Electromagnetic Field Theory

Cheatsheet Content

1. Fundamental Constants & Material Properties Permittivity of Free Space: $\epsilon_0 \approx 8.854 \times 10^{-12}$ F/m Permeability of Free Space: $\mu_0 = 4\pi \times 10^{-7}$ H/m Speed of Light in Vacuum: $c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3 \times 10^8$ m/s Intrinsic Impedance of Free Space: $\eta_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} \approx 376.7 \Omega$ Relative Permittivity: $\epsilon_r = \frac{\epsilon}{\epsilon_0}$ (Dielectric constant) Relative Permeability: $\mu_r = \frac{\mu}{\mu_0}$ Conductivity: $\sigma$ (S/m) - Ability to conduct current. 2. Vector Calculus & Coordinate Systems 2.1 Vector Operations Review Dot Product: $A \cdot B = |A||B|\cos\theta = A_x B_x + A_y B_y + A_z B_z$ (Scalar) Cross Product: $A \times B = |A||B|\sin\theta \hat{n}$ (Vector, perpendicular to $A, B$) Gradient ($\nabla f$): Max rate of change of scalar field $f$. Direction of steepest ascent. Divergence ($\nabla \cdot A$): Net outward flux per unit volume. "Source" or "sink" strength. Curl ($\nabla \times A$): Circulation per unit area. "Rotation" or "vorticity". Laplacian ($\nabla^2 f = \nabla \cdot (\nabla f)$): Measures concavity/convexity of a scalar field. 2.2 Coordinate Systems (Differential Elements) System $dl$ (Line) $dS$ (Surface) $dV$ (Volume) Cartesian $(x,y,z)$ $dx\hat{x} + dy\hat{y} + dz\hat{z}$ $dxdy, dydz, dzdx$ $dxdydz$ Cylindrical $(\rho,\phi,z)$ $d\rho\hat{\rho} + \rho d\phi\hat{\phi} + dz\hat{z}$ $\rho d\phi dz, d\rho dz, \rho d\rho d\phi$ $\rho d\rho d\phi dz$ Spherical $(r,\theta,\phi)$ $dr\hat{r} + r d\theta\hat{\theta} + r \sin\theta d\phi\hat{\phi}$ $r^2\sin\theta d\theta d\phi, r dr d\theta, r \sin\theta dr d\phi$ $r^2\sin\theta dr d\theta d\phi$ Note: For $dS$, choose appropriate components based on normal vector. E.g., $dS_z = \rho d\rho d\phi \hat{z}$. 2.3 Integral Theorems Divergence Theorem: $\oint_S A \cdot dS = \int_V (\nabla \cdot A) dV$ Stokes' Theorem: $\oint_C A \cdot dl = \int_S (\nabla \times A) \cdot dS$ 3. Electrostatics (Static Electric Fields) Governed by charges at rest. $\frac{\partial}{\partial t} = 0$. 3.1 Coulomb's Law & Electric Field Intensity ($E$) Force between point charges $Q_1, Q_2$: $F = \frac{Q_1 Q_2}{4\pi\epsilon R^2} \hat{a}_R$ (Newtons) Electric Field due to point charge $Q$: $E = \frac{Q}{4\pi\epsilon R^2} \hat{a}_R$ (V/m) Superposition Principle: $E_{total} = \sum E_i$ Charge Distributions: Line Charge $\rho_L$ (C/m): $E = \int_L \frac{\rho_L dl}{4\pi\epsilon R^2} \hat{a}_R$ Surface Charge $\rho_S$ (C/m$^2$): $E = \int_S \frac{\rho_S dS}{4\pi\epsilon R^2} \hat{a}_R$ Volume Charge $\rho_v$ (C/m$^3$): $E = \int_V \frac{\rho_v dV}{4\pi\epsilon R^2} \hat{a}_R$ 3.2 Electric Flux Density ($D$) Constitutive Relation: $D = \epsilon E = \epsilon_0 \epsilon_r E$ (C/m$^2$) 3.3 Gauss's Law Integral Form: $\oint_S D \cdot dS = Q_{enc}$ Differential (Point) Form: $\nabla \cdot D = \rho_v$ Example: For an infinite line charge $\rho_L$ along z-axis, $D = \frac{\rho_L}{2\pi\rho} \hat{a}_\rho$ (Cylindrical). 3.4 Electric Potential ($V$) Definition: $V = -\int E \cdot dl$ (V). Potential difference. Reference Point: Usually $V(\infty) = 0$. Point Charge $Q$: $V = \frac{Q}{4\pi\epsilon r}$ Relation to $E$: $E = -\nabla V$. Poisson's Equation: $\nabla^2 V = -\frac{\rho_v}{\epsilon}$ Laplace's Equation: $\nabla^2 V = 0$ (For charge-free regions). 3.5 Electric Dipole Two equal and opposite charges $\pm Q$ separated by distance $d$. Dipole Moment: $p = Qd$ (vector from $-Q$ to $+Q$). Potential: $V = \frac{p \cos\theta}{4\pi\epsilon r^2}$ Electric Field: $E = \frac{p}{4\pi\epsilon r^3} (2\cos\theta \hat{a}_r + \sin\theta \hat{a}_\theta)$ -Q +Q d 3.6 Conductors & Dielectrics Conductors: $\rho_v=0, E=0$ inside; $\rho_S$ on surface; $E$ normal to surface. Dielectrics: Material polarizes, reducing $E$. $D = \epsilon_0 E + P$, where $P = \epsilon_0 \chi_e E$. 3.7 Capacitance ($C$) Definition: $C = \frac{Q}{V}$ (Farads) Parallel Plate Capacitor: $C = \frac{\epsilon S}{d}$ (S=Area, d=separation) Coaxial Cable: $C = \frac{2\pi\epsilon L}{\ln(b/a)}$ Energy Stored: $W_E = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{1}{2}\int_V D \cdot E dV$ Energy Density: $w_E = \frac{1}{2} D \cdot E = \frac{1}{2}\epsilon E^2$ (J/m$^3$) 4. Magnetostatics (Static Magnetic Fields) Governed by steady currents. $\frac{\partial}{\partial t} = 0$. 4.1 Biot-Savart Law & Magnetic Field Intensity ($H$) Current Element $Idl$: $dH = \frac{I dl \times \hat{a}_R}{4\pi R^2}$ (A/m) Current Distributions: Line Current: $H = \int_L \frac{I dl \times \hat{a}_R}{4\pi R^2}$ Surface Current: $H = \int_S \frac{K dS \times \hat{a}_R}{4\pi R^2}$ ($K$ in A/m) Volume Current: $H = \int_V \frac{J dV \times \hat{a}_R}{4\pi R^2}$ ($J$ in A/m$^2$) Example: For infinite straight wire carrying current $I$: $H = \frac{I}{2\pi\rho} \hat{a}_\phi$ (Cylindrical). 4.2 Magnetic Flux Density ($B$) Constitutive Relation: $B = \mu H = \mu_0 \mu_r H$ (Teslas or Wb/m$^2$) Magnetic Flux: $\Psi = \int_S B \cdot dS$ (Webers) Gauss's Law for Magnetism: $\nabla \cdot B = 0$ (No magnetic monopoles) 4.3 Ampere's Circuital Law Integral Form: $\oint_C H \cdot dl = I_{enc}$ Differential (Point) Form: $\nabla \times H = J$ Example: For an infinite solenoid with $N$ turns/meter, current $I$: $H = NI \hat{a}_z$ inside, $0$ outside. 4.4 Magnetic Vector Potential ($A$) Definition: $B = \nabla \times A$ (Wb/m) For volume current $J$: $A = \int_V \frac{\mu J}{4\pi R} dV$ Relation to $J$: $\nabla^2 A = -\mu J$ (Poisson's Equation for $A$) 4.5 Inductance ($L$) Definition: $L = \frac{\Psi}{I}$ (Henrys) Solenoid: $L = \mu N^2 A/l$ (N=total turns, A=cross-sectional area, l=length) Coaxial Cable: $L = \frac{\mu L}{2\pi} \ln(b/a)$ Energy Stored: $W_M = \frac{1}{2}LI^2 = \frac{1}{2}\int_V B \cdot H dV$ Energy Density: $w_M = \frac{1}{2} B \cdot H = \frac{1}{2\mu} B^2$ (J/m$^3$) 4.6 Magnetic Force Force on a moving charge $q$: $F = q(E + v \times B)$ (Lorentz Force) Force on a current element $Idl$: $dF = I dl \times B$ Force between two current loops: $F = \frac{\mu_0 I_1 I_2}{2\pi R} L$ (for parallel infinite wires) Torque on a current loop: $\tau = m \times B$ where $m = IS\hat{n}$ (magnetic dipole moment). I B Current Loop in B-field 5. Time-Varying Fields & Maxwell's Equations Unifies electrostatics and magnetostatics, explains EM waves. 5.1 Faraday's Law of Induction Induced EMF: $\mathcal{E} = -\frac{d\Psi_m}{dt}$ (V) Integral Form: $\oint_C E \cdot dl = -\int_S \frac{\partial B}{\partial t} \cdot dS$ Differential Form: $\nabla \times E = -\frac{\partial B}{\partial t}$ Motional EMF: $\mathcal{E} = \oint_C (v \times B) \cdot dl$ (for conductor moving in static B-field) 5.2 Displacement Current ($J_D$) Introduced by Maxwell to ensure charge conservation and explain EM waves. $J_D = \frac{\partial D}{\partial t}$ (A/m$^2$) 5.3 Maxwell's Equations (Differential Form) Gauss's Law (Electric): $\nabla \cdot D = \rho_v$ Gauss's Law (Magnetic): $\nabla \cdot B = 0$ Faraday's Law: $\nabla \times E = -\frac{\partial B}{\partial t}$ Ampere-Maxwell Law: $\nabla \times H = J + \frac{\partial D}{\partial t}$ 5.4 Maxwell's Equations (Integral Form) $\oint_S D \cdot dS = \int_V \rho_v dV$ $\oint_S B \cdot dS = 0$ $\oint_C E \cdot dl = -\frac{d}{dt} \int_S B \cdot dS$ $\oint_C H \cdot dl = \int_S (J + \frac{\partial D}{\partial t}) \cdot dS$ 5.5 Continuity Equation $\nabla \cdot J = -\frac{\partial \rho_v}{\partial t}$ (Based on charge conservation) For steady currents, $\partial \rho_v/\partial t = 0 \implies \nabla \cdot J = 0$. Relaxation Time: $\tau_r = \frac{\epsilon}{\sigma}$ (Time for charge density to decay in a medium). 6. Electromagnetic Waves Solutions to Maxwell's equations in source-free media ($\rho_v=0, J=0$). 6.1 Wave Equations For $E$: $\nabla^2 E - \mu\epsilon \frac{\partial^2 E}{\partial t^2} = 0$ For $H$: $\nabla^2 H - \mu\epsilon \frac{\partial^2 H}{\partial t^2} = 0$ In phasor form (for time-harmonic fields $e^{j\omega t}$): $\nabla^2 E + \omega^2\mu\epsilon E = 0$ (Helmholtz Equation) 6.2 Plane Waves in Lossless Media ($\sigma=0$) Phase Velocity: $v_p = \frac{1}{\sqrt{\mu\epsilon}} = \frac{c}{\sqrt{\mu_r\epsilon_r}}$ Wavelength: $\lambda = \frac{v_p}{f} = \frac{2\pi}{\beta}$ Wave Impedance: $\eta = \sqrt{\frac{\mu}{\epsilon}}$ (For free space, $\eta_0 \approx 377 \Omega$) Relationship between $E$ and $H$: $E = -\eta H \times \hat{k}$ (where $\hat{k}$ is propagation direction) $|E| = \eta |H|$ $E$ and $H$ are perpendicular to each other and to the direction of propagation. k E H Plane Wave Propagation 6.3 Plane Waves in Lossy Dielectrics ($\sigma \ne 0$) Propagation Constant: $\gamma = \alpha + j\beta = \sqrt{j\omega\mu(\sigma + j\omega\epsilon)}$ Attenuation Constant ($\alpha$): (Np/m) - Rate of amplitude decay. Phase Constant ($\beta$): (rad/m) - Rate of phase change. Wave Impedance: $\eta = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\epsilon}}$ (Complex) Skin Depth ($\delta$): Distance at which wave amplitude reduces to $e^{-1}$ (approx. 37%). $\delta = \frac{1}{\alpha}$. Good Conductor ($\sigma \gg \omega\epsilon$): $\alpha \approx \beta \approx \sqrt{\frac{\omega\mu\sigma}{2}}$, $\delta = \sqrt{\frac{2}{\omega\mu\sigma}}$, $\eta \approx (1+j)\sqrt{\frac{\omega\mu}{2\sigma}}$ Good Dielectric ($\sigma \ll \omega\epsilon$): $\alpha \approx \frac{\sigma}{2}\sqrt{\frac{\mu}{\epsilon}}$, $\beta \approx \omega\sqrt{\mu\epsilon}$ 6.4 Poynting Vector & Power Flow Instantaneous Poynting Vector: $S = E \times H$ (W/m$^2$) - Direction and magnitude of power flow. Average Poynting Vector: $S_{avg} = \frac{1}{2} \text{Re}\{E \times H^*\}$ (For time-harmonic fields) Power Transmitted: $P = \int_S S_{avg} \cdot dS$ (Watts) 7. Reflection and Refraction of Plane Waves Behavior of EM waves at the interface between two different media. Snell's Law: $\frac{\sin\theta_i}{\sin\theta_t} = \frac{v_{p1}}{v_{p2}} = \frac{n_2}{n_1}$ (where $n$ is refractive index). Critical Angle: $\sin\theta_c = \frac{n_2}{n_1}$ (for total internal reflection, $n_1 > n_2$). 7.1 Normal Incidence Reflection Coefficient: $\Gamma = \frac{E_{r0}}{E_{i0}} = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}$ Transmission Coefficient: $\mathcal{T} = \frac{E_{t0}}{E_{i0}} = \frac{2\eta_2}{\eta_1 + \eta_2} = 1 + \Gamma$ Standing Wave Ratio (SWR): $SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|} = \frac{|E|_{max}}{|E|_{min}}$ Power Reflection Coefficient: $|\Gamma|^2$ Power Transmission Coefficient: $(1 - |\Gamma|^2)$ 7.2 Oblique Incidence (Fresnel's Equations) Decomposition into perpendicular (TE) and parallel (TM) polarizations. Reflection and transmission coefficients depend on incident angle and polarization. Brewster Angle: Angle at which there is no reflection for a specific polarization (e.g., TM wave). 8. Transmission Lines Used to guide electrical energy over distances, typically for RF/microwave frequencies. 8.1 Transmission Line Parameters Primary: $R, L, G, C$ (per unit length) Secondary: Propagation Constant: $\gamma = \alpha + j\beta = \sqrt{(R+j\omega L)(G+j\omega C)}$ Characteristic Impedance: $Z_0 = \sqrt{\frac{R+j\omega L}{G+j\omega C}}$ Lossless Line ($R=0, G=0$): $\alpha=0$, $\beta=\omega\sqrt{LC}$, $Z_0=\sqrt{L/C}$, $v_p=1/\sqrt{LC}$ Low Loss Line: Small $R, G$. 8.2 Wave Propagation on TL Voltage: $V(z) = V_0^+ e^{-\gamma z} + V_0^- e^{\gamma z}$ Current: $I(z) = \frac{V_0^+}{Z_0} e^{-\gamma z} - \frac{V_0^-}{Z_0} e^{\gamma z}$ Input Impedance: $Z_{in}(l) = Z_0 \frac{Z_L + Z_0 \tanh(\gamma l)}{Z_0 + Z_L \tanh(\gamma l)}$ For Lossless Line: $Z_{in}(l) = Z_0 \frac{Z_L + jZ_0 \tan(\beta l)}{Z_0 + jZ_L \tan(\beta l)}$ Special Cases (Lossless): Short Circuit ($Z_L=0$): $Z_{in} = jZ_0 \tan(\beta l)$ (Purely reactive) Open Circuit ($Z_L=\infty$): $Z_{in} = -jZ_0 \cot(\beta l)$ (Purely reactive) Quarter-wave transformer ($\beta l = \pi/2$): $Z_{in} = Z_0^2/Z_L$ 8.3 Reflection Coefficient & SWR Load Reflection Coefficient: $\Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0}$ Reflection Coefficient at $z$: $\Gamma(z) = \Gamma_L e^{-2\gamma z}$ Voltage Standing Wave Ratio (VSWR): $VSWR = \frac{|V|_{max}}{|V|_{min}} = \frac{1+|\Gamma_L|}{1-|\Gamma_L|}$ ZL Transmission Line 9. Waveguides Hollow metallic tubes that guide EM waves at microwave frequencies. Act as high-pass filters. Do NOT support TEM waves. 9.1 Waveguide Modes Transverse Electric (TE) Modes: $E_z = 0$, $H_z \ne 0$. Transverse Magnetic (TM) Modes: $H_z = 0$, $E_z \ne 0$. Cutoff Frequency ($f_c$): Lowest frequency a mode can propagate. Below $f_c$, wave is evanescent. Dominant Mode: Mode with the lowest $f_c$. 9.2 Rectangular Waveguides ($a \times b$, $a>b$) Cutoff Frequency for TE$_{mn}$ or TM$_{mn}$: $f_c = \frac{1}{2\pi\sqrt{\mu\epsilon}} \sqrt{(\frac{m\pi}{a})^2 + (\frac{n\pi}{b})^2}$ Dominant Mode (TE$_{10}$): $f_c = \frac{1}{2a\sqrt{\mu\epsilon}}$ Guide Wavelength: $\lambda_g = \frac{\lambda_0}{\sqrt{1 - (f_c/f)^2}}$ Phase Velocity: $v_p = \frac{c}{\sqrt{1 - (f_c/f)^2}} > c$ Group Velocity: $v_g = c\sqrt{1 - (f_c/f)^2} $v_p v_g = c^2$ a b k Rectangular Waveguide 10. Antennas Transducers converting guided waves to free-space EM waves, and vice-versa. 10.1 Key Antenna Parameters Radiation Pattern: Spatial distribution of radiated power. Isotropic Radiator: Hypothetical antenna radiating equally in all directions. Directive Gain ($D$): Ratio of radiation intensity in a given direction to an isotropic radiator. Directivity ($D_0$): Maximum directive gain. Antenna Efficiency ($e_0$): Ratio of total radiated power to input power. Power Gain ($G$): $G = e_0 D$. Radiation Resistance ($R_{rad}$): Equivalent resistance dissipating radiated power. Input Impedance: $Z_{in} = R_{rad} + R_{loss} + jX_{ant}$. Half-Wave Dipole: $R_{rad} \approx 73 \Omega$. Antenna Aperture ($A_e$): Effective area intercepting incident power. $A_e = \frac{\lambda^2}{4\pi} G$. Friis Transmission Formula: $P_r = P_t G_t G_r \left(\frac{\lambda}{4\pi R}\right)^2$ (Power received). 11. Boundary Conditions Specify how fields behave at the interface between two different media. Derived from integral forms of Maxwell's equations. 11.1 Dielectric-Dielectric Interface Tangential Electric Field: $E_{1t} = E_{2t}$ (Continuous) Normal Electric Flux Density: $D_{1n} - D_{2n} = \rho_S$ (Discontinuous if surface charge exists) Tangential Magnetic Field: $H_{1t} - H_{2t} = J_S$ (Discontinuous if surface current exists) Normal Magnetic Flux Density: $B_{1n} = B_{2n}$ (Continuous) 11.2 Conductor-Dielectric Interface (Perfect Conductor) $E_t = 0$ (Tangential electric field is zero) $D_n = \rho_S$ (Normal electric flux density equals surface charge density) $H_t = J_S$ (Tangential magnetic field equals surface current density) $B_n = 0$ (Normal magnetic flux density is zero) Inside a perfect conductor: $E=0, D=0, H=0, B=0, J=0, \rho_v=0$.