Electric Charges and Fields

$e \approx 1.6\times10^{-19}\text{ C}$ — a charge of $1.5e$ never occurs.

• Additive: total charge is the algebraic sum $q_1+q_2+\cdots+q_n$.
• Conserved: total charge of an isolated system cannot change; rubbing only transfers electrons.

$k = \dfrac{1}{4\pi\varepsilon_0} \approx 8.99\times10^9\text{ N m}^2\text{C}^{-2}$ (often rounded to $9\times10^9$); $\varepsilon_0 \approx 8.85\times10^{-12}\text{ C}^2\text{N}^{-1}\text{m}^{-2}$.

Like charges → repulsive (points away from source); unlike → attractive (points toward source). Newton’s third law: $\mathbf{F}_{21} = -\mathbf{F}_{12}$.

Field of a single point charge $Q$:

Radially outward if $Q>0$, inward if $Q<0$.

Vector sum — each charge contributes as if the others weren’t there.

• Start on positive charge (or infinity), end on negative charge (or infinity); never closed loops.
• Two field lines never cross.
• Closer lines ⇒ stronger field; spread apart ⇒ weaker.

Directed from $-q$ to $+q$; magnitude $p = q\times2a$.

Equatorial field points opposite to $\mathbf{p}$, exactly half the axial magnitude at the same $r$. Dipole field falls off as $1/r^3$ — a full power faster than a single charge’s $1/r^2$.

Net force on a dipole in a uniform field is always zero; the two equal, opposite forces form a couple.

1. $\theta = 0$ ($\mathbf{p}$ aligned with $\mathbf{E}$): stable equilibrium, $\tau=0$.
2. $\theta = 180^\circ$ ($\mathbf{p}$ anti-aligned): unstable equilibrium, $\tau=0$.

Depends only on enclosed charge — independent of the surface’s shape/size, where inside the charge sits, and any charge outside (its net flux contribution is zero).

1. Infinite line charge (linear density $\lambda$): $E = \dfrac{\lambda}{2\pi\varepsilon_0 r} = \dfrac{2k\lambda}{r}$ — falls off as $1/r$.
2. Infinite plane sheet (surface density $\sigma$): $E = \dfrac{\sigma}{2\varepsilon_0}$ — constant, independent of distance from the sheet.