Electromagnetism & Relativity
Course 2 · Chapter 2 · Electric Fields & Potential
Chapter 1 described the force between two charges directly, the way Chapter 6 of Course 1 first described gravity between two masses. This chapter takes the same step Course 1's own gravity chapter implicitly relied on: reframing that force as a field filling the space around a charge, and reintroducing potential energy — this time in electrical form.
The Electric Field
An electric field describes the force a charge would experience at any point in space, per unit of its own charge:
E = F/q
For a single point charge Q, Coulomb's law from Chapter 1 gives the field's exact real form directly:
E = kQ/r²
The electric field is measured in newtons per coulomb (N/C), equivalently volts per metre (V/m). Unlike a bare force, a field exists at every point in space regardless of whether a second charge happens to be there to feel it — it describes what would happen to a charge, if one were placed at that location.
💡 Michael Faraday's Real, Untrained Genius
The idea of a "field" filling empty space — rather than charges reaching out and acting on each other directly at a distance — comes from Michael Faraday, working in the early-to-mid 1800s. Faraday's own formal education was genuinely minimal: by his own later account, his mathematical ability "did not extend as far as trigonometry," and he was largely self-taught, having taken up reading and lectures while apprenticed as a bookbinder from age 14. Working almost entirely through visualised "lines of force" rather than equations, Faraday proposed that electric and magnetic forces genuinely extend into the empty space around a charge or magnet — an idea his own contemporaries initially rejected. James Clerk Maxwell, who later gave Faraday's field concept its full, rigorous mathematical form (Chapter 6 of this course), wrote of him with real, striking respect: that Faraday's own use of lines of force showed him to be "a mathematician of a very high order — one from whom the mathematicians of the future may derive valuable and fertile methods," despite having barely any formal mathematics at all.
Worked Example: The Field Around a Point Charge
What is the electric field 0.2 m from a +5 µC point charge?
E = kQ/r²
E = (8.99×10&sup9;) × (5×10&supminus;&sup6;) / (0.2)²
E ≈ 1.12×10&sup6; N/C
Electric Potential
Just as Course 1's own Chapter 3 defined gravitational potential energy from work done against gravity, electric potential (commonly called voltage) measures electrical potential energy per unit charge:
V = kQ/r
The electric potential energy of a charge q placed at a location with potential V is:
U = qV
Electric potential is measured in volts (V), where 1 V = 1 J/C — one joule of potential energy per coulomb of charge.
💡 Alessandro Volta and the First Steady Current
The volt is named for Alessandro Volta, who in 1800 built the voltaic pile — genuinely the first device able to produce a steady, continuous electric current, rather than the brief static discharges every earlier electrical device (like the Leyden jar) could manage. Volta's own real motivation was a scientific dispute with fellow Italian Luigi Galvani, who had proposed that a frog's leg twitching when touched by two different metals revealed a genuine "animal electricity" produced by living tissue. Volta's crucial, correct insight was that the frog's leg was acting only as a conductor and a sensitive detector — the electricity itself came from the two dissimilar metals in contact, not from anything biological. Replacing the frog's leg with brine-soaked paper between stacked pairs of different metals, Volta both refuted Galvani's specific claim and, in the same stroke, invented the world's first battery.
Worked Example: Potential and Potential Energy
Using the same +5 µC point charge from the field example, what is the electric potential at r = 0.2 m, and what potential energy would a +2 µC test charge have there?
V = kQ/r
V = (8.99×10&sup9;) × (5×10&supminus;&sup6;) / 0.2
V ≈ 2.25×10&sup5; V
U = qV = (2×10&supminus;&sup6;) × (2.25×10&sup5;)
U ≈ 0.45 J
🔗 A Direct Real Relationship, Not a Coincidence
Multiplying this chapter's own field result by the distance, E × r = (1.12×10&sup6;) × 0.2 ≈ 2.25×10&sup5; V, exactly reproduces the potential calculated above. This is the real, general relationship E = −dV/dr (the field is the negative rate of change of potential with distance), simplified for this specific case: since a point charge's own potential is exactly zero infinitely far away, the field and potential around it are directly linked by a single factor of r.
Gravity and Electricity: The Field/Potential Parallel
| Property | Gravity (Course 1) | Electricity (This Chapter) |
|---|
| Force | F = Gm1m2/r² | F = kq1q2/r² |
| Field | g = GM/r² | E = kQ/r² |
| Potential energy | PE = mgh (near a surface) | U = qV |
| Sign | Always attractive | Attractive or repulsive |
Hands-On Exercises
Exercise 1
Calculate the electric field 0.5 m from a +8 uC point charge.
→ Solution
Exercise 2
A +3 uC charge is placed at a point where the electric potential is 150,000 V. Calculate its electric potential energy at that location.
→ Solution
Exercise 3
Explain, in your own words, why Volta's real insight about Galvani's frog-leg experiment - that the two dissimilar metals, not the frog's own tissue, were the true source of the electricity - was the key step that led directly to inventing the first battery.
→ Solution
Quick Reference
- Electric field: E = F/q = kQ/r², in N/C (equivalently V/m)
- Electric potential: V = kQ/r, in volts (1 V = 1 J/C)
- Electric potential energy: U = qV
- Relationship: E = −dV/dr
- Faraday (self-taught, minimal formal mathematics) originated the field/"lines of force" concept; Maxwell later gave it rigorous mathematical form (Chapter 6)
- Volta's 1800 voltaic pile was the first steady-current source, invented while refuting Galvani's "animal electricity"
Next chapter: Circuits: Current, Resistance & Ohm's Law — where Volta's own voltaic pile finally gets to do what it was built for, driving a steady, continuous current through a real circuit.