The physics behind engineering and the physical sciences: motion, forces, energy, momentum, torque, circuits, induction, waves and heat, with Python exercises and a home pendulum lab.

Physics for Engineering & Science

## Why this matters

Every circuit, motor, sensor, battery and chip runs on electric charge. Electrical engineers design with voltages every day; chemists use the same ideas for electrochemistry and batteries; materials scientists study how charges move through solids. The single most useful idea in this lesson is that **voltage is energy per unit charge**. Once that clicks, batteries, circuits and power stop being mysterious.

## Electric charge

Charge comes in two kinds, positive and negative. Like charges repel; opposite charges attract. Protons carry +e and electrons carry −e, where

```text
e = 1.602 × 10⁻¹⁹ C     (coulombs)
```

A coulomb is a lot of charge: about 6.24 × 10¹⁸ electrons. Charge is **conserved**: it can move from one object to another, but the total never changes. Rub a balloon on your hair and electrons move from your hair to the balloon; the balloon becomes negative and your hair positive by the same amount.

## Coulomb's law

The force between two point charges q₁ and q₂ a distance r apart is

```text
F = k · |q₁ · q₂| / r²       k = 8.99 × 10⁹ N·m²/C²
```

Two charges of 1.0 µC (10⁻⁶ C) held 0.10 m apart push or pull with

```text
F = 8.99 × 10⁹ × (10⁻⁶ × 10⁻⁶) / 0.10² = 0.90 N
```

It is an inverse-square law, just like gravity: double the distance and the force falls to a quarter.

## The electric field

Rather than working out every pair of charges, physicists describe the effect of charges with an **electric field** E: the force per unit charge that a small test charge would feel at a point.

```text
E = F / q          units: N/C, which are the same as V/m
```

The field points in the direction a positive charge would be pushed: away from positive charges and toward negative ones. Around a single point charge, E = k·|q| / r². At 0.10 m from a 1.0 µC charge, E = 8.99 × 10⁵ N/C.

Between two large parallel metal plates with opposite charges, the field is nearly **uniform**: the same size and direction everywhere between them. That arrangement is a **capacitor**, and it appears in almost every circuit board.

## Potential energy and voltage

Lifting a mass against gravity stores potential energy. Pushing a positive charge toward another positive charge, against the electric force, stores **electric potential energy** U in the same way.

**Electric potential** V is potential energy per unit charge:

```text
V = U / q          units: volts, where 1 V = 1 J/C
```

What matters in practice is the **potential difference** ΔV between two points, usually just called the voltage. Moving a charge q through a potential difference ΔV changes its energy by

```text
ΔU = q · ΔV
```

So a 9 V battery gives every coulomb that passes through it 9 J of energy; a 1.5 V AA cell gives each coulomb 1.5 J. The battery does not supply charge (the charges are already in the wires); it supplies energy to the charges.

### The gravity analogy

| Gravity | Electricity |
|---|---|
| mass m | charge q |
| gravitational field g (N/kg) | electric field E (N/C) |
| height difference | potential difference ΔV |
| PE per kg = g·h | potential V = U / q |

Water at the top of a dam has energy per kilogram because of its height; charge at the + terminal of a battery has energy per coulomb because of its potential.

## Field and voltage in a uniform field

For two plates a distance d apart with voltage ΔV between them:

```text
E = ΔV / d
```

12 V across a 2.0 mm gap gives E = 12 / 0.002 = 6,000 V/m. This is why the field is quoted in V/m and why thin insulation layers in chips must withstand huge fields even at small voltages.

## The electron-volt

Physicists often measure tiny energies in **electron-volts** (eV): the energy an electron gains moving through 1 V.

```text
1 eV = 1.602 × 10⁻¹⁹ J
```

Chemical bonds are a few eV; visible-light photons carry about 1.7–3.1 eV; the electron beams in X-ray tubes carry tens of keV.

## Common mistakes

- **Thinking voltage "flows".** Voltage is a difference between two points; current flows (next lesson).
- **Confusing E and V.** E is a field (force per charge, a vector). V is potential (energy per charge, a scalar). In a uniform field they are linked by E = ΔV / d.
- **Dropping powers of ten.** µC is 10⁻⁶ C, nC is 10⁻⁹ C, mm is 10⁻³ m.
- **Assuming low voltage means low danger everywhere, or high voltage always kills.** Harm depends on current through the body, path and duration. Treat every mains or high-voltage source as dangerous; work on them requires training, supervision and, in many jobs, a licence.

## Quick recap

- Charge is quantised in units of e and is conserved.
- Coulomb's law: F = k·q₁·q₂ / r².
- Electric field E = F / q, in N/C or V/m.
- Voltage = potential energy per unit charge: 1 V = 1 J/C, and ΔU = q·ΔV.
- In a uniform field, E = ΔV / d.

## Check yourself

This lesson has no coding activity. Use the quiz in the **Do** tab to check that you can link voltage, energy and charge.


Explain electric charge, field and potential, and why a volt is a joule of energy per coulomb of charge.

Charge, field, and potential

Electric Fields: Crash Course Physics #26

Voltage, Electric Energy, and Capacitors: Crash Course Physics #27

## Why this matters

Voltage is the energy each charge carries; current is how many charges go past per second; resistance is how hard the material makes them work to get through. Ohm's law ties these three together and is the first calculation in almost every circuit problem, from choosing a resistor for an LED to estimating how warm a wire gets. Electrical engineers, physicists and lab scientists use it daily, often without writing it down.

## Current

**Current** I is the rate at which charge flows past a point:

```text
I = Q / t           units: amperes (A), where 1 A = 1 C/s
```

A current of 2 A moves 2 coulombs every second. Circuits often use milliamps: 20 mA = 0.020 A.

By convention, current is drawn flowing from the + terminal of a source, through the circuit, to the − terminal. (In metal wires the moving charges are actually electrons drifting the other way. The convention predates that discovery, and the maths works the same either way.)

Current only flows around a **closed loop**. Break the loop anywhere, with a switch or a broken wire, and current stops everywhere in that loop.

## Resistance and Ohm's law

**Resistance** R, measured in **ohms** (Ω), describes how much voltage it takes to push a given current through a component. For many materials (metals, carbon film resistors) at a steady temperature, current is proportional to voltage. That is **Ohm's law**:

```text
V = I · R       I = V / R       R = V / I
```

A 100 Ω resistor with 5 V across it carries I = 5 / 100 = 0.05 A = 50 mA.

Not everything obeys Ohm's law: diodes, LEDs and transistors do not, and a light bulb's filament resistance rises a lot as it heats up. For those, V = I·R still defines the resistance at one operating point, but R is not a constant.

## What sets resistance

For a uniform wire:

```text
R = ρ · L / A
```

ρ (rho) is the material's **resistivity** in Ω·m, L the length and A the cross-sectional area. Copper has ρ ≈ 1.68 × 10⁻⁸ Ω·m. So 10 m of copper wire with a 1.0 mm² (10⁻⁶ m²) cross-section has

```text
R = 1.68 × 10⁻⁸ × 10 / 10⁻⁶ = 0.168 Ω
```

Longer wire means more resistance; thicker wire means less. That is why high-current cables are thick.

## Series and parallel

Components in **series** share one path, so the same current flows through each, and their resistances add:

```text
R_series = R₁ + R₂ + R₃ + ...
```

Components in **parallel** sit across the same two points, so they share the same voltage, and the currents add. The combined resistance is smaller than the smallest branch:

```text
1 / R_parallel = 1 / R₁ + 1 / R₂ + ...
```

Two equal resistors in parallel give half the resistance of one.

## Power

A current through a resistance turns electrical energy into heat at the rate

```text
P = V · I = I² · R = V² / R        (watts)
```

Every resistor has a power rating (commonly 0.25 W for small through-hole parts). Exceed it and the part overheats.

## Worked example 1: a series circuit

A 9.0 V battery drives a 100 Ω and a 200 Ω resistor in series.

```text
total resistance   R = 100 + 200       = 300 Ω
current            I = 9.0 / 300       = 0.030 A = 30 mA
voltage across 100 Ω:   0.030 × 100     = 3.0 V
voltage across 200 Ω:   0.030 × 200     = 6.0 V    (3.0 + 6.0 = 9.0 V)
power in 100 Ω:         0.030² × 100    = 0.09 W
power in 200 Ω:         0.030² × 200    = 0.18 W
total power             9.0 × 0.030     = 0.27 W
```

The voltage divides in proportion to the resistances. That is a **voltage divider**, one of the most common sub-circuits in electronics.

## Worked example 2: the same resistors in parallel

```text
1 / R = 1/100 + 1/200 = 0.015   →   R = 66.7 Ω
I from the battery = 9.0 / 66.7 = 0.135 A
```

Each branch has the full 9.0 V: 90 mA flows through the 100 Ω branch and 45 mA through the 200 Ω branch, 135 mA in total.

## Worked example 3: an LED resistor

A red LED needs about 20 mA and drops about 2.0 V. From a 9.0 V battery, the series resistor must drop the other 7.0 V:

```text
R = 7.0 / 0.020 = 350 Ω    →   use the next standard value up, 390 Ω
```

Rounding up keeps the current slightly below the maximum.

## In code

```python
def parallel(*resistances):
    return 1 / sum(1 / r for r in resistances)

print(round(parallel(100, 200), 1))   # 66.7
```

## Common mistakes

- **Forgetting to convert mA.** 30 mA is 0.030 A.
- **Adding parallel resistances.** Add reciprocals instead.
- **Assuming the same current flows in parallel branches.** Same voltage, different currents.
- **Treating mains like a battery.** Home wiring can kill. Practise on batteries of 9 V or less or in a simulator; real installation work needs training, supervision and, for electricians, a licence.

## Quick recap

- I = Q / t, in amperes.
- V = I·R; R = ρL/A.
- Series resistances add; parallel reciprocals add.
- P = VI = I²R = V²/R.

## Try it

In the **Do** tab, write the series-current function and use it to analyse a battery circuit.


Apply Ohm's law, series and parallel rules and the power formulas to simple battery circuits.

Current and resistance

Electric Current: Crash Course Physics #28

Ohm's law | Physics | Khan Academy

## Why this matters

Almost all the electricity you use was made by a **generator**: a coil and a magnet moving relative to each other, turned by steam, water, wind or a gas turbine. The same physics runs electric motors, transformers, induction cooktops, wireless chargers, guitar pickups and MRI scanners. Electrical and mechanical engineers meet it in power systems and machines; physicists and materials scientists meet it in magnetic materials and instruments. This lesson explains the two key facts: currents make magnetic fields, and changing magnetic fields make voltages.

## Magnetic fields

A **magnetic field** B is measured in **tesla** (T). Some reference sizes:

- Earth's field at the surface: about 25–65 µT (millionths of a tesla).
- A fridge magnet near its surface: a few thousandths of a tesla.
- A hospital MRI scanner: 1.5–3 T.

Field lines leave a magnet's north pole and enter its south pole. Like poles repel and opposite poles attract, but unlike electric charges, magnetic poles always come in pairs: cut a bar magnet in half and you get two smaller magnets.

## Moving charges feel magnetic forces

A charge q moving with velocity v through a field B feels a force

```text
F = q · v · B · sin θ
```

where θ is the angle between v and B. Two things are unusual about this force. It is zero for a charge at rest, and it always acts **perpendicular** to the motion, so it bends a charge's path without speeding it up. That is how old CRT televisions steered their electron beams and how mass spectrometers sort ions by mass.

A wire carrying current I, with length L inside the field, feels

```text
F = B · I · L · sin θ
```

That force on current-carrying wires is what turns an **electric motor**.

## Currents make magnetic fields

In 1820 Hans Christian Ørsted noticed that a wire carrying current deflected a compass needle. Every current produces a magnetic field that circles around it. Wind the wire into a coil and the fields from each turn add up, giving a strong field through the middle: an **electromagnet**. Wrap the coil around iron and the field becomes much stronger still. Relays, door locks, scrapyard cranes and loudspeakers all use electromagnets.

## Magnetic flux

To talk about induction we need **magnetic flux** Φ: how much field passes through a loop.

```text
Φ = B · A · cos θ        units: webers (Wb), 1 Wb = 1 T·m²
```

A is the loop's area and θ is the angle between the field and the line perpendicular to the loop. Flux is largest when the field passes straight through the loop and zero when the loop is edge-on to the field.

## Faraday's law of induction

In 1831 Michael Faraday found that a **changing** flux through a coil produces a voltage (an electromotive force, emf) around it:

```text
emf = − N · ΔΦ / Δt
```

N is the number of turns. It does not matter how the flux changes: move a magnet, change the current in a nearby coil, or rotate the coil. A steady flux, however strong, produces nothing.

**Example.** A 50-turn coil of area 0.010 m² sits in a field that rises from 0 to 0.20 T in 0.10 s:

```text
ΔΦ  = 0.20 × 0.010 = 0.0020 Wb
emf = 50 × 0.0020 / 0.10 = 1.0 V
```

The minus sign is **Lenz's law**: the induced current flows in the direction that opposes the change that caused it. Push a magnet toward a coil and the induced current makes a field that pushes back. That opposition is why it takes mechanical work to drive a generator: energy is conserved, and the electrical energy out comes from the work you put in.

## How a generator works

1. A coil is turned inside a magnetic field (or a magnet is turned inside a set of coils, as in large power-station generators).
2. As the coil rotates, the angle between the coil and the field changes continuously, so the flux through it rises and falls.
3. The changing flux induces an emf that swings positive and negative in a smooth sine wave: **alternating current** (AC).
4. One full turn of a simple two-pole generator gives one full AC cycle. The grid frequency is 60 Hz in North America and 50 Hz in most other regions, so a simple two-pole generator must turn 60 (or 50) times per second.

For a coil of N turns and area A spinning at angular speed ω in field B, the peak voltage is

```text
emf_peak = N · B · A · ω
```

100 turns of area 0.010 m² in 0.50 T, spinning at 60 Hz (ω = 2π × 60 = 377 rad/s), gives a peak of 100 × 0.50 × 0.010 × 377 ≈ 190 V.

A **motor** is the same machine run backwards: feed it current and the force on the wires turns the shaft.

## Transformers

A **transformer** is two coils wound on a shared iron core. AC in the first coil makes a changing flux in the core, which induces a voltage in the second coil. The voltages scale with the turns ratio:

```text
V_secondary / V_primary = N_secondary / N_primary
```

Power companies step voltage up for long-distance lines (less current, so less I²R loss in the wires) and step it down near homes. Transformers need a changing flux, which is why the grid uses AC. Grid equipment carries lethal voltages; working on it requires specialist training and authorisation.

## Common mistakes

- **Expecting a steady field to induce a voltage.** Only a change in flux does.
- **Forgetting the number of turns.** More turns, proportionally more emf.
- **Thinking a generator "makes" energy.** It converts mechanical work into electrical energy; Lenz's law makes you pay for it.
- **Expecting a transformer to work on DC.** Steady DC gives a steady flux and no induced voltage.

## Quick recap

- B is measured in tesla; moving charges and currents feel magnetic forces.
- Currents create magnetic fields; coils make electromagnets.
- Faraday: emf = −N·ΔΦ/Δt. Lenz: the induced current opposes the change.
- A generator rotates a coil in a field to make AC; a motor is the reverse.
- Transformers change AC voltage by the turns ratio.

## Check yourself

This lesson has no coding activity. Use the quiz in the **Do** tab to check that you can explain how a generator produces a voltage.


Explain magnetic forces, Faraday's law of induction and how generators and transformers work.

Magnetism and induction

Induction - An Introduction: Crash Course Physics #34

AC Electrical Generator Basics - How electricity is generated

Charge, electric field and voltage as energy per charge; current, resistance and Ohm's law; and how magnetism and induction make generators and transformers work.