Electricity and Magnetism

§1.3 Point-like Charge

1. Electric Charge — the “stuff” that creates electric forces

Electric charge is a basic property of matter, just like mass.

  • Two kinds: positive (+) and negative (–).
  • Like charges repel, unlike charges attract.
  • Unit: coulomb (C). The smallest amount any particle can carry is one elementary charge    e=1.602×10−19 Ce = 1.602 \times 10^{-19}\,{\rm C}.
  • Conservation: charge can move around and cancel in pairs, but the total in a closed system never changes.

1.1 Static electricity (why a sweater crackles)

Rubbing materials (amber & cloth, balloon & hair) scrapes electrons off one surface and lets them pile up on another. The imbalance produces attractive or repulsive forces even though nothing is touching.

2. Conductors, Insulators & Induction — how charge behaves in materials

TermWhat it meansEveryday example
ConductorContains “loose” conduction electrons that can wander freely.Metals like copper or aluminum wire
InsulatorElectrons are tightly held; charge stays put.Plastic, glass, rubber
PolarizationCharges inside an insulator shift slightly when an external charge is nearby, creating a tiny positive-negative separation (an electric dipole).A charged balloon sticking to a wall
Charging by inductionBring a charged object near (but don’t touch) a conductor, let charges rearrange, then separate the conductor into two pieces. One part ends up +, the other –.The classic “two metal spheres on stands” demo

3. Point-Like Charges — the simplest model

When an object is:

  1. Extremely small (electron, positron) or
  2. So far away that its size doesn’t matter (a star seen from Earth) or
  3. Spherically symmetric in its charge distribution (so it “looks” like all charge sits at the centre)

…we treat it as a point charge. That lets us describe the force with one neat formula instead of messy geometry.

4. Coulomb’s Law — the rule for the electric force

  F⃗  =  ke q1q2r2  r^  \boxed{\;\vec F \;=\; k_e\,\dfrac{q_1 q_2}{r^{2}}\;\hat r\;}
SymbolMeaning
q1,q2q_1,q_2the two charges (C)
rrdistance between them (m)
r^\hat runit vector pointing from the source charge toward the charge that feels the force
ke=14πε0=8.99×109 N⋅m2/C2k_e = \dfrac{1}{4\pi\varepsilon_0}= 8.99\times10^{9}\,{\rm N·m^2/C^2}Coulomb constant

Key ideas

  1. Inverse-square: doubling the distance makes the force 4 × weaker.

  2. Sign decides direction:

    • If q1q2>0q_1 q_2 > 0 (both + or both –) → F⃗\vec F points away ⇒ repulsion.
    • If q1q2<0q_1 q_2 < 0 (opposite signs) → F⃗\vec F points toward ⇒ attraction.

  3. Vector form: r^\hat r keeps the magnitude positive while the direction (±) lives in the unit vector, so you never have to juggle negative numbers in the formula.

  4. Superposition: when many charges act, find each F⃗i\vec F_i and add the vectors tip-to-tail: F⃗net=∑iF⃗i\displaystyle \vec F_{\text{net}} = \sum_i \vec F_i.

5. Worked Examples (from the slides)

5.1 Electron–Proton force inside a hydrogen atom

Data: r=5.292×10−11 mr = 5.292\times10^{-11}\,\text{m} (Bohr radius).

F=kee2r2=(8.99×109) (1.602×10−19)2(5.292×10−11)2≈8.2×10−8 NF = k_e \dfrac{e^2}{r^2} = (8.99\times10^{9})\, \dfrac{(1.602\times10^{-19})^2}{(5.292\times10^{-11})^2} \approx 8.2\times10^{-8}\,\text{N}

Direction: attraction (opposite signs), so the electron is pulled toward the proton.

5.2 Two protons separated by a human-hair width

Data: r=52.00 μm=5.20×10−5 mr = 52.00\,\mu\text{m} = 5.20\times10^{-5}\,\text{m}.

  1. Force
F=kee2r2≈8.5×10−20 NF = k_e \dfrac{e^2}{r^2} \approx 8.5\times10^{-20}\,\text{N}
  1. Acceleration of each proton
a=Fmp=8.5×10−201.673×10−27≈5.1×107 m/s2a = \dfrac{F}{m_p} = \dfrac{8.5\times10^{-20}}{1.673\times10^{-27}} \approx 5.1\times10^{7}\,\text{m/s}^2

(Yes—tiny force, but the proton’s mass is even tinier!)

6. Best-Practice Checklist for Problem Solving

  1. Draw a picture; mark all charges and expected force directions.
  2. List knowns/unknowns (charges, distances, masses).
  3. Compute magnitudes with ∣q∣|q| values only.
  4. Assign directions afterwards (attract vs. repel).
  5. Add vectors carefully (x- and y-components or tip-to-tail).
  6. Check units: forces in newtons (N), distances in metres (m), charges in coulombs (C).

7. What to remember

  • Charge is quantized, conserved, and comes in two signs.
  • Conductors let charge move; insulators don’t.
  • Coulomb’s law gives you both the size and the direction of the electric force.
  • Use superposition for many-charge problems.
  • Always separate magnitude calculations from reasoning about direction to avoid sign mistakes.
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