Circuits: Current, Resistance & Ohm's Law

Electromagnetism & Relativity

Course 2 · Chapter 3 · Circuits: Current, Resistance & Ohm's Law

Chapter 2 closed by naming Volta's 1800 voltaic pile as the first source of a steady, continuous current. This chapter finally puts that current to work — and along the way, meets a second real scientist whose own most important discovery was, at first, badly received.

Electric Current

Electric current is the rate at which charge flows past a point:

I = Q/t

Current is measured in amperes (A), where 1 A = 1 C/s — one coulomb of charge passing per second.

💡 André-Marie Ampère: "The Newton of Electricity" The ampere is named for André-Marie Ampère, who founded the entire field of electrodynamics almost immediately after Danish physicist Hans Christian Ørsted's real 1820 discovery: a compass needle placed near a current-carrying wire deflects, meaning an electric current itself creates a magnetic field — the very connection Chapter 4 and Chapter 5 of this course build on directly (Chapter 4 covers the real, genuinely deliberate story behind Ørsted's own discovery). Ampère seized on Ørsted's discovery immediately, developing the mathematical theory (including what's now called Ampère's force law, describing the attraction or repulsion between two current-carrying wires) that turned an isolated observation into a real, predictive science. James Clerk Maxwell — already met in Chapter 2 giving Faraday's field concept mathematical form — later called Ampère "the Newton of electricity." The ampere itself was formally adopted as the international unit of current in 1881.

Ohm's Law

For most real conductors, voltage and current are directly proportional, related by the conductor's resistance:

V = IR

Resistance (R) is measured in ohms (Ω), where 1 Ω = 1 V/A.

⚠ A Discovery That Was Coldly Received Georg Ohm published this exact relationship in 1827, in a book titled Die galvanische Kette, mathematisch bearbeitet ("The Galvanic Circuit Investigated Mathematically"). It was, by real historical account, "coldly received" by his own scientific community — his colleagues at the Jesuit Gymnasium in Cologne did not value the work, and Ohm resigned his position there as a direct result. His career then genuinely stalled for years, until he found a new post at the Polytechnic School of Nuremberg in 1833 — six years after publishing the law that now carries his name. Real recognition came only much later: the Royal Society awarded him the Copley Medal in 1841 and elected him a foreign member in 1842, and he did not reach a prestigious professorship (at Munich) until 1852, just two years before his death in 1854. This joins a real, recurring pattern across this two-course project — Carnot (Classical Mechanics & Thermodynamics, Ch.8), Cavendish and Priestley (this course, Ch.1) — of major discoveries taking real, documented years or decades to be properly recognised.

Worked Example: Current Through a Resistor

A 12 V battery is connected across a 4 Ω resistor. What current flows?

I = V/R
I = 12/4
I = 3 A

Resistance of a Wire

A real conductor's resistance depends on its own material and geometry:

R = ρL/A

Here ρ (rho) is the material's resistivity (an intrinsic property, measured in Ω·m), L is the conductor's length, and A is its cross-sectional area. Copper, a real, excellent conductor widely used in wiring, has a resistivity of roughly 1.68×10&supminus;&sup8; Ω·m — compared to a genuinely good insulator like PTFE (Teflon), whose real conductivity is roughly 10³&sup0; times lower than copper's, making it an effectively total non-conductor for any everyday electrical purpose.

Worked Example: Resistance of a Copper Wire

What is the resistance of a 2 m length of copper wire with a cross-sectional area of 1 mm² (1×10&supminus;&sup6; m²)?

R = ρL/A
R = (1.68×10&supminus;&sup8;) × 2 / (1×10&supminus;&sup6;)
R ≈ 0.0336 Ω

A real, everyday length of copper wire has genuinely tiny resistance — exactly why copper is the standard choice for household and electronic wiring, where minimising unwanted resistance (and the power it wastes as heat) matters.

Power in a Circuit

Course 1's own definition of power (P = W/t) applies directly to circuits, combined with Ohm's law into three equivalent forms:

P = IV = I²R = V²/R

Worked Example: Power Dissipated by the Resistor

Using this chapter's own 12 V, 4 Ω, 3 A example:

P = IV = 3 × 12 = 36 W
P = I²R = 3² × 4 = 36 W

Both forms agree exactly, as they must — this power is dissipated as real heat in the resistor, the same conversion of organised electrical energy into disordered thermal energy that Classical Mechanics & Thermodynamics Chapter 8's own entropy discussion covers in general terms.

Series vs. Parallel Circuits

Property	Series Circuit	Parallel Circuit
Current	Same through every component	Splits between branches
Voltage	Splits across components	Same across every branch
Total resistance	R_total = R₁ + R₂ + …	1/R_total = 1/R₁ + 1/R₂ + …
If one component fails (open)	Entire circuit stops	Other branches keep working

Hands-On Exercises

Exercise 1

A 9 V battery drives a current of 0.5 A through a resistor. Using Ohm's law, calculate the resistor's resistance.

→ Solution

Exercise 2

A copper wire has a resistivity of 1.68 x 10^-8 ohm.m, a length of 5 m, and a cross-sectional area of 2 x 10^-6 m^2. Calculate its resistance.

→ Solution

Exercise 3

A heating element carries a current of 5 A and has a resistance of 20 ohms. Calculate the power it dissipates, using P = I^2 R.

→ Solution

Quick Reference

Current: I = Q/t, measured in amperes (A)
Ohm's law: V = IR, measured in ohms (Ω)
Resistance of a wire: R = ρL/A
Power: P = IV = I²R = V²/R
Ørsted's 1820 discovery (current deflects a compass needle) launched Ampère's electrodynamics
Ohm's 1827 law was coldly received; real recognition (Copley Medal) came only in 1841

Next chapter: Magnetism & Magnetic Fields — where Ørsted's own 1820 accidental discovery, already named twice in this chapter, finally gets the full treatment.