Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

3.1.9 Recap

Recap

Electrostatics rests on two experimental facts (Table 3.1.1): charges interact, and they do so by Coulomb's law with superposition.
Charge is quantised, $q = \pm N e$ , $e = 1.602 \times 1 0^{- 19} C$ . Macroscopically we use a point charge $q$ or densities $ρ_{L}$ [C/m], $ρ_{S}$ [C/m²], $ρ_{v}$ [C/m³], with $d q^{'} = ρ_{L} d l^{'}$ , $ρ_{S} d S^{'}$ , $ρ_{v} d V^{'}$ and $Q = \int d q^{'}$ .
Source point $Q (x^{'}, y^{'}, z^{'})$ (primed), field point $P (x, y, z)$ , displacement vector $R = r - r^{'}$ from Q to P.
Coulomb's law (3.1.5), (3.1.7): $F = \frac{q q ^{'}}{4 π ε _{0}} \frac{R}{R ^{3}}$ , like charges repel, unlike attract, with $ε_{0} = 1 0^{- 9} / (36 π) \approx 8.842 \times 1 0^{- 12} F/m$ , $1/ (4 π ε_{0}) = 9 \times 1 0^{9} m/F$ .
Superposition: forces and fields of several charges add as vectors (3.1.15), (3.1.25). For continuous sources the sums become integrals over the primed coordinates (3.1.26) to (3.1.28).
Electric field: $F = q E$ , $E = lim_{q \to 0} F / q$ [V/m]. Point charge: $E = q^{'} a_{R} / (4 π ε_{0} R^{2})$ . In general: $E = \frac{1}{4 π ε _{0}} \int \frac{R}{R ^{3}} d q^{'}$ (3.1.29).
Recipe: points Q and P $\to$ $R$ and $R$ $\to$ $d q^{'}$ with the right element $\to$ symmetry, integrate in $a_{x}, a_{y}, a_{z}$ , check limits.
Field lines are tangent to $E$ , $d x / E_{x} = d y / E_{y} = d z / E_{z}$ . They start on $+$ , end on $-$ , never cross, and are dense where $∣ E ∣$ is large.
Results to remember (Problems 3.1.1 to 3.1.3): infinite line $ρ_{L} / (2 π ε_{0} ρ)$ . Ring on its axis: maximum at $h = b / 2$ . Disk on its axis: $\to ρ_{S} / (2 ε_{0})$ near it (infinite sheet), $\to Q / (4 π ε_{0} h^{2})$ far away.
Applications: precipitators, powder coating, xerography, beam deflection, inkjet coding and ESD all rest on $F = q E$ .

Coulomb's law with superposition answers every question of electrostatics in principle, but the integrals can be laborious. In Section 3.2 we shall derive from it Gauss's law, which gives the fields of highly symmetric distributions (the infinite line of Problem 3.1.1, the infinite sheet that appears as a limit in Problem 3.1.3, spheres and cylinders) in a few lines, and in Section 3.3 the electrostatic potential, a scalar from which $E$ follows by differentiation.