Module 03 — Electricity & Fields
Duration: 3–4 weeks | Prereq: Modules 00–02
Electricity is arguably the most practically consequential area of physics. Fields are among the most conceptually deep. This module covers both — circuits first (concrete, calculable), then field theory (more abstract, but the gateway to electromagnetism, Maxwell's equations, and ultimately special relativity and quantum mechanics).
Part A: Electric Circuits
A1. Charge, Current, and Voltage
Electric charge
The fundamental quantity. Measured in Coulombs (C).
- Electrons carry a charge of −1.6 × 10⁻¹⁹ C (the elementary charge, e)
- Protons carry +1.6 × 10⁻¹⁹ C
- Like charges repel, opposite charges attract
Current
Electric current (I) is the rate of flow of charge:
I = ΔQ / Δt (current = charge / time)
Units: Amperes (A) = Coulombs per second (C/s)
Conventional current flows from positive to negative terminal (the direction a positive charge would move). Electrons actually flow the other way — from negative to positive. This historical convention causes occasional confusion but doesn't affect calculations.
Potential difference (voltage)
Potential difference (V) between two points is the energy transferred per unit charge flowing between them:
V = W / Q (voltage = energy / charge)
Units: Volts (V) = Joules per Coulomb (J/C)
Analogy: voltage is like water pressure difference; current is like flow rate. High pressure (voltage) drives high flow (current).
Electromotive force (EMF)
EMF (ε) is the energy given to each coulomb of charge by a source (battery, generator). Not technically a force despite the name.
V = ε − Ir where r is the battery's internal resistance.
A2. Resistance and Ohm's Law
Resistance (R) opposes the flow of current. Units: Ohms (Ω)
Ohm's Law: V = IR
This is the fundamental relationship in circuit analysis. Rearranged:
- I = V/R (higher resistance → less current for same voltage)
- R = V/I (resistance is measured by this ratio)
Ohm's Law only holds for ohmic conductors — materials where resistance stays constant regardless of current or voltage. Metals at constant temperature are approximately ohmic. Filament bulbs, diodes, and thermistors are non-ohmic.
Factors affecting resistance
For a uniform conductor: R = ρL / A
Where:
- ρ (rho) = resistivity of the material (Ω m) — intrinsic property of the material
- L = length of the conductor (m)
- A = cross-sectional area (m²)
Longer wire = more resistance. Thicker wire = less resistance. Makes intuitive sense.
Power in circuits
P = IV (power = current × voltage)
Combined with V = IR:
- P = I²R (useful when you know current and resistance)
- P = V²/R (useful when you know voltage and resistance)
A3. Series and Parallel Circuits
This is where a lot of circuit problems live.
Series circuits
Components connected end-to-end in a single loop.
- Current is the same through all components
- Voltages add: V_total = V₁ + V₂ + V₃ + ...
- Resistances add: R_total = R₁ + R₂ + R₃ + ...
Parallel circuits
Components connected across the same two nodes.
- Voltage is the same across all components
- Currents add: I_total = I₁ + I₂ + I₃ + ...
- Resistances: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...
For two resistors in parallel: R_total = R₁R₂ / (R₁ + R₂) — this shortcut is worth knowing.
Key insight: Adding more resistors in parallel always decreases total resistance. Intuitively: more paths means less total obstruction to flow.
Kirchhoff's Laws
Two laws that formalise circuit analysis:
Kirchhoff's Current Law (KCL): At any junction, the sum of currents in = sum of currents out. (Conservation of charge)
Kirchhoff's Voltage Law (KVL): Around any closed loop, the sum of voltages = 0. (Conservation of energy)
A4. Capacitors
A capacitor stores electric charge. Two conducting plates separated by an insulator.
Capacitance (C): C = Q / V
Units: Farads (F). In practice, microfarads (μF) or picofarads (pF) are common.
Energy stored in a capacitor: E = ½CV² = ½QV = Q²/2C
Charging and discharging
When a capacitor charges through a resistor, it doesn't fill instantly. The charge (and voltage) rises exponentially:
Q(t) = Q_max(1 − e^(−t/RC))
Where Q_max = CV is the final maximum charge the capacitor reaches.
When it discharges:
Q(t) = Q₀ e^(−t/RC)
Where Q₀ is the initial charge at the start of discharge.
Time constant: τ = RC (seconds)
After one time constant, the capacitor has discharged to about 37% of its initial charge (or charged to 63% of maximum).
Applications: Camera flashes (store energy, release it quickly), smoothing circuits in power supplies, timing circuits, touchscreens.
A5. Alternating Current (AC)
Mains electricity uses alternating current — the voltage and current oscillate sinusoidally.
UK mains: 230 V, 50 Hz
The peak voltage is higher than the quoted (RMS) value: V_peak = V_rms × √2 ≈ 325 V for UK mains.
RMS (root mean square) values are used because they give the equivalent steady DC that would deliver the same power:
P = V_rms² / R = I_rms² × R
Part B: Fields
Fields are a profound conceptual leap. Instead of thinking about forces between objects directly, we say each object creates a field in the space around it, and other objects respond to the local field. This framework — introduced by Faraday and mathematised by Maxwell — underlies all of modern physics.
B1. Gravitational Fields
Every mass creates a gravitational field that attracts other masses.
Field strength (g): force per unit mass at a point g = F/m (units: N/kg = m/s² — same as acceleration)
On Earth's surface: g = 9.81 N/kg
Newton's Law of Gravitation
F = GMm / r²
Where:
- G = 6.67 × 10⁻¹¹ N m² kg⁻² (the gravitational constant)
- M and m are the two masses
- r is the distance between their centres
This is an inverse square law — double the distance, quarter the force.
Gravitational field strength at radius r from mass M:
g = GM / r²
Gravitational potential
Gravitational potential (V_g): the work done per unit mass by an external agent moving a test mass from infinity to that point at constant velocity (i.e., against gravity).
V_g = −GM / r (negative because gravity is attractive — you gain energy moving inward; you must do work to escape back to infinity)
Potential energy: E_p = mV_g = −GMm / r
Escape velocity: the minimum speed needed to escape a gravitational field.
At escape: KE = |PE| → ½mv² = GMm/r → v_esc = √(2GM/r)
For Earth: v_esc ≈ 11.2 km/s. This doesn't depend on the mass of the escaping object.
B2. Electric Fields
Every charge creates an electric field that exerts forces on other charges.
Field strength (E): force per unit positive charge at a point E = F/Q (units: N/C = V/m)
Coulomb's Law
F = kQ₁Q₂ / r² (or F = Q₁Q₂ / 4πε₀r²)
Where k = 8.99 × 10⁹ N m² C⁻² and ε₀ = 8.85 × 10⁻¹² F/m is the permittivity of free space.
Same form as Newton's gravity — also an inverse square law.
The symmetry is not a coincidence. Both laws share the same mathematical form because in three-dimensional space, any effect spreading out isotropically from a point source gets diluted over the surface area of a sphere (4πr²) — so intensity falls off as 1/r². This geometric argument applies to light, gravity, sound, and any other quantity that radiates equally in all directions.
Uniform electric fields
Between two parallel plates, the field is uniform (constant magnitude and direction):
E = V/d (field strength = voltage across plates / separation)
A charged particle in a uniform field experiences constant force → constant acceleration → projectile-style motion (just as in Module 01, but driven by electric force rather than gravity).
Electric potential
Analogous to gravitational potential:
V = kQ / r = Q / 4πε₀r
Work done moving charge q through potential difference ΔV: W = qΔV
B3. Magnetic Fields
Magnetic fields arise from moving charges (currents). They exert forces on other moving charges.
The key rule: The force on a moving charge in a magnetic field is perpendicular to both the velocity and the field. This is described by the cross product, but at A Level you use:
F = BQv sin θ
Where B is the magnetic flux density (Tesla, T), Q is the charge, v is the speed, and θ is the angle between v and B.
For maximum force: θ = 90° (velocity perpendicular to field) → F = BQv
Force on a current-carrying conductor:
F = BIL sin θ
Where I is current and L is the length of the conductor in the field.
This is the principle behind electric motors.
Direction of magnetic force: Fleming's Left-Hand Rule
For a conventional current in a magnetic field:
- Index finger → magnetic Field direction (B)
- Middle finger → conventional Current direction (I)
- Thumb → Motion (force direction)
Circular motion in magnetic fields
Because the magnetic force is always perpendicular to velocity, it does no work but constantly changes direction — producing circular motion.
r = mv / BQ (radius of circular path)
Application: Particle accelerators use magnetic fields to bend particle beams in circles. Mass spectrometers use this to identify particles by their mass/charge ratio.
B4. Electromagnetic Induction
Faraday's law is one of the most important results in physics — it connects changing magnetic fields to electric fields, and vice versa. This symmetry is the foundation of all electrical generation, motors, and transformers.
Faraday's Law: A changing magnetic flux induces an EMF.
EMF = −NΔΦ/Δt (induced EMF = number of turns × rate of change of magnetic flux)
For a single loop, N = 1. For a coil with N turns (as in generators and transformers), multiply by N — each turn contributes independently.
Magnetic flux (Φ): Φ = BA cos θ (flux density × area × cos of angle between field and normal to area)
Units: Webers (Wb) = T m²
Lenz's Law: The induced current always flows in a direction that opposes the change causing it. (This is conservation of energy — you can't get free electricity.)
Generators and transformers
Generator: coil rotating in a magnetic field → flux changes → EMF induced → alternating current produced.
Transformer: two coils sharing an iron core. Changing current in primary coil → changing flux → EMF induced in secondary coil.
V_s / V_p = N_s / N_p (voltage ratio = turns ratio)
I_p / I_s = N_s / N_p (for an ideal transformer, power in = power out → step-up voltage = step-down current)
Transformers only work with AC (changing flux needed). This is why mains electricity is AC — it can be stepped up to high voltage for efficient long-distance transmission, then stepped down for safe home use.
B5. Maxwell's Equations and the Big Picture
James Clerk Maxwell (1860s) unified electricity, magnetism, and optics into four equations. You don't need to use them at A Level, but understanding what they say conceptually is very much worth it:
- Gauss's Law for electricity: electric fields spread outward from charges (inverse square)
- Gauss's Law for magnetism: there are no magnetic monopoles — field lines always form closed loops
- Faraday's Law: changing magnetic fields create electric fields
- Ampère-Maxwell Law: changing electric fields create magnetic fields (and so do currents)
Laws 3 and 4 together say: a changing electric field creates a changing magnetic field, which creates a changing electric field, which... — a self-sustaining wave that travels through space. Maxwell calculated the speed of this wave: 3 × 10⁸ m/s. Exactly the speed of light. Light is an electromagnetic wave.
This was one of the greatest unifications in the history of science. Maxwell connected electricity, magnetism, optics, and radio waves as different manifestations of the same thing.
Why this matters for everything else: Einstein's special relativity was born partly from thinking carefully about what Maxwell's equations implied about the speed of light. Module 05 picks this up.
9. Self-Check Questions
- A 12 V battery drives current through a 6 Ω resistor. What is the current?
- Three resistors — 3 Ω, 6 Ω, and 9 Ω — are connected in parallel. What is the total resistance?
- A capacitor of 100 μF is charged to 20 V. How much energy does it store?
- At what distance from a 5 × 10⁻⁶ C charge is the electric field strength 4500 N/C?
- A conductor of length 0.3 m carries a current of 5 A at 90° to a magnetic field of 0.4 T. What force acts on it?
Answers:
- I = V/R = 12/6 = 2 A
- 1/R = 1/3 + 1/6 + 1/9 = 6/18 + 3/18 + 2/18 = 11/18 → R = 18/11 ≈ 1.64 Ω
- E = ½CV² = ½ × 100×10⁻⁶ × 400 = 0.02 J = 20 mJ
- E = kQ/r² → r² = kQ/E = (9×10⁹ × 5×10⁻⁶) / 4500 = 45000/4500 = 10 → r = √10 ≈ 3.16 m
- F = BIL = 0.4 × 5 × 0.3 = 0.6 N
Go Deeper
- Maxwell's Equations — the equations themselves aren't frightening to look at. A good introductory treatment: Griffiths' Introduction to Electrodynamics (the standard undergraduate text, and unusually readable for a physics textbook).
- Special Relativity and Electromagnetism — the speed of light is the same for all observers. This seemingly bizarre fact is the direct consequence of Maxwell's equations being consistent in all reference frames. Once you have Module 05 under your belt, come back to this.
- Quantum Electrodynamics (QED) — the quantum version of Maxwell's theory. Feynman's QED: The Strange Theory of Light and Matter is a popular-level book that is both accurate and completely mind-bending.
- Electromagnetism and motors — if the practical side interests you, electronics and motor theory extend naturally from this module. The Practical Electronics Handbook is a good reference.