Electromagnetics Essentials

Cheatsheet Content

Charge Density Volume Charge Density ($\rho_v$): Charge per unit volume. $$ \rho_v = \frac{dQ}{dV} \quad (\text{C/m}^3) $$ Surface Charge Density ($\rho_s$): Charge per unit area. $$ \rho_s = \frac{dQ}{dS} \quad (\text{C/m}^2) $$ Line Charge Density ($\rho_l$): Charge per unit length. $$ \rho_l = \frac{dQ}{dL} \quad (\text{C/m}) $$ Total Charge (Q): $$ Q = \int_V \rho_v dV = \int_S \rho_s dS = \int_L \rho_l dL $$ Del Operator ($\nabla$) A vector differential operator. In Cartesian coordinates $(x, y, z)$: $$ \nabla = \hat{a}_x \frac{\partial}{\partial x} + \hat{a}_y \frac{\partial}{\partial y} + \hat{a}_z \frac{\partial}{\partial z} $$ Used to define gradient, divergence, and curl. Gradient ($\nabla f$) Applies to a scalar field $f$. $$ \nabla f = \hat{a}_x \frac{\partial f}{\partial x} + \hat{a}_y \frac{\partial f}{\partial y} + \hat{a}_z \frac{\partial f}{\partial z} $$ Result is a vector field that points in the direction of the maximum rate of increase of $f$. Magnitude is the maximum rate of change. Physical interpretation: For electric potential $V$, $\mathbf{E} = -\nabla V$. Divergence ($\nabla \cdot \mathbf{A}$) Applies to a vector field $\mathbf{A}$. In Cartesian coordinates $\mathbf{A} = A_x \hat{a}_x + A_y \hat{a}_y + A_z \hat{a}_z$: $$ \nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} $$ Result is a scalar field. Measures the "outward flux" per unit volume from an infinitesimal volume. Indicates whether a vector field converges (negative divergence) or diverges (positive divergence) from a point. Physical interpretation: $\nabla \cdot \mathbf{D} = \rho_v$ (Gauss's Law in differential form). Curl ($\nabla \times \mathbf{A}$) Applies to a vector field $\mathbf{A}$. In Cartesian coordinates: $$ \nabla \times \mathbf{A} = \begin{vmatrix} \hat{a}_x & \hat{a}_y & \hat{a}_z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix} $$ Result is a vector field. Measures the "circulation" or "rotation" of a vector field around a point. Indicates the tendency of the field to swirl. Physical interpretation: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ (Faraday's Law in differential form). Fundamental Theorem of Calculus Relates differentiation and integration. $$ \int_a^b F'(x) dx = F(b) - F(a) $$ Generalizes to vector calculus theorems. Fundamental Theorem for Gradient Relates the line integral of a gradient field to the difference in the scalar function at the endpoints. $$ \int_L (\nabla f) \cdot d\mathbf{l} = f(\mathbf{r}_b) - f(\mathbf{r}_a) $$ Where $L$ is a path from $\mathbf{r}_a$ to $\mathbf{r}_b$. This means the line integral of a conservative field (a gradient field) is path-independent. Gauss’s or Green’s Theorem (Divergence Theorem) Relates the flux of a vector field through a closed surface to the divergence of the field within the enclosed volume. $$ \oint_S \mathbf{A} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{A}) dV $$ Here $S$ is a closed surface enclosing volume $V$. Green's Theorem is a special 2D case of Stokes' or Divergence Theorem. Stokes’ Theorem Relates the circulation of a vector field around a closed loop to the flux of the curl of the field through any surface bounded by that loop. $$ \oint_L \mathbf{A} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S} $$ Here $L$ is a closed path bounding an open surface $S$. The direction of $d\mathbf{l}$ and $d\mathbf{S}$ follow the right-hand rule. Electric Field ($\mathbf{E}$) and Electric Potential ($V$) Electric Field: Force per unit charge ($\mathbf{E} = \mathbf{F}/q$). $$ \mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^2} \hat{a}_R \quad (\text{for a point charge}) $$ Electric Potential: Work done per unit charge to bring a charge from reference to a point. $$ V = -\int \mathbf{E} \cdot d\mathbf{l} $$ Relationship: $\mathbf{E} = -\nabla V$. Units: $\mathbf{E}$ in V/m or N/C, $V$ in Volts (J/C). Poisson’s and Laplace’s Equations Derived from Gauss's Law ($\nabla \cdot \mathbf{D} = \rho_v$) and $\mathbf{D} = \epsilon \mathbf{E}$, $\mathbf{E} = -\nabla V$. $$ \nabla \cdot (\epsilon \mathbf{E}) = \rho_v \implies \nabla \cdot (-\epsilon \nabla V) = \rho_v $$ Poisson's Equation: For a medium with uniform permittivity $\epsilon$. $$ \nabla^2 V = -\frac{\rho_v}{\epsilon} $$ Laplace's Equation: For regions where there is no free charge ($\rho_v = 0$). $$ \nabla^2 V = 0 $$ $\nabla^2$ is the Laplacian operator: $\nabla^2 = \nabla \cdot \nabla$. $$ \nabla^2 V = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} $$ These equations are fundamental for solving electrostatic problems. Capacitor A device that stores electric energy in an electric field. Capacitance (C): Ratio of charge stored on one plate to the potential difference between the plates. $$ C = \frac{Q}{V} \quad (\text{Farads, F}) $$ For a parallel-plate capacitor: $$ C = \frac{\epsilon S}{d} $$ where $\epsilon$ is permittivity, $S$ is plate area, $d$ is separation. Energy Stored: $$ W_E = \frac{1}{2} C V^2 = \frac{1}{2} Q V = \frac{1}{2} \frac{Q^2}{C} $$ Energy Density: $$ w_e = \frac{1}{2} \epsilon |\mathbf{E}|^2 $$ Magnetic Flux Density ($\mathbf{B}$) A measure of the strength of a magnetic field. It is a vector field. Units: Tesla (T) or Weber per square meter (Wb/m$^2$). Magnetic flux $\Phi_B = \int_S \mathbf{B} \cdot d\mathbf{S}$. $\mathbf{B}$ is related to the force on a moving charge: $\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$. Magnetic Field Strength ($\mathbf{H}$) A measure of the intensity of the magnetizing field. Units: Ampere per meter (A/m). Relationship with $\mathbf{B}$ in linear, isotropic, homogeneous media: $$ \mathbf{B} = \mu \mathbf{H} $$ where $\mu$ is the magnetic permeability ($\mu = \mu_0 \mu_r$). $\mu_0 = 4\pi \times 10^{-7}$ H/m (permeability of free space). Ampere’s Circuital Law Integral Form (for steady currents): $$ \oint_L \mathbf{H} \cdot d\mathbf{l} = I_{enc} $$ where $I_{enc}$ is the total current enclosed by the closed path $L$. Differential Form (for steady currents): $$ \nabla \times \mathbf{H} = \mathbf{J} $$ where $\mathbf{J}$ is the current density. Relates the circulation of the magnetic field strength around a closed loop to the total current passing through the surface bounded by the loop. Electrostatic Boundary Conditions Describe how electric fields and potentials behave at the interface between two different media. Normal Component of Electric Flux Density ($\mathbf{D}$): $$ D_{1n} - D_{2n} = \rho_s $$ If no surface charge ($\rho_s = 0$), then $D_{1n} = D_{2n}$. Tangential Component of Electric Field ($\mathbf{E}$): $$ E_{1t} = E_{2t} $$ Using $\mathbf{D} = \epsilon \mathbf{E}$: $$ \epsilon_1 E_{1n} - \epsilon_2 E_{2n} = \rho_s $$ Using $E_t = -\frac{\partial V}{\partial l}$: $$ \frac{1}{\epsilon_1} D_{1n} - \frac{1}{\epsilon_2} D_{2n} = \frac{\rho_s}{\epsilon_1} - \frac{\rho_s}{\epsilon_2} \quad (\text{if } \rho_s \text{ at interface}) $$ Scalar and Vector Potentials Scalar Electric Potential (V): $$ \mathbf{E} = -\nabla V $$ Used in electrostatics, as $\nabla \times \mathbf{E} = 0$. Vector Magnetic Potential ($\mathbf{A}$): $$ \mathbf{B} = \nabla \times \mathbf{A} $$ Used in magnetostatics, as $\nabla \cdot \mathbf{B} = 0$. These potentials simplify calculations by reducing vector operations to scalar or simpler vector operations. Relationship for dynamic fields: $$ \mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t} $$ Continuity Equation Expresses the conservation of charge. $$ \nabla \cdot \mathbf{J} = -\frac{\partial \rho_v}{\partial t} $$ If current flows out of a differential volume, the charge within that volume must decrease. For steady currents (DC), $\frac{\partial \rho_v}{\partial t} = 0$, so $\nabla \cdot \mathbf{J} = 0$. Maxwell’s Equations: Differential Form Gauss's Law for Electric Fields: $$ \nabla \cdot \mathbf{D} = \rho_v $$ (Source of electric field is electric charge) Gauss's Law for Magnetic Fields: $$ \nabla \cdot \mathbf{B} = 0 $$ (No magnetic monopoles; magnetic field lines are always closed loops) Faraday's Law of Induction: $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ (A time-varying magnetic field produces an electric field) Ampere-Maxwell Law: $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$ (Currents and time-varying electric fields produce magnetic fields; includes displacement current term) Constitutive relations: $\mathbf{D} = \epsilon \mathbf{E}$, $\mathbf{B} = \mu \mathbf{H}$, $\mathbf{J} = \sigma \mathbf{E}$ (for linear, isotropic media). Maxwell’s Equations: Integral Form Gauss's Law for Electric Fields: $$ \oint_S \mathbf{D} \cdot d\mathbf{S} = \int_V \rho_v dV = Q_{enc} $$ Gauss's Law for Magnetic Fields: $$ \oint_S \mathbf{B} \cdot d\mathbf{S} = 0 $$ Faraday's Law of Induction: $$ \oint_L \mathbf{E} \cdot d\mathbf{l} = -\frac{\partial}{\partial t} \int_S \mathbf{B} \cdot d\mathbf{S} $$ Ampere-Maxwell Law: $$ \oint_L \mathbf{H} \cdot d\mathbf{l} = \int_S \mathbf{J} \cdot d\mathbf{S} + \frac{\partial}{\partial t} \int_S \mathbf{D} \cdot d\mathbf{S} $$ Significance of Maxwell’s Equations They unify all classical electromagnetic phenomena. Predict the existence of electromagnetic waves, which travel at the speed of light. Form the foundation of radio, television, radar, optics, wireless communication, etc. Demonstrate the interconnectedness of electric and magnetic fields. Maxwell’s Displacement Current and Correction in Ampere’s Law Maxwell added the displacement current term $\frac{\partial \mathbf{D}}{\partial t}$ to Ampere's Law. $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$ This term is crucial for: Ensuring charge conservation (continuity equation is satisfied). Explaining the propagation of electromagnetic waves in free space (where $\mathbf{J}=0$). Explaining how capacitors work with AC circuits (current flows even without conduction). Displacement current acts as a source for magnetic fields, just like conduction current. Electromagnetic (EM) Wave Propagation in Free Space In free space, $\rho_v = 0$, $\mathbf{J} = 0$, $\epsilon = \epsilon_0$, $\mu = \mu_0$. Maxwell's equations simplify to: $$ \nabla \cdot \mathbf{E} = 0 $$ $$ \nabla \cdot \mathbf{H} = 0 $$ $$ \nabla \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} $$ $$ \nabla \times \mathbf{H} = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} $$ Taking the curl of Faraday's Law and substituting Ampere-Maxwell Law leads to wave equations: $$ \nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 $$ $$ \nabla^2 \mathbf{H} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{H}}{\partial t^2} = 0 $$ These are wave equations of the form $\nabla^2 \mathbf{F} - \frac{1}{v^2} \frac{\partial^2 \mathbf{F}}{\partial t^2} = 0$, where $v$ is the wave speed. In free space, the speed of EM waves is $v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = c \approx 3 \times 10^8$ m/s. Plane wave solutions: $$ \mathbf{E}(z,t) = E_0 \cos(\omega t - kz) \hat{a}_x $$ $$ \mathbf{H}(z,t) = H_0 \cos(\omega t - kz) \hat{a}_y $$ where $k = \omega \sqrt{\mu_0 \epsilon_0} = \omega/c$ (wave number). Transverse Nature of Electromagnetic Waves In a uniform, isotropic medium, EM waves are transverse. The electric field $\mathbf{E}$, magnetic field $\mathbf{H}$, and direction of propagation are mutually orthogonal. For a plane wave propagating in the $+z$ direction: $\mathbf{E}$ is in the $xy$-plane, $\mathbf{H}$ is in the $xy$-plane, and both are perpendicular to $\hat{a}_z$. Also, $\mathbf{E}$ and $\mathbf{H}$ are perpendicular to each other. The ratio of their magnitudes is the intrinsic impedance of the medium: $\eta = \frac{|\mathbf{E}|}{|\mathbf{H}|} = \sqrt{\frac{\mu}{\epsilon}}$. For free space: $\eta_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} \approx 377 \, \Omega$. Maxwell’s Equations in Isotropic Dielectric Medium: EM Wave Propagation For a lossless, non-conducting dielectric medium: $\rho_v = 0$, $\mathbf{J} = 0$, $\epsilon = \epsilon_r \epsilon_0$, $\mu = \mu_r \mu_0$. The wave equations become: $$ \nabla^2 \mathbf{E} - \mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 $$ $$ \nabla^2 \mathbf{H} - \mu \epsilon \frac{\partial^2 \mathbf{H}}{\partial t^2} = 0 $$ The speed of propagation is $v = \frac{1}{\sqrt{\mu \epsilon}} = \frac{c}{\sqrt{\mu_r \epsilon_r}}$. The intrinsic impedance is $\eta = \sqrt{\frac{\mu}{\epsilon}} = \eta_0 \sqrt{\frac{\mu_r}{\epsilon_r}}$. The wave number is $k = \omega \sqrt{\mu \epsilon}$. Maxwell’s Equations in Conducting Medium: EM Wave Propagation and Skin Depth For a conducting medium, $\mathbf{J} = \sigma \mathbf{E}$ (Ohm's Law), $\rho_v$ is often assumed to be zero for time-harmonic fields. Maxwell's equations lead to modified wave equations: $$ \nabla^2 \mathbf{E} - \mu \sigma \frac{\partial \mathbf{E}}{\partial t} - \mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 $$ For time-harmonic fields ($e^{j\omega t}$ dependence), the wave equation for $\mathbf{E}$ becomes: $$ \nabla^2 \mathbf{E} + \omega^2 \mu \epsilon \left(1 - j \frac{\sigma}{\omega \epsilon}\right) \mathbf{E} = 0 $$ $$ \nabla^2 \mathbf{E} + \gamma^2 \mathbf{E} = 0 $$ where $\gamma = \sqrt{j\omega\mu(\sigma+j\omega\epsilon)}$ is the complex propagation constant. $\gamma = \alpha + j\beta$, where $\alpha$ is the attenuation constant and $\beta$ is the phase constant. The wave is attenuated as it propagates through the conductor. $$ \mathbf{E}(z,t) = E_0 e^{-\alpha z} \cos(\omega t - \beta z) \hat{a}_x $$ Skin Depth ($\delta$): The depth at which the field amplitude reduces to $1/e$ (approx. 37%) of its surface value. $$ \delta = \frac{1}{\alpha} $$ For good conductors ($\sigma \gg \omega \epsilon$): $$ \alpha \approx \sqrt{\frac{\omega \mu \sigma}{2}} = \beta $$ $$ \delta \approx \sqrt{\frac{2}{\omega \mu \sigma}} $$ High frequency currents in good conductors tend to flow near the surface (skin effect). Electromagnetic Energy Density The total energy stored in an electromagnetic field per unit volume. $$ w = w_e + w_m = \frac{1}{2} \mathbf{D} \cdot \mathbf{E} + \frac{1}{2} \mathbf{B} \cdot \mathbf{H} $$ For linear, isotropic media: $$ w = \frac{1}{2} \epsilon |\mathbf{E}|^2 + \frac{1}{2} \mu |\mathbf{H}|^2 $$ Units: Joules per cubic meter (J/m$^3$). Poynting Vector and Poynting Theorem Poynting Vector ($\mathbf{P}$): Represents the directional energy flux density of an electromagnetic field. $$ \mathbf{P} = \mathbf{E} \times \mathbf{H} $$ Units: Watts per square meter (W/m$^2$). Represents power flow per unit area. For a plane wave, $\mathbf{P}$ points in the direction of wave propagation. Poynting Theorem: Expresses the conservation of energy in electromagnetic fields. $$ \oint_S (\mathbf{E} \times \mathbf{H}) \cdot d\mathbf{S} = -\frac{\partial}{\partial t} \int_V \left(\frac{1}{2}\mathbf{D} \cdot \mathbf{E} + \frac{1}{2}\mathbf{B} \cdot \mathbf{H}\right) dV - \int_V \mathbf{E} \cdot \mathbf{J} dV $$ $$ \oint_S \mathbf{P} \cdot d\mathbf{S} = -\frac{\partial W}{\partial t} - P_{diss} $$ The net power flowing out of a closed surface $S$ (LHS) is equal to the rate of decrease of energy stored in the EM fields within volume $V$ plus the power dissipated as heat (Joule heating, $P_{diss} = \int_V \mathbf{E} \cdot \mathbf{J} dV$). Wave Propagation in Bounded System (Waveguide) Waveguides are structures that guide EM waves, typically hollow metal tubes of specific cross-sections (rectangular, circular). Unlike free space, waves in waveguides cannot propagate at arbitrary frequencies or modes. Modes: Specific field configurations that can propagate. Transverse Electric (TE) Modes: $\mathbf{E}$ field is entirely transverse to the direction of propagation ($E_z = 0$). Transverse Magnetic (TM) Modes: $\mathbf{H}$ field is entirely transverse to the direction of propagation ($H_z = 0$). Transverse ElectroMagnetic (TEM) Modes: Both $\mathbf{E}$ and $\mathbf{H}$ are entirely transverse to the direction of propagation ($E_z = 0, H_z = 0$). TEM modes can only exist in waveguides with at least two conductors (e.g., coaxial cable), not in hollow metal waveguides. Cutoff Frequency ($f_c$): The minimum frequency below which a given mode cannot propagate in the waveguide. For a rectangular waveguide ($a \times b$, $a>b$): $$ f_c = \frac{1}{2\pi\sqrt{\mu\epsilon}} \sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2} $$ where $m, n$ are mode integers (not both zero for TE/TM). Dominant Mode: The mode with the lowest cutoff frequency (e.g., $TE_{10}$ for a rectangular waveguide). Phase Velocity ($v_p$): Speed of a point of constant phase. $$ v_p = \frac{\omega}{\beta} = \frac{c}{\sqrt{1-(f_c/f)^2}} $$ Note $v_p > c$ for propagating modes, but this does not violate relativity as energy travels at group velocity. Group Velocity ($v_g$): Speed at which energy or information propagates. $$ v_g = \frac{d\omega}{d\beta} = c \sqrt{1-(f_c/f)^2} $$ Note $v_g The propagation constant $\beta$ becomes imaginary below cutoff, meaning the wave is evanescent (attenuated).