Rub a balloon on your hair and stick it to the wall. It hangs there. Gravity, the pull of an entire planet, is trying to drop it, and a few billion stray electrons on its surface are winning. Between a proton and an electron the electric attraction is about 2.3 × 10³⁹ times stronger than their gravitational attraction. Gravity only seems strong because the Earth is enormous and its charge is balanced out.
The second surprise is quieter. When you flip a switch, the light comes on at once, yet the electrons in the copper behind the wall drift at a fraction of a millimetre per second. What travels at nearly light speed is not the electrons. It is a field. Physics II is the course where those two sentences stop being puzzling.
The library has an easier first read, How Electricity Reaches Your Wall (13/04), about the grid and Ohm's law. This piece goes further: into fields, capacitors, induction, and Maxwell's discovery that light is made of the same stuff.
Who takes it: second-term first-year or second-year students of Mechanical & Aerospace Engineering and, in many programmes, Computer Science.
What it's for: predicting what charges and currents do, from a battery to a radio wave.
Comes after: Physics I: Mechanics (34/06) and Multivariable Calculus (34/02).
Leads to: the motors, sensors and actuators of Mechanical & Aerospace Engineering, then Control Systems (34/20); for Computer Science, Computer Architecture (34/25) and Computer Networks (34/27), which are electricity at tiny and large scales.
Honest line: you won't design a power supply after this. You will know why a thin wire gets hot and a thick one doesn't, and why a transformer hums.
Matter carries electric charge, positive or negative. Like charges push apart and opposite charges pull together, with a force that falls with the square of the distance, the same shape as gravity. A charge doesn't need to "reach" another: it fills the space around it with a field, and any other charge placed there feels it.
Worked number: 1 nC (a billionth of a coulomb) makes a field of 900 V/m at 10 cm. Drag the gold probe below. Put a positive and a negative charge side by side and the field curves from one to the other.
Charges are 30 cm apart; the grid shows field direction. Equal and opposite charges: the voltage along the midline between them is exactly 0.
Lift a book and it gains energy; the same book on a higher shelf has more. Voltage is the electrical version of height: the energy a charge gains per coulomb moved. A 1.5 V battery gives every coulomb that crosses it 1.5 joules, whatever the battery's size.
That makes electrons easy to launch. Accelerate one through 100 V and it leaves at about 5.9 × 10⁶ m/s, two per cent of the speed of light. At 10,000 V its speed is 0.195 c, and the ordinary formula (½mv²) overestimates it by about 1.5%. The old cathode-ray TV worked this way, aiming such electrons at a screen.
Computed with the relativistic energy formula, and compared with the ordinary one.
Current is charge flowing past a point each second (1 ampere = 1 coulomb per second). Push a voltage across a wire and a current flows; the ratio is the resistance. For most metals it's constant, which is Ohm's law, and a wire's resistance grows with its length and shrinks with its thickness. Resistance turns electrical energy into heat at a rate P = I²R.
Now the surprise. The electrons' average drift speed is I ÷ (n·A·e), with n about 8.5 × 10²⁸ free electrons per cubic metre of copper. Ten amps through a 2.5 mm² house wire gives a drift of 0.29 mm/s. An electron takes nearly an hour to cover one metre. The lamp lights at once because the wire is already full of electrons and the field pushes them all together, like water in a pipe that is already full.
With those defaults the wire wastes 13 W as heat. Halve the thickness and the heat doubles. That is why extension cords are rated by thickness, and why power lines run at very high voltage: a given power needs a smaller current, and heat goes as current squared.
Every circuit, however tangled, obeys two bookkeeping rules from Gustav Kirchhoff. The junction rule: current flowing into a junction equals current flowing out, because charge doesn't pile up. The loop rule: going around any closed loop, the voltage rises and drops add to zero, because energy is conserved. They are conservation of charge and of energy, again.
From them follow the two shapes. Resistors in series share one current, so their resistances add. Resistors in parallel share one voltage, so their currents add, and the total resistance gets smaller as you add branches. That's why plugging more appliances into one socket pulls more current from the wall, not less.
With 100, 200 and 300 Ω: in series the battery supplies 20 mA; in parallel the total resistance falls to 54.5 Ω and the current rises to 220 mA, eleven times as much.
A capacitor is two plates close together that store charge, and with it energy in the field between them. Its capacity C is how much charge it holds per volt. Connect one through a resistor to a battery and the plates fill, but ever more slowly, because the more it holds the less voltage is left to push more in.
After one τ the capacitor is 63% full, after three τ 95%, after five 99%. A 10 kΩ resistor with a 100 µF capacitor makes τ = 1 second. Camera flashes, phone touch screens, power-supply smoothing and the blink of a turn signal all use this clock.
A 5 V supply.
A current makes a magnetic field that circles the wire. Ten amps at 5 cm makes about 40 µT, the same size as Earth's own field (about 25 to 65 µT at the surface). A magnetic field pushes only on moving charge, and sideways: perpendicular to both the motion and the field. A sideways push with constant speed is the circular motion of Physics I.
A wire carrying current in a field gets the same push: that is a motor. And a particle's lap time, 2πm/(qB), doesn't depend on its speed, which is how cyclotron accelerators work.
In 1831 Michael Faraday showed that a changing magnetic field through a loop of wire drives a voltage around it. A steady field does nothing; only change counts. Faster change, more loops, and a stronger field all give a bigger voltage.
This one rule explains three machines. Spin a coil in a field and you have a generator: the voltage rises and falls as a sine wave, peak N·B·A·ω. Run the same machine backwards and you have a motor. Wind two coils on one iron core and an alternating current in the first induces an alternating voltage in the second, with output voltage in the ratio of the turns: a transformer. It works only with alternating current, since only change induces, and it is why the grid uses AC and why a charger hums.
Coil area 10 cm × 10 cm. Ideal transformer, no losses.
At the defaults the generator gives 31.4 V peak at 10 Hz, and the transformer steps 240 V down to 12 V: 20 times fewer turns, 20 times the current.
In the 1860s James Clerk Maxwell gathered everything known into four equations. In words: (1) electric charges make electric fields that spread outward; (2) there are no isolated magnetic charges, so magnetic field lines never end; (3) a changing magnetic field makes a circulating electric field (Faraday); (4) an electric current, and, Maxwell's own addition, a changing electric field, make a circulating magnetic field.
Put (3) and (4) together and something remarkable happens: a changing electric field makes a changing magnetic field, which makes a changing electric field, and the disturbance can sustain itself and travel. Maxwell calculated its speed from two measured constants of electricity and magnetism, 1/√(μ₀ε₀), and got the measured speed of light, 299,792,458 m/s. In his 1865 paper he concluded that light is such a wave. Heinrich Hertz produced and detected radio waves in the late 1880s, and the rest of the spectrum followed.
Radio, microwaves, infrared, visible light and X-rays differ only in frequency, and the wavelength is c ÷ f. A 100 MHz FM station has a wavelength of 3 m; Wi-Fi at 2.4 GHz, 12.5 cm; red light at 700 nm, 430 THz. This also explains antenna sizes: a quarter wavelength works well, so an FM antenna is about 75 cm.
Charges make fields; a voltage is energy per charge; a current is charge per second, resisted into heat. Two rules, conservation of charge and of energy, solve any circuit. A changing magnetic field makes a voltage, which gives generators, motors and transformers, and Maxwell showed that a changing electric field and a changing magnetic field can sustain each other as a wave travelling at the speed of light. That wave is light.
Try this: look at the charger of a laptop or phone. Find the voltage and amperes printed on it, multiply them, and you have the power it can deliver. Then feel how warm the cable and the brick get, and work out which part is I²R.
Sources. Constants from the NIST CODATA 2022 listing (elementary charge 1.602 176 634 × 10⁻¹⁹ C and speed of light 299 792 458 m/s are exact; electron and proton masses; G; μ₀; ε₀). Copper's resistivity (1.68 × 10⁻⁸ Ω·m at 20 °C) from the tabulation in Wikipedia's "Electrical resistivity and conductivity"; free-electron density 8.5 × 10²⁸ m⁻³ and the drift-velocity worked example from Wikipedia's "Drift velocity". Earth's field 25–65 µT: Wikipedia's "Earth's magnetic field". Faraday's 1831 demonstration: Wikipedia's "Faraday's law of induction" (Joseph Henry independently, 1832). Maxwell's paper read to the Royal Society 8 December 1864 and published 1865: Wikipedia's "A Dynamical Theory of the Electromagnetic Field". Hertz produced and detected radio waves in 1886 and published his evidence in 1887–88: Wikipedia's "Heinrich Hertz". Computed in node and checked: electric to gravitational force ratio 2.27 × 10³⁹; drift speed 0.294 mm/s and 13.4 W; electron speeds (5.93 × 10⁶ m/s at 100 V; 0.195 c at 10 kV); 54.5 Ω and 220 mA; 63.2%, 95.0%, 99.3% RC fractions; electron orbit radius 5.7 cm and period 35.7 ns at 10⁷ m/s and 1 mT; 31.4 V peak; 1/√(μ₀ε₀) = 299,792,458 m/s to 8 digits.