Potential energy of dipole: $U = -pE\cos\theta$
Stable equilibrium: $\theta_i = 0°$
$$U_i = -pE\cos 0° = -pE = -(5\times10^{-6})(4\times10^5) = -2 \text{ J}$$Final position: $\theta_f = 60°$
$$U_f = -pE\cos 60° = -pE \cdot \frac{1}{2} = -1 \text{ J}$$$$\Delta U = U_f - U_i = -1 - (-2) = \boxed{1 \text{ J}}$$An electric dipole consists of two equal and opposite charges $+q$ and $-q$ separated by distance $2l$ (or $d$). Dipole moment: $\vec{p} = q\vec{d}$ (from $-q$ to $+q$). SI unit: C·m (or Debye: $1$ D $= 3.33\times10^{-30}$ C·m). Torque in uniform field: $\vec{\tau} = \vec{p}\times\vec{E}$, magnitude $\tau = pE\sin\theta$. Potential energy: $U = -\vec{p}\cdot\vec{E} = -pE\cos\theta$. Work done rotating from $\theta_1$ to $\theta_2$: $W = pE(\cos\theta_1 - \cos\theta_2)$.
Stable equilibrium: $\theta = 0°$ (aligned with field). $U = -pE$ (minimum). Small perturbation → restoring torque → returns to $\theta = 0°$. Unstable equilibrium: $\theta = 180°$ (anti-aligned). $U = +pE$ (maximum). Small perturbation → torque pushes further away. Neutral: no torque at $\theta = 0°$ and $180°$. Maximum torque at $\theta = 90°$: $\tau = pE$. In non-uniform field: net force on dipole (towards stronger field for aligned dipole).
At axial point (on dipole axis, end-on position, distance $r >> l$): $E_{axial} = \frac{2kp}{r^3}$ (along $\vec{p}$). At equatorial point (perpendicular bisector, broad-side): $E_{equatorial} = \frac{kp}{r^3}$ (opposite to $\vec{p}$). Both fall off as $1/r^3$ (faster than monopole $1/r^2$). Ratio: $E_{axial}/E_{equatorial} = 2$ (at same distance). These formulas apply when $r >> l$ (dipole approximation).
Net charge of dipole $= 0$. So electric flux through any closed surface completely enclosing the dipole $= 0$ (Gauss law). But the field of a dipole is non-zero; flux cancels because field lines leave through one face and return through another. A dipole inside a Gaussian surface contributes zero net flux. This is why the electric field of a magnetic dipole (bar magnet) also follows $1/r^3$ at large distances (analogous structure).
Dielectric in electric field: polar molecules align with field (permanent dipoles). Non-polar molecules: induced dipoles. Both reduce net field inside dielectric. Dielectric constant $K = E_0/E$ where $E_0$ = field without dielectric, $E$ = field with dielectric. Capacitance with dielectric: $C = K\epsilon_0 A/d = KC_0$. Energy stored: $U = Q^2/(2C) = Q^2/(2KC_0)$. Inserting dielectric with battery connected: $V$ constant, $C$ increases, $Q$ increases, $U$ increases. With battery disconnected: $Q$ constant, $C$ increases, $V$ decreases, $U$ decreases.
For point charge system: $U = \sum_{i
Electrostatic generator producing very high voltages (millions of volts). Working: positive charges sprayed onto moving belt at bottom. Belt carries charges up inside hollow sphere. Charges transferred to outer surface of sphere (charges always reside on outer surface of conductor). Potential builds up: $V = kQ/R$ (limited by corona discharge in air). Uses: accelerate charged particles (nuclear physics). Medical: cancer treatment. Demonstrate high-voltage phenomena. Can create potential difference of several MV.
Polar molecules: permanent dipole moments. Water: $p = 1.85$ D (large polar molecule). HCl: $p = 1.08$ D. CO$_2$: $p = 0$ (linear, symmetric, dipoles cancel). Microwave cooking: water molecules rotate to align with oscillating microwave field ($2.45$ GHz resonance) → molecular friction → heat. Dielectrics: alignment of molecular dipoles reduces field. Electrophoresis: charged biomolecules (DNA, proteins) move in electric field based on charge and size. NMR (Nuclear Magnetic Resonance): nuclear magnetic dipoles in magnetic field (analogous to electric dipole in E field).