Electric Charge
The force that holds every atom together, sparks from a doorknob, lifts toner onto a page, and — written as one inverse-square law — turns out to be the same shape as gravity, only far stronger and able to both pull and push
- That there are two kinds of electric charge, and the rule that like charges repel while opposite charges attract.
- The difference between conductors and insulators, and how charging by induction and grounding work.
- Coulomb's law \(F = \dfrac{1}{4\pi\varepsilon_{0}}\dfrac{|q_{1}||q_{2}|}{r^{2}}\) with \(k = 1/4\pi\varepsilon_{0} = 8.99\times10^{9}\,\mathrm{N\cdot m^{2}/C^{2}}\).
- The superposition principle — net force is the vector sum of pairwise forces — and the two electrostatic shell theorems.
- That charge is quantized, \(q = ne\) with \(e = 1.602\times10^{-19}\,\mathrm{C}\), and that charge is conserved.
- How charge and current relate, \(i = dq/dt\), so that \(1\,\mathrm{C} = (1\,\mathrm{A})(1\,\mathrm{s})\).
What Is Physics?
You are surrounded by devices that run on electromagnetism — the union of electric and magnetic phenomena. It powers computers, television, radio, lighting, and even the cling of plastic wrap. It is also the glue of the natural world: it binds every atom and molecule together and produces lightning, auroras, and rainbows.
The Greeks knew that rubbed amber attracts straw (an electric force) and that lodestone attracts iron (a magnetic force), but the two sciences grew up separately for centuries — until 1820, when Oersted noticed that an electric current deflects a compass needle, linking them. Faraday's brilliant physical intuition and Maxwell's mathematics then welded them into a single theory. Our study of it runs through the next sixteen chapters, and it begins here, with the nature of electric charge.
Electric Charge
Walk across a carpet in dry weather and you can draw a spark to a doorknob — proof that your body carries electric charge. In fact every object holds a vast amount of it. Charge is an intrinsic property of the fundamental particles, and it comes in two kinds, positive and negative. Most objects are electrically neutral because they hold equal amounts of both; only a tiny imbalance makes an object "charged."
Rub a glass rod with silk and it loses a little negative charge, becoming slightly positive; rub a plastic rod with fur and it gains negative charge. Bring two like-charged rods together and they push apart; bring opposite ones together and they pull. This is the central rule:
The labels "positive" and "negative" were Benjamin Franklin's arbitrary choice — he could have picked any pair of opposites. Section 4 makes this rule quantitative as Coulomb's law. The term electrostatic signals that the charges are at rest or moving only slowly. This attraction and repulsion drives real technology: electrostatic paint spraying, smokestack fly-ash collection, ink-jet printing, and the toner-and-drum cycle of a photocopier.
Conductors and Insulators
Materials divide by how freely charge moves through them. In conductors (metals, the human body, tap water) charge flows easily; in insulators or nonconductors (rubber, plastic, glass, pure water) it cannot. Semiconductors (silicon, germanium) sit between, and superconductors conduct with no resistance at all.
This explains a familiar frustration: rub a copper rod while holding it and touching a faucet, and it will not charge — because you, the rod, and the plumbing form a conducting path to Earth (a giant conductor), and the excess charge simply drains away. Setting up such a path is called grounding; neutralizing the object this way is discharging it. Hold the rod by an insulating handle instead, and the charge stays put.
The cause lies in atomic structure. In a metal, the outermost electrons break free and roam as conduction electrons, leaving fixed positive ions behind. Bring a charged rod near a neutral conductor and those mobile electrons rearrange — pushed away or pulled near — giving the conductor an induced charge that makes it attract the rod even though it stays neutral overall.
Coulomb's Law
Charles-Augustin de Coulomb measured the electrostatic force in 1785 and found it follows an inverse-square law. For two point charges \(q_{1}\) and \(q_{2}\) separated by \(r\), the magnitude of the force each exerts on the other is:
The electrostatic constant is \(k = 1/4\pi\varepsilon_{0} = 8.99\times10^{9}\,\mathrm{N\cdot m^{2}/C^{2}}\), and the permittivity constant is \(\varepsilon_{0} = 8.85\times10^{-12}\,\mathrm{C^{2}/N\cdot m^{2}}\). The force points along the line joining the charges — apart for like signs, together for opposite signs.
The Shell Theorems
The shell theorems that simplified gravitation have exact electrostatic analogs — a direct payoff of the shared inverse-square form:
These let us treat a charged sphere as a point charge at its center for any external charge. On a spherical conductor, excess charge spreads uniformly over the outer surface — the mutual repulsion pushes charges as far apart as possible — so the sphere again behaves like a point charge at its center.
Charge Is Quantized
Franklin imagined charge as a continuous fluid, but matter is made of discrete particles — and so is charge. Every charge that can be detected is a whole-number multiple of an elementary charge \(e\):
| Particle | Symbol | Charge |
|---|---|---|
| Electron | e or e⁻ | −e |
| Proton | p | +e |
| Neutron | n | 0 |
Charge Is Conserved
Rub glass with silk and the rod gains exactly the positive charge the silk gains in negative — rubbing only transfers charge, never creates it. This conservation of charge, first proposed by Franklin, has never been violated, from everyday objects down to nuclei and elementary particles. Charge joins energy and momentum on the list of conserved quantities.
In radioactive decay, \(^{238}\mathrm{U} \to {}^{234}\mathrm{Th} + {}^{4}\mathrm{He}\): the net charge is \(+92e\) before and \(+90e + 2e\) after. In pair annihilation, \(e^{-} + e^{+} \to \gamma + \gamma\) (net zero before and after); in pair production, \(\gamma \to e^{-} + e^{+}\). Always add charges algebraically, signs included, and the total holds fixed.
Putting It to Work
Problem. Charges \(q_{1} = 1.60\times10^{-19}\,\mathrm{C}\) and \(q_{2} = 3.20\times10^{-19}\,\mathrm{C}\) sit on an x axis a distance \(R = 0.0200\,\mathrm{m}\) apart. Then a negative charge \(q_{3} = -3.20\times10^{-19}\,\mathrm{C}\) is placed at \(\tfrac{3}{4}R\) from particle 1, between the two. Find the net force on particle 1.
Solution. Both forces lie on the x axis. \(q_{2}\) repels particle 1 (toward −x); \(q_{3}\), opposite in sign, attracts it (toward +x).
Because both forces lie on one line, we add signed magnitudes — no vector decomposition needed. The closer charge wins even though both source charges are equal.
Problem. Keep \(q_{1}, q_{2}\) as above, but now add \(q_{4} = -3.20\times10^{-19}\,\mathrm{C}\) at \(\tfrac{3}{4}R\) from particle 1, along a line at \(\theta = 60^{\circ}\) above the x axis. Find the net force on particle 1.
Solution. \(\vec{F}_{12} = 1.15\times10^{-24}\,\mathrm{N}\) along −x; \(\vec{F}_{14} = 2.05\times10^{-24}\,\mathrm{N}\) toward \(q_{4}\) at 60°. Add by components.
When forces are not collinear, resolve into components first. The direction (94°) must lie between the two contributing forces — a useful sanity check that catches the common arctangent sign error.
Problem. A charge \(q_{1} = -8q\) sits at the origin and \(q_{2} = +2q\) at \(x = L\). Where can a proton be placed so the net force on it is zero? Stable or unstable?
Solution. The forces can cancel only where they point oppositely and match in size. That is on the x axis beyond \(q_{2}\) (right of it), where the larger but farther \(q_{1}\) can balance the smaller but nearer \(q_{2}\). Set \(F_{1} = F_{2}\):
The equilibrium at \(x = 2L\) is unstable: nudge the proton left and \(F_{2}\) grows faster than \(F_{1}\), driving it further left; nudge it right and the net force drives it further right. A stable equilibrium would push it back.
Problem. Sphere A has charge \(+Q\); identical sphere B is neutral, far away. (a) They are joined briefly by a wire, then separated — what is the force afterward? (b) Instead, A is grounded momentarily — what then?
Solution. (a) Identical spheres share charge equally, so each ends with \(Q/2\) (charge is conserved). (b) Grounding a positively charged sphere draws electrons up from Earth until it is neutral.
Sharing leaves both spheres positive, so they repel. Grounding instead neutralizes A, and a neutral sphere feels (and exerts) no electrostatic force.
Problem. Two protons in an iron nucleus are \(r = 4.0\times10^{-15}\,\mathrm{m}\) apart. Compare the electrostatic and gravitational forces between them.
Solution. Use Coulomb's law for the repulsion and Newton's law for the attraction (\(m_{p} = 1.67\times10^{-27}\,\mathrm{kg}\)).
The electrostatic repulsion exceeds gravity by some \(10^{36}\) — gravity cannot possibly hold the nucleus together. What does is the strong nuclear force, which overpowers the proton repulsion at these tiny separations.
Chapter Summary
Like charges repel, opposites attract. Equal amounts = neutral; an imbalance = charged.
Charge moves freely in conductors (mobile electrons), not in insulators.
A nearby charge separates charge in a neutral conductor; a path to Earth discharges it.
\(F = \dfrac{1}{4\pi\varepsilon_{0}}\dfrac{|q_{1}||q_{2}|}{r^{2}}\), \(k = 8.99\times10^{9}\).
Net force = vector sum of the pairwise Coulomb forces from all other charges.
Outside a uniform shell: acts as a point charge at center. Inside: zero net force.
\(q = ne\), \(e = 1.602\times10^{-19}\,\mathrm{C}\). Electron \(-e\), proton \(+e\).
Net charge of an isolated system is fixed — seen in decay, annihilation, pair production.
\(i = dq/dt\), so \(1\,\mathrm{C} = (1\,\mathrm{A})(1\,\mathrm{s})\).
Problems
Take \(k = 8.99\times10^{9}\,\mathrm{N\cdot m^{2}/C^{2}}\) and \(e = 1.602\times10^{-19}\,\mathrm{C}\). For multi-charge problems, find each pairwise force from Coulomb's law and add as vectors; for identical conducting spheres in contact, split the total charge equally; for quantization, count in units of \(e\).
- What must be the distance between point charges \(q_{1} = 26.0\,\mu\mathrm{C}\) and \(q_{2} = -47.0\,\mu\mathrm{C}\) for the electrostatic force between them to have magnitude 5.70 N?
- A particle of charge \(+3.00\times10^{-6}\,\mathrm{C}\) is 12.0 cm from a second particle of charge \(-1.50\times10^{-6}\,\mathrm{C}\). Find the magnitude of the electrostatic force between them.
- In a lightning bolt, a current of \(2.5\times10^{4}\,\mathrm{A}\) flows for 20 µs. How much charge is transferred?
- Two identical conducting spheres attract with a force of 0.108 N at 50.0 cm separation. After being joined by a wire and separated, they repel with 0.0360 N. Find the original charges (one was positive overall).
- Three charged particles lie on an x axis: particles 1 and 2 fixed, particle 3 free but feeling zero net force. If \(L_{23} = L_{12}\), find the ratio \(q_{1}/q_{2}\).
- Particle 1 (\(+1.0\,\mu\mathrm{C}\)) and particle 2 (\(-3.0\,\mu\mathrm{C}\)) are held a distance \(L = 10.0\,\mathrm{cm}\) apart on an x axis. Where must particle 3 be placed so the net force on it is zero?
- How many electrons must be removed from a coin to leave it with a charge of \(+1.0\times10^{-7}\,\mathrm{C}\)?
- Find the electrostatic force between a singly charged sodium ion (\(+e\)) and an adjacent chlorine ion (\(-e\)) separated by \(2.82\times10^{-10}\,\mathrm{m}\) in a salt crystal.
- Two identical isolated water drops carry charges of \(-1.00\times10^{-16}\,\mathrm{C}\) each, 1.00 cm apart. Find (a) the force between them and (b) the number of excess electrons on each.
- A current of 0.300 A passes through your chest for 2.00 min. How many conduction electrons pass through?
- Sphere A starts with charge \(-50e\) and sphere B with \(+20e\). The identical conducting spheres touch and separate. What is the resulting charge on A?
- Identify X in the nuclear reaction \(^{1}\mathrm{H} + {}^{9}\mathrm{Be} \to \mathrm{X} + \mathrm{n}\) using charge and mass-number conservation.
- Four particles lie on an x axis separated by \(d = 2.00\,\mathrm{cm}\), with charges \(q_{1} = +2e\), \(q_{2} = -e\), \(q_{3} = +e\), \(q_{4} = +4e\). Find the net force on particle 1 (in unit-vector notation).
- What equal positive charges placed on Earth and the Moon would neutralize their gravitational attraction? (Why is the lunar distance not needed?)