field formulaes

Here is the exhaustive Master Formula Sheet for your ECE 2105 course, extracted directly from your syllabus, class notes, and PYQs. Every formula, derivation step, numerical relation, and approximation has been broken out into its own dedicated block according to your strict formatting rules.

👨‍🏫 Instructor 1: Field Theory, Electrostatics, Maxwell’s Equations & Potentials

[Differential Length in Cartesian Coordinates]

Formula: $d l = \overset{a}{^}_{x} d x + \overset{a}{^}_{y} d y + \overset{a}{^}_{z} d z$ Concept: Represents an infinitesimally small vector length element in 3D Cartesian space. Symbols:

$d l$ = Differential length vector [m]
$\overset{a}{^}_{x}, \overset{a}{^}_{y}, \overset{a}{^}_{z}$ = Unit vectors in x, y, and z directions
$d x, d y, d z$ = Differential lengths along respective axes [m]

[Differential Volume in Cartesian Coordinates]

Formula: $d v = d x, d y, d z$ Concept: Represents an infinitesimally small volume element in Cartesian coordinates. Symbols:

$d v$ = Differential volume [m³]
$d x, d y, d z$ = Differential lengths [m]

[Differential Length in Cylindrical Coordinates]

Formula: $d l = \overset{a}{^}_{r} d r + \overset{a}{^}_{θ} r d θ + \overset{a}{^}_{z} d z$ Concept: Represents an infinitesimally small vector length element in Cylindrical space. Used for integrating line charges. Symbols:

$d l$ = Differential length vector [m]
$\overset{a}{^}_{r}, \overset{a}{^}_{θ}, \overset{a}{^}_{z}$ = Unit vectors in radial, azimuthal, and vertical directions
$d r$ = Differential radius [m]
$d θ$ = Differential angle [rad]
$d z$ = Differential height [m]

[Differential Volume in Cylindrical Coordinates]

Formula: $d v = r, d r, d θ, d z$ Concept: Represents an infinitesimally small volume element in Cylindrical coordinates. Symbols:

$d v$ = Differential volume [m³]
$r$ = Radial distance [m]
$d r, d θ, d z$ = Differential coordinate changes

[Differential Length in Spherical Coordinates]

Formula: $d l = \overset{a}{^}_{R} d R + \overset{a}{^}_{θ} R d θ + \overset{a}{^}_{ϕ} R sin θ d ϕ$ Concept: Represents an infinitesimally small vector length element in Spherical space. Symbols:

$d l$ = Differential length vector [m]
$\overset{a}{^}_{R}, \overset{a}{^}_{θ}, \overset{a}{^}_{ϕ}$ = Unit vectors in radial, elevation, and azimuthal directions
$R$ = Radial distance from origin [m]
$d R, d θ, d ϕ$ = Differential coordinate changes

[Differential Volume in Spherical Coordinates]

Formula: $d v = R^{2} sin θ, d R, d θ, d ϕ$ Concept: Represents an infinitesimally small volume element in Spherical coordinates. Used heavily in charge cloud numericals. Symbols:

$d v$ = Differential volume [m³]
$R$ = Radial distance [m]
$θ$ = Elevation angle [rad]

[Differential Form of Gauss’s Law for Electrostatics] [Heavily Tested]

Formula: $\nabla \cdot D = ρ_{v}$ Concept: The first fundamental postulate of electrostatics. States that the volume charge density is the source of the electric flux density divergence. Symbols:

$\nabla \cdot D$ = Divergence of electric flux density [C/m³]
$ρ_{v}$ = Volume charge density [C/m³]

[Integral Form of Gauss’s Law for Electrostatics] [Heavily Tested]

Formula: $\oint_{S} D \cdot d s = Q$ Concept: The total outward electric flux through any closed surface is equal to the total charge enclosed within that surface. Symbols:

$\oint_{S}$ = Closed surface integral
$D$ = Electric flux density [C/m²]
$d s$ = Differential surface area vector [m²]
$Q$ = Total enclosed charge [C]

[Differential Form of Faraday’s Law for Electrostatics] [Heavily Tested]

Formula: $\nabla \times E = 0$ Concept: The second fundamental postulate of electrostatics. States that a static electric field is conservative (curl-free or irrotational). Symbols:

$\nabla \times E$ = Curl of the electric field intensity
$E$ = Electric field intensity [V/m]

[Integral Form of Faraday’s Law for Electrostatics] [Heavily Tested]

Formula: $\oint_{C} E \cdot d l = 0$ Concept: The line integral of the static electric field along any closed path is exactly zero. (Kirchhoff’s Voltage Law in field theory). Symbols:

$\oint_{C}$ = Closed line integral
$E$ = Electric field intensity [V/m]
$d l$ = Differential length vector [m]

[Coulomb’s Law] [PYQ: 2018, 2021]

Formula: $F_{12} = \frac{q _{1} q _{2}}{4 π ϵ _{0} R _{12}^{2}} \overset{a}{^}_{R_{12}}$ Concept: Calculates the electrostatic force exerted on point charge $q_{2}$ by point charge $q_{1}$ separated by distance $R_{12}$ . Symbols:

$F_{12}$ = Force on charge 2 due to charge 1 [N]
$q_{1}, q_{2}$ = Point charges [C]
$ϵ_{0}$ = Permittivity of free space ( $8.854 \times 1 0^{- 12}$ F/m)
$R_{12}$ = Distance between charges [m]
$\overset{a}{^}_{R_{12}}$ = Unit vector pointing from $q_{1}$ to $q_{2}$

[Electric Field Intensity of a Point Charge] [Heavily Tested]

Formula: $E = \frac{q}{4 π ϵ _{0} R ^{2}} \overset{a}{^}_{R}$ Concept: Defines the electric field generated by a single point charge at a distance $R$ . Symbols:

$E$ = Electric field intensity [V/m]
$q$ = Point charge [C]
$R$ = Distance from charge to observation point [m]
$\overset{a}{^}_{R}$ = Unit vector pointing from charge to observation point

[Superposition of Electric Field for Multiple Point Charges] [Heavily Tested]

Formula: $E = \frac{1}{4 π ϵ _{0}} \sum_{k = 1}^{n} \frac{q _{k} ( R - R _{k} )}{∣ R - R _{k} ∣ ^{3}}$ Concept: Calculates the net electric field at point $R$ due to $n$ individual point charges located at points $R_{k}$ . Heavily used in specific Cartesian axis numericals. Symbols:

$E$ = Total electric field intensity [V/m]
$q_{k}$ = $k$ -th point charge [C]
$R$ = Position vector of observation point [m]
$R_{k}$ = Position vector of the $k$ -th charge [m]

[Electric Field Intensity of a Continuous Volume Charge]

Formula: $E = \frac{1}{4 π ϵ _{0}} \int_{v} \frac{ρ _{v} d V ^{'}}{R ^{2}} \overset{a}{^}_{R}$ Concept: General formula to calculate the electric field generated by a 3D charged cloud or solid body. Symbols:

$E$ = Electric field intensity [V/m]
$ρ_{v}$ = Volume charge density [C/m³]
$d V^{'}$ = Differential volume element of the source [m³]
$R$ = Distance from source element to observation point [m]

[Electric Field of an Infinite Line Charge] [PYQ: 2016]

Formula: $E = \frac{ρ _{L}}{2 π ϵ _{0} r} \overset{a}{^}_{r}$ Concept: Derived via Gauss’s Law for an infinitely long straight wire carrying uniform line charge. Symbols:

$E$ = Electric field intensity [V/m]
$ρ_{L}$ = Uniform line charge density [C/m]
$r$ = Perpendicular radial distance from the line [m]
$\overset{a}{^}_{r}$ = Unit vector pointing radially outward

[Electric Field of an Infinite Sheet of Charge] [PYQ: 2015, 2023]

Formula: $E = \frac{ρ _{s}}{2 ϵ _{0}} \overset{a}{^}_{n}$ Concept: Derived via Gauss’s law for an infinitely large charged plane. Notice the field is uniform and independent of distance. Symbols:

$E$ = Electric field intensity [V/m]
$ρ_{s}$ = Surface charge density [C/m²]
$\overset{a}{^}_{n}$ = Unit vector normal (outward) from the sheet

[Electric Field Inside a Uniformly Charged Spherical Cloud] [Heavily Tested]

Formula: $E = \frac{ρ _{v} R}{3 ϵ _{0}} (for R \leq a)$ Concept: Proves that the electric field inside a charged cloud grows linearly from zero at the center up to the surface. Symbols:

$E$ = Magnitude of electric field intensity [V/m]
$ρ_{v}$ = Volume charge density [C/m³]
$R$ = Radial distance from the center [m]
$a$ = Total radius of the charge cloud [m]

[Electric Potential Difference] [Heavily Tested]

Formula: $V_{2} - V_{1} = - \int_{P_{1}}^{P_{2}} E \cdot d l$ Concept: Calculates the absolute work done moving a unit positive charge from point 1 to point 2 against the electric field. Symbols:

$V_{2} - V_{1}$ = Potential difference [V]
$E$ = Electric field intensity [V/m]
$d l$ = Differential path vector [m]

[Electric Potential of a Point Charge] [Heavily Tested]

Formula: $V = \frac{q}{4 π ϵ _{0} R}$ Concept: Absolute electric potential at a distance $R$ from a point charge, assuming zero reference potential at infinity. Symbols:

$V$ = Electric potential [V]
$q$ = Point charge [C]
$R$ = Radial distance [m]

[Relationship between Electric Field and Potential] [PYQ: 2016, 2021]

Formula: $E = - \nabla V$ Concept: Shows that the electric field intensity is the negative gradient of the scalar electric potential. Symbols:

$E$ = Electric field intensity [V/m]
$\nabla V$ = Gradient of potential field [V/m]

[Electric Dipole Moment] [Heavily Tested]

Formula: $p = q d$ Concept: Defines the vector strength of an electric dipole. Symbols:

$p$ = Electric dipole moment vector [C·m]
$q$ = Magnitude of one of the charges [C]
$d$ = Distance vector directed from negative to positive charge [m]

[Electric Potential of a Dipole at a Distant Point] [Heavily Tested]

Formula: $V = \frac{p \cdot a ^ _{R}}{4 π ϵ _{0} R ^{2}} = \frac{p c o s θ}{4 π ϵ _{0} R ^{2}}$ Concept: Calculates the potential generated by a dipole at a point far away ( $R ≫ d$ ). Symbols:

$V$ = Electric potential [V]
$p$ = Dipole moment vector [C·m]
$R$ = Distance from the dipole center to observation point [m]
$θ$ = Angle between dipole axis and observation vector [rad]

[Electric Field Intensity of a Dipole] [PYQ: 2016, 2023]

Formula: $E = \frac{p}{4 π ϵ _{0} R ^{3}} (2 cos θ \overset{a}{^}_{R} + sin θ \overset{a}{^}_{θ})$ Concept: The vector electric field of a dipole derived by taking the negative gradient of the dipole’s potential. Symbols:

$E$ = Electric field intensity [V/m]
$p$ = Magnitude of dipole moment [C·m]
$R$ = Radial distance [m]
$θ$ = Elevation angle [rad]

[Electric Flux Density with Polarization] [Heavily Tested]

Formula: $D = ϵ_{0} E + P$ Concept: Defines total electric flux density inside a dielectric material, accounting for free space flux and the material’s induced polarization. Symbols:

$D$ = Total electric flux density [C/m²]
$ϵ_{0}$ = Free space permittivity [F/m]
$E$ = Electric field intensity [V/m]
$P$ = Polarization vector (dipole moment per unit volume) [C/m²]

[Electric Boundary Condition - Tangential Component] [Heavily Tested]

Formula: $E_{1 t} = E_{2 t}$ Concept: The tangential component of the electric field intensity is always continuous across the boundary between two distinct dielectric media. Symbols:

$E_{1 t}$ = Tangential electric field in medium 1 [V/m]
$E_{2 t}$ = Tangential electric field in medium 2 [V/m]

[Electric Boundary Condition - Normal Component] [Heavily Tested]

Formula: $D_{1 n} - D_{2 n} = ρ_{s}$ Concept: The normal component of the electric flux density is discontinuous by an amount equal to the free surface charge density existing on the boundary. Symbols:

$D_{1 n}$ = Normal electric flux density in medium 1 [C/m²]
$D_{2 n}$ = Normal electric flux density in medium 2 [C/m²]
$ρ_{s}$ = Free surface charge density [C/m²]

[Law of Refraction for Electric Fields] [Heavily Tested]

Formula: $\frac{t a n α _{1}}{t a n α _{2}} = \frac{ϵ _{1}}{ϵ _{2}}$ Concept: Relates the bending angles of electric field lines as they cross a charge-free boundary between two dielectrics. Used heavily in boundary angle numericals. Symbols:

$α_{1}$ = Angle $E_{1}$ makes with the boundary normal [rad/deg]
$α_{2}$ = Angle $E_{2}$ makes with the boundary normal [rad/deg]
$ϵ_{1}, ϵ_{2}$ = Permittivities of medium 1 and 2 [F/m]

[Poisson’s Equation] [Heavily Tested]

Formula: $\nabla^{2} V = - \frac{ρ _{v}}{ϵ}$ Concept: Relates the spatial variation (Laplacian) of the electric potential directly to the volume charge density in that region. Symbols:

$\nabla^{2} V$ = Laplacian of scalar electric potential [V/m²]
$ρ_{v}$ = Volume charge density [C/m³]
$ϵ$ = Permittivity of the medium [F/m]

[Laplace’s Equation] [Heavily Tested]

Formula: $\nabla^{2} V = 0$ Concept: A special case of Poisson’s equation applied to regions completely free of any charge density ( $ρ_{v} = 0$ ). Symbols:

$\nabla^{2} V$ = Laplacian of scalar electric potential [V/m²]

[Capacitance Definition]

Formula: $C = \frac{Q}{V}$ Concept: The fundamental definition of capacitance: the ratio of total charge on one conductor to the potential difference between conductors. Symbols:

$C$ = Capacitance [Farads, F]
$Q$ = Total charge [C]
$V$ = Potential difference [V]

[Capacitance of a Parallel Plate Capacitor] [Heavily Tested]

Formula: $C = \frac{ϵ A}{d}$ Concept: Derived capacitance for two parallel plates assuming negligible fringing effect. Symbols:

$C$ = Capacitance [F]
$ϵ$ = Permittivity of the dielectric material [F/m]
$A$ = Surface area of the plates [m²]
$d$ = Distance separating the plates [m]

[Electric Field Inside Parallel Plate Capacitor] [PYQ: 2021, 2022, 2025]

Formula: $E = - \overset{a}{^}_{y} \frac{V _{0}}{d}$ Concept: The uniform electric field intensity generated between two plates maintained at $V = 0$ and $V = V_{0}$ . Symbols:

$E$ = Electric field intensity [V/m]
$\overset{a}{^}_{y}$ = Unit vector directed from bottom to top plate
$V_{0}$ = Applied voltage [V]
$d$ = Plate separation [m]

[Surface Charge Density of Parallel Plate Capacitor] [PYQ: 2021, 2022, 2025]

Formula: $ρ_{s} = \frac{ϵ V _{0}}{d}$ Concept: Calculates the uniform charge spread over the plates given a fixed voltage. Symbols:

$ρ_{s}$ = Surface charge density [C/m²]
$ϵ$ = Permittivity [F/m]
$V_{0}$ = Applied voltage [V]
$d$ = Distance between plates [m]

[Capacitance of a Cylindrical Capacitor] [PYQ: 2015, 2017]

Formula: $C = \frac{2 π ϵ L}{l n ( b / a )}$ Concept: Capacitance of a coaxial cable with inner radius $a$ , outer radius $b$ , and length $L$ . Symbols:

$C$ = Capacitance [F]
$ϵ$ = Permittivity of dielectric [F/m]
$L$ = Length of the cylinder [m]
$b$ = Outer conductor inner radius [m]
$a$ = Inner conductor outer radius [m]

[Electrostatic Energy of Discrete Charges] [Heavily Tested]

Formula: $W_{e} = \frac{1}{2} \sum_{k = 1}^{n} Q_{k} V_{k}$ Concept: Calculates the total work done to assemble $n$ individual point charges from infinity to their final configuration. Symbols:

$W_{e}$ = Electrostatic potential energy [Joules, J]
$Q_{k}$ = The $k$ -th point charge [C]
$V_{k}$ = Electric potential at the location of $Q_{k}$ due to all other charges [V]

[Electrostatic Energy Density in a Continuous Field]

Formula: $w_{e} = \frac{1}{2} ϵ E^{2} = \frac{1}{2} D \cdot E$ Concept: The amount of electrostatic energy stored per unit volume in a continuous electric field. Symbols:

$w_{e}$ = Energy density [J/m³]
$ϵ$ = Permittivity [F/m]
$E$ = Magnitude of electric field intensity [V/m]

[Equation of Continuity] [Heavily Tested]

Formula: $\nabla \cdot J = - \frac{\partial ρ _{v}}{\partial t}$ Concept: Mathematical statement of the principle of conservation of charge. States that current diverging from a point must equal the rate of decrease of charge density at that point. Symbols:

$\nabla \cdot J$ = Divergence of current density [A/m³]
$ρ_{v}$ = Volume charge density [C/m³]
$t$ = time [s]

[Maxwell’s Equations - Faraday’s Law (Time-Varying)] [Heavily Tested]

Formula: $\nabla \times E = - \frac{\partial B}{\partial t}$ Concept: A time-varying magnetic field produces a circulating, non-conservative electric field. Symbols:

$\nabla \times E$ = Curl of electric field [V/m²]
$B$ = Magnetic flux density [Tesla, T]

[Maxwell’s Equations - Ampere’s Law with Displacement Current] [Heavily Tested]

Formula: $\nabla \times H = J + \frac{\partial D}{\partial t}$ Concept: Both conduction current and a time-varying electric flux (displacement current) generate a circulating magnetic field. Symbols:

$\nabla \times H$ = Curl of magnetic field intensity [A/m²]
$J$ = Conduction/Convection current density [A/m²]
$D$ = Electric flux density [C/m²]

[Displacement Current Density] [PYQ: 2016, 2017, 2023]

Formula: $J_{d} = \frac{\partial D}{\partial t} = ϵ \frac{\partial E}{\partial t}$ Concept: Fictitious current density introduced by Maxwell to satisfy the continuity equation in time-varying fields (e.g., through a capacitor). Symbols:

$J_{d}$ = Displacement current density [A/m²]
$D$ = Electric flux density [C/m²]
$ϵ$ = Permittivity [F/m]

[Dynamic Boundary Condition - Tangential E] [Heavily Tested]

Formula: $E_{1 t} = E_{2 t}$ Concept: The tangential component of the electric field remains completely continuous across a boundary, exactly identical to the static case. Symbols:

$E_{1 t}, E_{2 t}$ = Tangential E field in medium 1 and 2 [V/m]

[Electric Field from Potentials (Dynamic Case)] [Heavily Tested]

Formula: $E = - \nabla V - \frac{\partial A}{\partial t}$ Concept: In a dynamic (time-varying) system, the electric field is caused by both the gradient of scalar potential and the time-derivative of the vector magnetic potential. Symbols:

$E$ = Electric field intensity [V/m]
$V$ = Scalar electric potential [V]
$A$ = Vector magnetic potential [Wb/m]

[Lorentz Gauge Condition] [PYQ: 2016]

Formula: $\nabla \cdot A + μ ϵ \frac{\partial V}{\partial t} = 0$ Concept: A mathematical choice of gauge applied to decouple the uncoupled wave equations for scalar and vector potentials. Symbols:

$A$ = Vector magnetic potential [Wb/m]
$μ, ϵ$ = Permeability and Permittivity
$V$ = Scalar electric potential [V]

[Non-Homogeneous Wave Equation for Scalar Potential] [PYQ: 2016]

Formula: $\nabla^{2} V - μ ϵ \frac{\partial ^{2} V}{\partial t ^{2}} = - \frac{ρ _{v}}{ϵ}$ Concept: Derived using Lorentz gauge. Describes how scalar potential propagates as a wave driven by a time-varying charge density source. Symbols:

$V$ = Scalar electric potential [V]
$ρ_{v}$ = Time-varying volume charge density [C/m³]

[Non-Homogeneous Wave Equation for Vector Potential] [PYQ: 2016]

Formula: $\nabla^{2} A - μ ϵ \frac{\partial ^{2} A}{\partial t ^{2}} = - μ J$ Concept: Describes how the vector magnetic potential propagates as a wave driven by a time-varying current density source. Symbols:

$A$ = Vector magnetic potential [Wb/m]
$J$ = Time-varying current density [A/m²]

[Homogeneous Vector Wave Equation for Electric Field] [Heavily Tested]

Formula: $\nabla^{2} E - μ ϵ \frac{\partial ^{2} E}{\partial t ^{2}} = 0$ Concept: The foundational differential equation proving that an electric field propagates as an unattenuated wave in a source-free ( $ρ = 0, J = 0$ ), lossless medium. Symbols:

$E$ = Electric field intensity [V/m]
$μ ϵ$ = Defines wave velocity ( $1/ u^{2}$ )

[Speed of Electromagnetic Wave Propagation] [Heavily Tested]

Formula: $u = \frac{1}{μ ϵ} = \frac{c}{μ _{r} ϵ _{r}}$ Concept: The velocity at which the wavefront travels through a simple medium. Proves that light is an electromagnetic wave ( $c = 3 \times 1 0^{8}$ m/s in free space). Symbols:

$u$ = Wave velocity [m/s]
$μ$ = Permeability [H/m]
$ϵ$ = Permittivity [F/m]
$c$ = Speed of light in vacuum [m/s]

[Retarded Scalar Potential] [PYQ: 2018]

Formula: $V (R, t) = \frac{1}{4 π ϵ} \int_{v^{'}} \frac{ρ ( t - R / u )}{R} d v^{'}$ Concept: The scalar potential at distance $R$ at time $t$ is caused by the source charge density that existed at an earlier time ( $t - R / u$ ) due to finite propagation speed. Symbols:

$V (R, t)$ = Potential at observation point at time $t$ [V]
$ρ (t - R / u)$ = Source charge density evaluated at retarded time [C/m³]
$u$ = Velocity of wave propagation [m/s]

[Helmholtz Equation for Electric Field] [PYQ: 2016, 2021, 2023, 2025]

Formula: $\nabla^{2} E + k^{2} E = 0$ Concept: The time-harmonic (phasor) form of the homogeneous wave equation. Assumes sinusoidal steady-state variations. Symbols:

$E$ = Phasor electric field intensity [V/m]
$k$ = Wave number or phase constant [rad/m]

[Wave Number (Phase Constant) in Lossless Media]

Formula: $k = ω μ ϵ$ Concept: Represents the spatial frequency of the wave (radians per meter) in a lossless dielectric. Symbols:

$k$ = Wave number [rad/m]
$ω$ = Angular frequency ( $2 π f$ ) [rad/s]

[Plasma Frequency of Ionosphere] [PYQ: 2019, 2023]

Formula: $f_{p} \approx 9 N$ Concept: An empirical/derived formula calculating the natural resonant frequency of ionospheric plasma based on electron density. Waves below $f_{p}$ are totally reflected. Symbols:

$f_{p}$ = Plasma frequency [Hz]
$N$ = Electron density [electrons/m³]

👨‍🏫 Instructor 2: Vector Calculus, Magnetostatics, Plane Waves & Power

[Gradient of a Scalar Field]

Formula: $\nabla V = \overset{a}{^}_{x} \frac{\partial V}{\partial x} + \overset{a}{^}_{y} \frac{\partial V}{\partial y} + \overset{a}{^}_{z} \frac{\partial V}{\partial z}$ Concept: A vector operation representing the maximum spatial rate of change of a scalar potential field. Points in direction of max increase. Symbols:

$\nabla V$ = Gradient of potential [V/m]
$\partial / \partial x$ = Partial derivative operators

[Divergence of a Vector Field] [PYQ: 2022, 2025]

Formula: $\nabla \cdot A = \frac{\partial A _{x}}{\partial x} + \frac{\partial A _{y}}{\partial y} + \frac{\partial A _{z}}{\partial z}$ Concept: A scalar operation measuring the net outward flow (flux) of a vector field from an infinitesimally small volume. Symbols:

$\nabla \cdot A$ = Divergence of vector A
$A_{x}, A_{y}, A_{z}$ = Cartesian components of A

[Divergence Theorem]

Formula: $\int_{v} (\nabla \cdot A) d v = \oint_{S} A \cdot d s$ Concept: Converts a volume integral of divergence into a closed surface integral of the vector field. Fundamental for deriving Gauss’s and Continuity laws. Symbols:

$\int_{v}$ = Volume integral
$\oint_{S}$ = Closed surface integral

[Stokes’s Theorem]

Formula: $\int_{S} (\nabla \times A) \cdot d s = \oint_{C} A \cdot d l$ Concept: Converts an open surface integral of the curl of a vector into a closed line integral bounding that surface. Fundamental for deriving Ampere’s and Faraday’s laws. Symbols:

$\nabla \times A$ = Curl of vector A
$\oint_{C}$ = Line integral along bounding contour C

[Lorentz Force Equation] [PYQ: 2018, 2021]

Formula: $F = q (E + u \times B)$ Concept: Calculates the total electromagnetic force exerted on a charged particle moving through both electric and magnetic fields. Symbols:

$F$ = Total force [N]
$q$ = Charge magnitude [C]
$E$ = Electric field [V/m]
$u$ = Velocity vector of the charge [m/s]
$B$ = Magnetic flux density [T]

[Magnetic Force on a Current Element]

Formula: $d F_{m} = I d l \times B$ Concept: Calculates the differential magnetic force exerted on a tiny segment of a current-carrying wire. Symbols:

$d F_{m}$ = Differential magnetic force [N]
$I$ = Current [A]
$d l$ = Differential length vector indicating current direction [m]
$B$ = External magnetic flux density [T]

[Biot-Savart Law] [Heavily Tested]

Formula: $B = \frac{μ _{0} I}{4 π} \oint_{C} \frac{d l ^{'} \times a ^ _{R}}{R ^{2}}$ Concept: General formula to calculate the exact magnetic flux density generated by an arbitrary current-carrying wire loop. Symbols:

$B$ = Magnetic flux density [Tesla, T]
$μ_{0}$ = Permeability of free space ( $4 π \times 1 0^{- 7}$ H/m)
$I d l^{'}$ = Current source element [A·m]
$R$ = Distance from source element to observation point [m]
$\overset{a}{^}_{R}$ = Unit vector from source to observation point

[Magnetic Field of a Finite Straight Wire] [PYQ: 2015, 2016, 2021, 2023, 2025]

Formula: $B = \overset{a}{^}_{ϕ} \frac{μ _{0} I L}{2 π r L ^{2} + r ^{2}}$ Concept: Derived from Biot-Savart Law. Finds the magnetic flux density at distance $r$ in the bisecting plane of a wire of total length $2 L$ . Symbols:

$B$ = Magnetic flux density [T]
$L$ = Half-length of the wire [m]
$r$ = Perpendicular distance from wire [m]
$\overset{a}{^}_{ϕ}$ = Azimuthal unit vector (Right-Hand Rule)

[Magnetic Field of an Infinite Straight Wire] [Heavily Tested]

Formula: $B_{ϕ} = \frac{μ _{0} I}{2 π r}$ Concept: Limits the finite wire formula for $L ≫ r$ or derived directly from Ampere’s Law. Symbols:

$B_{ϕ}$ = Magnitude of magnetic flux density [T]
$r$ = Radial distance from wire [m]

[Ampere’s Circuital Law - Integral Form] [Heavily Tested]

Formula: $\oint_{C} B \cdot d l = μ_{0} I_{e n c l ose d}$ Concept: The circulation of magnetic flux density around any closed path is equal to $μ_{0}$ times the total steady current passing through the surface bounded by the path. Symbols:

$\oint_{C} B \cdot d l$ = Magnetic circulation [T·m]
$I_{e n c l ose d}$ = Total current enclosed by contour C [A]

[Magnetic Field Inside a Solid Conductor] [PYQ: 2017, 2018, 2022]

Formula: $B_{ϕ} = \frac{μ _{0} I r}{2 π b ^{2}} (for r \leq b)$ Concept: Derived using Ampere’s Law for an infinitely long solid wire of radius $b$ . Field increases linearly from the center. Symbols:

$B_{ϕ}$ = Magnetic flux density inside wire [T]
$r$ = Radial distance from center [m]
$b$ = Radius of the conductor [m]
$I$ = Total steady current [A]

[Magnetic Field at Center of Square Loop] [PYQ: 2017]

Formula: $B_{z} = \frac{2 2 μ _{0} I}{π w}$ Concept: Evaluated via Biot-Savart by summing the contributions of 4 finite wire segments of length $w$ . Symbols:

$B_{z}$ = Magnetic field at the center [T]
$w$ = Side length of the square loop [m]

[Magnetic Field on Axis of Circular Loop] [Heavily Tested]

Formula: $B = \overset{a}{^}_{z} \frac{μ _{0} I b ^{2}}{2 ( z ^{2} + b ^{2} ) ^{3/2}}$ Concept: Magnetic flux density along the z-axis generated by a circular loop of radius $b$ lying in the xy-plane. Symbols:

$B$ = Magnetic flux density [T]
$b$ = Radius of circular loop [m]
$z$ = Height above the center of the loop [m]

[Magnetic Dipole Moment] [Heavily Tested]

Formula: $m = I S = \overset{a}{^}_{z} I π b^{2}$ Concept: Defines the vector strength of a magnetic dipole (a small current loop). Direction is given by Right-Hand Rule. Symbols:

$m$ = Magnetic dipole moment [A·m²]
$I$ = Loop current [A]
$S$ = Vector area of the loop [m²]

[Vector Magnetic Potential Definition] [Heavily Tested]

Formula: $B = \nabla \times A$ Concept: An auxiliary mathematical field whose curl yields the true magnetic flux density, assuring that $\nabla \cdot B = 0$ . Symbols:

$B$ = Magnetic flux density [T]
$A$ = Vector magnetic potential [Weber/m, Wb/m]

[Vector Magnetic Potential Integral Formula] [PYQ: 2020, 2024]

Formula: $A = \frac{μ _{0} I}{4 π} \oint_{C} \frac{d l ^{'}}{R}$ Concept: Calculates the vector magnetic potential generated directly by a closed current loop (analogous to $V$ in electrostatics). Symbols:

$A$ = Vector magnetic potential [Wb/m]
$d l^{'}$ = Differential source line element [m]

[Magnetic Flux and Vector Potential Relation] [PYQ: 2019]

Formula: $Φ = \int_{S} B \cdot d s = \oint_{C} A \cdot d l$ Concept: Proves via Stokes’s theorem that the line integral of vector magnetic potential around a closed path equals total magnetic flux through the area. Symbols:

$Φ$ = Magnetic flux [Webers, Wb]

[Magnetic Force Between Two Parallel Wires] [Heavily Tested]

Formula: $\frac{F}{L} = \frac{μ _{0} I _{1} I _{2}}{2 π d}$ Concept: Calculates the force per unit length exerted between two infinitely long, parallel wires separated by distance $d$ . Symbols:

$F / L$ = Force per unit length [N/m]
$I_{1}, I_{2}$ = Currents in the respective wires [A]
$d$ = Separation distance [m]

[Volume Magnetization Current Density] [Heavily Tested]

Formula: $J_{m} = \nabla \times M$ Concept: Represents the equivalent internal circulating volume current density resulting from non-uniform spatial magnetization. Symbols:

$J_{m}$ = Volume magnetization current density [A/m²]
$M$ = Magnetization vector (dipole moment per unit volume) [A/m]

[Surface Magnetization Current Density] [Heavily Tested]

Formula: $J_{m s} = M \times \overset{a}{^}_{n}$ Concept: Represents the equivalent surface current density on the boundary of a magnetized material. Symbols:

$J_{m s}$ = Surface magnetization current density [A/m]
$\overset{a}{^}_{n}$ = Outward unit normal vector from the material surface

[Magnetic Field Intensity Definition] [Heavily Tested]

Formula: $H = \frac{B}{μ _{0}} - M$ Concept: Defines the fundamental magnetic field intensity independent of material magnetization. Symbols:

$H$ = Magnetic field intensity [A/m]
$B$ = Magnetic flux density [T]
$M$ = Magnetization vector [A/m]

[Magnetic Susceptibility and Relative Permeability]

Formula: $M = χ_{m} H ⟹ μ_{r} = 1 + χ_{m}$ Concept: Relates induced magnetization linearly to the applied H-field, leading to the relative permeability of the material. Symbols:

$χ_{m}$ = Magnetic susceptibility [dimensionless]
$μ_{r}$ = Relative permeability [dimensionless]

[Magnetic Boundary Condition - Normal Component] [Heavily Tested]

Formula: $B_{1 n} = B_{2 n}$ Concept: The normal component of magnetic flux density is perfectly continuous across any interface. (Consequence of no magnetic monopoles). Symbols:

$B_{1 n}, B_{2 n}$ = Normal magnetic flux density in media 1 and 2 [T]

[Magnetic Boundary Condition - Tangential Component] [Heavily Tested]

Formula: $H_{1 t} - H_{2 t} = J_{s}$ Concept: The tangential component of magnetic field intensity is discontinuous exactly by the amount of free surface current density at the interface. Symbols:

$H_{1 t}, H_{2 t}$ = Tangential magnetic field intensity [A/m]
$J_{s}$ = Free surface current density [A/m]

[Law of Refraction for Magnetic Fields] [PYQ: 2016, 2018, 2021, 2023, 2024]

Formula: $\frac{t a n α _{1}}{t a n α _{2}} = \frac{μ _{1}}{μ _{2}}$ Concept: Calculates the exact bending angle of magnetic field lines crossing a current-free boundary between two magnetic materials. Symbols:

$α_{1}, α_{2}$ = Angles made with the normal in medium 1 and 2
$μ_{1}, μ_{2}$ = Permeabilities of medium 1 and 2 [H/m]

[Total Magnetic Energy] [Heavily Tested]

Formula: $W_{m} = \frac{1}{2} \int_{v} B \cdot H d v$ Concept: Calculates the total magnetic energy stored in a continuous magnetic field system. Symbols:

$W_{m}$ = Magnetic energy [J]
$B$ = Magnetic flux density [T]
$H$ = Magnetic field intensity [A/m]

[Total Magnetic Energy of Mutually Coupled Circuits] [Heavily Tested]

Formula: $W_{m} = \frac{1}{2} L_{1} I_{1}^{2} + \frac{1}{2} L_{2} I_{2}^{2} + L_{12} I_{1} I_{2}$ Concept: Derives the energy stored when two inductor coils (L1, L2) carry currents and interact via mutual inductance ( $L_{12}$ ). Symbols:

$L_{1}, L_{2}$ = Self-inductances [Henries, H]
$L_{12}$ = Mutual inductance [H]
$I_{1}, I_{2}$ = Circuit currents [A]

[Hall Voltage] [PYQ: 2025, 2021, 2019]

Formula: $V_{h} = u_{0} B_{0} d$ Concept: Derived potential difference across a semiconductor material placed in a transverse magnetic field due to deflected charge carriers. Symbols:

$V_{h}$ = Hall voltage [V]
$u_{0}$ = Drift velocity of charge carriers [m/s]
$B_{0}$ = Applied uniform magnetic flux density [T]
$d$ = Transverse width of the material [m]

[Poynting Vector] [Heavily Tested]

Formula: $P = E \times H$ Concept: Represents the instantaneous magnitude and direction of the flow of electromagnetic power per unit area. Symbols:

$P$ = Poynting vector [W/m²]
$E$ = Electric field intensity [V/m]
$H$ = Magnetic field intensity [A/m]

[Time-Average Poynting Vector] [Heavily Tested]

Formula: $P_{a v e} = \frac{1}{2} R e E \times H^{*}$ Concept: Calculates the practical, measurable average power density flowing over a full sinusoidal cycle. Symbols:

$P_{a v e}$ = Time-average power density [W/m²]
$H^{*}$ = Complex conjugate of phasor magnetic field intensity

[Complex Permittivity] [Heavily Tested]

Formula: $ϵ_{c} = ϵ^{'} - j ϵ^{''} = ϵ - j \frac{σ}{ω}$ Concept: Mathematical construct incorporating both the ideal capacitive energy storage ( $ϵ^{'}$ ) and the conductive/heating energy losses ( $ϵ^{''}$ ) into a single parameter. Symbols:

$ϵ_{c}$ = Complex permittivity [F/m]
$σ$ = Conductivity of the medium [S/m]
$ω$ = Angular frequency [rad/s]

[Loss Tangent] [Heavily Tested]

Formula: $tan δ_{c} = \frac{ϵ ^{''}}{ϵ ^{'}} = \frac{σ}{ω ϵ}$ Concept: The strict ratio of conduction current density to displacement current density. Measures how “lossy” a dielectric medium is. Symbols:

$tan δ_{c}$ = Loss tangent [dimensionless]
$σ$ = Conductivity [S/m]
$ω ϵ$ = Displacement term

[Average Power Dissipated in a Lossy Medium] [Heavily Tested]

Formula: $P_{d i ss i p a t e d} = \frac{1}{2} σ E^{2} = \frac{1}{2} ω ϵ^{''} E^{2}$ Concept: Calculates the actual physical heat/power lost per cubic meter (Joule heating) when an EM wave travels through a lossy material. Symbols:

$P_{d i ss i p a t e d}$ = Volume power density dissipated [W/m³]
$σ$ = Conductivity [S/m]
$E$ = Amplitude of sinusoidal electric field [V/m]

[Wave Propagation Constant]

Formula: $γ = α + j β = jω μ ϵ_{c}$ Concept: Describes exactly how an EM wave attenuates (shrinks) and oscillates (phases) as it travels through any generalized medium. Symbols:

$γ$ = Complex propagation constant [m⁻¹]
$α$ = Attenuation constant [Nepers/m]
$β$ = Phase constant [rad/m]

[Intrinsic Impedance of a Lossy Medium] [Heavily Tested]

Formula: $η_{c} = \frac{jω μ}{σ + jω ϵ}$ Concept: The complex ratio of $E$ magnitude to $H$ magnitude in a lossy medium. It shows that $H$ lags $E$ in phase. Symbols:

$η_{c}$ = Complex intrinsic impedance [Ohms, $Ω$ ]

[Good Conductor Approximation: Attenuation & Phase] [Heavily Tested]

Formula: $α \approx β \approx \frac{ω μ σ}{2} = π f μ σ$ Concept: Simplifies the complex propagation constant when $σ ≫ ω ϵ$ . Shows that attenuation is extremely high and equal to phase shift. Symbols:

$α$ = Attenuation constant [Np/m]
$β$ = Phase constant [rad/m]
$f$ = Frequency [Hz]

[Good Conductor Approximation: Skin Depth] [Heavily Tested]

Formula: $δ = \frac{1}{α} = \frac{2}{ω μ σ}$ Concept: Calculates the exact depth at which an EM wave’s amplitude is severely decayed to $e^{- 1}$ (36.8%) inside a good conductor. Symbols:

$δ$ = Skin depth [m]

[Good Conductor Approximation: Intrinsic Impedance]

Formula: $η_{c} \approx \frac{ω μ}{σ} ∠4 5^{\circ}$ Concept: Shows that in a perfect or good conductor, the magnetic field mathematically lags the electric field by exactly 45 degrees. Symbols:

$η_{c}$ = Intrinsic impedance [ $Ω$ ]

[Low-Loss Dielectric Approximation: Attenuation & Phase] [Heavily Tested]

Formula: $α \approx \frac{σ}{2} \frac{μ}{ϵ}, β \approx ω μ ϵ$ Concept: Simplifies parameters for media acting as good insulators ( $σ ≪ ω ϵ$ ). Shows minimal attenuation. Symbols:

$α$ = Attenuation constant [Np/m]
$β$ = Phase constant [rad/m]

[Wave Phase Velocity] [PYQ: 2024]

Formula: $u_{p} = \frac{ω}{β}$ Concept: Calculates the specific speed at which a single constant-phase point of the wave travels. Symbols:

$u_{p}$ = Phase velocity [m/s]
$ω$ = Angular frequency [rad/s]
$β$ = Phase constant [rad/m]

[Wave Group Velocity] [PYQ: 2024]

Formula: $u_{g} = \frac{1}{\frac{d β}{d ω}}$ Concept: Calculates the speed at which an entire packet of waves (and therefore energy/information) travels. Symbols:

$u_{g}$ = Group velocity [m/s]
$d β / d ω$ = Rate of change of phase constant w.r.t frequency

[Non-Dispersive Medium Condition] [PYQ: 2018]

Formula: $u_{p} u_{g} = c^{2} and u_{p} = u_{g}$ Concept: Mathematical condition proving that an ideal lossless medium does not separate or distort a multi-frequency wave packet over distance. Symbols:

$u_{p}, u_{g}$ = Phase and Group velocities [m/s]

[Reflection Coefficient for Normal Incidence] [Heavily Tested]

Formula: $Γ = \frac{E _{ro}}{E _{i o}} = \frac{η _{2} - η _{1}}{η _{2} + η _{1}}$ Concept: Calculates the ratio of the reflected electric field amplitude to the incident electric field amplitude at a boundary. Symbols:

$Γ$ = Reflection coefficient [dimensionless]
$E_{ro}, E_{i o}$ = Reflected and Incident E-field magnitudes
$η_{1}, η_{2}$ = Intrinsic impedances of media 1 and 2 [ $Ω$ ]

[Transmission Coefficient for Normal Incidence] [Heavily Tested]

Formula: $τ = \frac{E _{t o}}{E _{i o}} = \frac{2 η _{2}}{η _{2} + η _{1}}$ Concept: Calculates the ratio of the transmitted electric field amplitude to the incident electric field amplitude. Symbols:

$τ$ = Transmission coefficient [dimensionless]
$E_{t o}$ = Transmitted E-field magnitude

[Reflection / Transmission Mathematical Relation] [Heavily Tested]

Formula: $1 + Γ = τ$ Concept: A strict boundary condition requirement proving that the sum of incident and reflected tangential fields must exactly equal the transmitted field. Symbols:

$Γ$ = Reflection coefficient
$τ$ = Transmission coefficient

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