of charge and signal propagation

Although individual charges drift slowly, when a battery is connected the electric field propagates through the conductor at a substantial fraction of the speed of light, so the effect (current starting to flow) appears almost instantaneous throughout the circuit. A useful analogy is pushing a line of marbles in a tube: pushing at one end causes a marble to emerge at the other end without each marble having to travel the full length.

Measuring current: the ammeter

An ammeter measures the rate of flow of charge in a circuit. An ammeter must be connected in series with the component whose current is to be measured. Ideal ammeters have very low resistance so they do not significantly change the current they are measuring.

Activity suggestions

Construct simple circuits using batteries, resistors, switches, connecting leads, an ammeter and a voltmeter. Use the ammeter in series to measure current through a component and the voltmeter in parallel to measure potential difference across it. If a meter has multiple ranges, always start with the largest range to avoid damaging the instrument.

Worked examples

Resistance

Definition - Resistance

Resistance R is a measure of how strongly a component opposes the flow of electric charge. Its SI unit is the ohm (Ω). One ohm is one volt per ampere (1 Ω = 1 V·A^-1).

Resistance arises microscopically because moving charge carriers (electrons) collide with atoms and other imperfections in a conductor, transferring energy and producing heating.

Physical factors affecting resistance

The resistance of a uniform conductor depends on:

• Length L: resistance increases with length. Doubling L doubles R.
• Cross-sectional area A: resistance decreases as area increases. Doubling A halves R.
• Material: different materials have different resistivities ρ; a greater resistivity gives greater resistance.

For a uniform conductor of length L and cross-sectional area A,

R = ρ · L / A

where ρ (rho) is the electrical resistivity of the material. Materials with very low resistivity can be superconducting at sufficiently low temperatures, in which case R = 0 for practical purposes.

Resistors in series

When resistors are connected in series:

• There is only a single path for current; the current is the same through every series element.
• The total potential difference across the series combination equals the sum of potential differences across each resistor: V_total = V₁ + V₂ + ...
• The total resistance is the sum of the resistances:

R_S = R₁ + R₂ + ...

Resistors in parallel

When resistors are connected in parallel:

• There are multiple paths for current; the total current splits among the branches.
• The potential difference across each parallel branch is the same: V_battery = V₁ = V₂ = ...
• The total resistance decreases as branches are added. The reciprocal of the total resistance is the sum of reciprocals of the individual resistances:

1 / R_P = 1 / R₁ + 1 / R₂ + ...

For two resistors in parallel this simplifies to

R_P = (R₁·R₂) / (R₁ + R₂).

Practical experiments and observations

Suggested experiments include measuring how voltages and currents change as you add resistors in series and parallel. Typical observations:

• In series circuits the current is the same everywhere and the battery voltage is divided among the resistors (voltage divider).
• In parallel circuits the voltage across each branch is the same and currents divide among branches (current divider); the total current equals the sum of branch currents.
• Adding resistors in series increases total resistance and decreases current; adding resistors in parallel decreases total resistance and increases total current (for the same applied voltage).

Common components and symbols

Instrument connection summary

Exercises

1. What is the unit of resistance called and what is its symbol?
2. Explain what happens to the total resistance of a circuit when resistors are added in series.
3. Explain what happens to the total resistance of a circuit when resistors are added in parallel.
4. Why do batteries go flat?

Summary

• The potential difference across the terminals of a battery when it is not delivering current is the emf (measured in volts).
• The potential difference across the terminals when current flows is the terminal potential difference; it is less than or equal to the emf because of internal battery effects.
• Voltage measures work done per unit charge (J·C^-1), current measures charge flow per unit time (C·s^-1).
• Conventional current flows from the positive terminal to the negative terminal.
• Ammeters measure current and are connected in series; voltmeters measure potential difference and are connected in parallel.
• Resistance (Ω) measures opposition to current; R = ρ·L/A for a uniform conductor.
• In series circuits the current is the same through all components and the total resistance is the sum of individual resistances.
• In parallel circuits the voltage across each branch is the same and the reciprocal of total resistance equals the sum of reciprocals of branch resistances; for two resistors R_P = R₁R₂/(R₁ + R₂).

Units and physical quantities