Electrostatics

Charge Properties: Quantized ($q = ne$, $e = 1.6 \times 10^{-19}$ C), conserved, additive. Like charges repel; unlike attract.

Coulomb's Law: Force between two point charges $q_1, q_2$ separated by $r$: $$F = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2} = k\frac{q_1 q_2}{r^2}$$ $k = 9 \times 10^9$ N·m²/C²; $\epsilon_0 = 8.85 \times 10^{-12}$ F/m.

In medium with relative permittivity $\epsilon_r$: $F_{\text{medium}} = F_{\text{vacuum}}/\epsilon_r$.

Electric Field at a point: $\vec{E} = \vec{F}/q_0$ (force per unit positive test charge). Units: N/C = V/m.

Field due to common charge configurations:

Electric Dipole: Two equal opposite charges $\pm q$ separated by $2a$. Dipole moment $\vec{p} = q \cdot 2a$ (from $-$ to $+$).

Torque on dipole in uniform field: $\vec{\tau} = \vec{p} \times \vec{E}$, $\tau = pE\sin\theta$.

Potential energy of dipole: $U = -\vec{p}\cdot\vec{E} = -pE\cos\theta$.

Electric Flux through a surface: $\Phi_E = \int \vec{E}\cdot d\vec{A}$. For uniform field through flat surface: $\Phi = EA\cos\theta$. SI unit: N·m²/C = V·m.

Gauss's Law: Net electric flux through any closed surface equals $1/\epsilon_0$ times enclosed charge: $$\oint \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}$$

Valid only for closed (Gaussian) surfaces. The charge outside the surface does not contribute to net flux but affects field at points.

Strategy for Applying Gauss's Law: Use symmetry to choose Gaussian surface where $\vec{E}$ is either constant or perpendicular/parallel to surface elements.

Applications:

Conductor in electrostatic equilibrium:

• Charges reside on surface
• Electric field inside conductor = $0$
• Field at surface = $\sigma/\epsilon_0$, perpendicular to surface
• Potential same throughout conductor (equipotential)
• Field is more intense near sharp points (basis of lightning rod)

Electric Potential $V$ at a point = work done per unit positive test charge to bring it from infinity to that point against electric forces. $$V = \frac{W_{\text{ext}}}{q_0} = -\int_\infty^P \vec{E}\cdot d\vec{r}$$ SI unit: volt (V) = J/C. Scalar quantity.

Potential due to point charge: $V = kq/r$ (taking $V = 0$ at infinity).

Potential due to dipole: $V = (kp\cos\theta)/r^2$ at angle $\theta$ from dipole axis.

Potential due to common configurations:

Relation between field and potential: $$E_x = -\frac{\partial V}{\partial x}, \quad \vec{E} = -\nabla V$$

Equipotential Surfaces: Surfaces of constant $V$. Always perpendicular to electric field. No work done in moving charge along equipotential surface.

Potential Energy of System of Charges: $$U = \frac{1}{4\pi\epsilon_0}\sum_{i

For two charges: $U = kq_1q_2/r$. For three charges: sum of three pair potentials.

Energy density in electric field: $u = \dfrac{1}{2}\epsilon_0 E^2$ (J/m³).

Capacitor: A device storing electrical energy. Capacitance $C = Q/V$ (SI: farad, F). Stores energy $U = \dfrac{1}{2}CV^2 = \dfrac{Q^2}{2C} = \dfrac{1}{2}QV$.

Capacitance of Standard Geometries:

Combination of Capacitors:

In series, all capacitors have the same charge $Q$; total voltage = sum of voltages. In parallel, all have same voltage; total charge = sum of charges.

Dielectric: Insulator that reduces electric field inside ($K = \epsilon_r > 1$). Effects:

• Capacitance multiplied by $K$
• Field reduced by factor $K$
• Voltage reduced by factor $K$ (if charge constant) OR
• Charge increased by factor $K$ (if voltage constant)

Dielectric Slab in Capacitor: Slab of thickness $t < d$ inserted: $$C = \frac{\epsilon_0 A}{d - t + t/K}$$

Energy density: $u = \dfrac{1}{2}\epsilon_0 E^2 = \dfrac{1}{2}\epsilon_0 K E^2$ in dielectric.

Force between plates of parallel plate capacitor: $F = Q^2/(2\epsilon_0 A) = (1/2)\epsilon_0 E^2 A$.