Coulomb's Law & Vector Formulation
The electrostatic force between two stationary point charges $q_1$ and $q_2$ in vacuum separated by distance $r$:
$$\vec{F}_{12} = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \hat{r}_{21}$$
Where $\varepsilon_0 = 8.854 \times 10^{-12}\text{ C}^2\text{N}^{-1}\text{m}^{-2}$ is vacuum permittivity, and $\frac{1}{4\pi\varepsilon_0} \approx 9 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2$.
Electric Dipole & Torque
An electric dipole consists of two equal and opposite charges $\pm q$ separated by distance $2a$. Dipole moment vector: $\vec{p} = q(2a)\hat{p}$ (directed from $-q$ to $+q$).
- Electric Field on Axial Line ($r \gg a$):
$$E_{\text{axial}} = \frac{1}{4\pi\varepsilon_0} \frac{2p}{r^3}$$
- Electric Field on Equatorial Line ($r \gg a$):
$$E_{\text{equatorial}} = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3} \implies E_{\text{axial}} = 2 E_{\text{equatorial}}$$
- Torque in Uniform Electric Field $\vec{E}$:
$$\vec{\tau} = \vec{p} \times \vec{E} \implies \tau = pE\sin\theta$$
Gauss's Law & Applications
The total electric flux passing through any closed Gaussian surface equals $\frac{1}{\varepsilon_0}$ times the net charge enclosed:
$$\Phi_E = \oint_S \vec{E} \cdot d\vec{A} = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$
Derivations:
- Infinitely Long Straight Uniformly Charged Wire ($\lambda$):
$$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$
- Uniformly Charged Infinite Thin Sheet ($\sigma$):
$$E = \frac{\sigma}{2\varepsilon_0} \quad (\text{independent of distance } r!)$$
