PhysicsElectromagnetismJEE Main & Advanced

Electromagnetism Formulas

Complete chapter-wise formula sheet for Electromagnetism (Physics) covering key equations, parameters, traps, and concept diagrams for JEE Main & Advanced.

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Chapter Directory (24 Chapters)

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Electromagnetism817 Qs

Electrostatics

Coulomb's law, Gauss's theorem, electric field, potential, dipoles, and electrostatic potential energy.

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Electromagnetism431 Qs

Capacitance & Dielectrics

Parallel plate capacitors, dielectric insertion, stored energy, common potential, and transient RC circuits.

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Electromagnetism902 Qs

Current Electricity

Ohm's law, drift velocity, resistivity, Kirchhoff's laws, Wheatstone bridge, potentiometer, and electrical power.

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Electromagnetism480 Qs

Magnetic Effects of Current

Biot-Savart law, Ampere's circuital law, magnetic field of straight wire, circular loop, solenoid, and Lorentz magnetic force.

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Electromagnetism490 Qs

Electromagnetic Induction (EMI)

Faraday's law, Lenz's law, motional EMF, self & mutual inductance, and LR growth/decay circuits.

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Electromagnetism606 Qs

Alternating Current (AC)

RMS and peak values, reactance, impedance of LCR circuits, resonance, Q-factor, and power factor.

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Electromagnetism299 Qs

Electromagnetic Waves (EM Waves)

Maxwell equations, displacement current, wave speed in vacuum, Poynting vector, and electromagnetic spectrum.

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Formulas Reference (Electromagnetism)

7 Chapters match

ElectromagnetismTop Priority (3-4 Qs in JEE)817 PYQs

Electrostatics

Coulomb's law, Gauss's theorem, electric field, potential, dipoles, and electrostatic potential energy.

Chapter

Coulomb's Law & Permittivity

F = \frac{1}{4\pi\varepsilon_0 \varepsilon_r} \frac{q_1 q_2}{r^2}, \quad \frac{1}{4\pi\varepsilon_0} \approx 9 \times 10^9 \text{ N m}^2/\text{C}^2

Inverse-square force between point charges in medium of relative permittivity ε_r.

Force in medium of dielectric constant K is F_air / K.

Gauss's Law & Electric Flux

\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\varepsilon_0}

Total electric flux through closed Gaussian surface equals enclosed charge divided by ε₀.

Charges outside the Gaussian surface contribute zero net flux.

Electric Field of Standard Geometries

E_{\text{wire}} = \frac{2k\lambda}{r}, \quad E_{\text{sheet}} = \frac{\sigma}{2\varepsilon_0}, \quad E_{\text{ring, axis}} = \frac{kqx}{(R^2 + x^2)^{3/2}}

Field formulas derived via Gauss's law for high-symmetry distributions.

Maximum field on ring axis occurs at x = R / √2.

Electric Dipole Fields & Torque

E_{\text{axial}} = \frac{2kp}{r^3}, \quad E_{\text{equatorial}} = \frac{kp}{r^3}, \quad \vec{\tau} = \vec{p} \times \vec{E}, \quad U = -\vec{p} \cdot \vec{E}

Field and potential energy of electric dipole p = q(2a) at distant points (r >> a).

Axial field is exactly twice the equatorial field: E_axial = 2 E_eq.

Relation Between Field and Potential

\vec{E} = -\vec{\nabla} V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right), \quad \Delta V = -\int \vec{E} \cdot d\vec{r}

Electric field points in direction of steepest decrease of potential.

Equipotential surfaces are always perpendicular to electric field lines.

Uniform Solid Dielectric Sphere (Field & Potential)

r \le R: \; E(r) = \frac{k Q r}{R^3} = \frac{\rho r}{3\varepsilon_0}, \quad V(r) = \frac{kQ}{2R^3}(3R^2 - r^2) \implies V_{\text{center}} = 1.5 \, V_{\text{surface}}

Linear internal electric field and parabolic internal potential distribution for uniform volume charge density ρ.

Potential at the exact center of a uniformly charged solid non-conducting sphere is 1.5 times the surface potential.

Electrostatic Self-Energy of Spheres

U_{\text{conducting shell}} = \frac{k Q^2}{2R}, \quad U_{\text{solid insulating sphere}} = \frac{3}{5}\frac{k Q^2}{R}

Work done in assembling a continuous spherical charge distribution against mutual electrostatic repulsion.

For concentric shells: total energy is sum of individual self-energies plus mutual interaction energy.

Earthing of Concentric Conducting Shells

V_{\text{earthed shell}} = 0 \implies \sum_{i} \frac{k q_i}{R_{\text{effective}}} = 0

When a conducting shell is grounded to Earth, charge flows to/from Earth until its net electrostatic potential becomes exactly zero.

Earthing a shell does NOT mean its charge is zero; it means its net potential V = 0.

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Capacitance & Dielectrics

Parallel plate capacitors, dielectric insertion, stored energy, common potential, and transient RC circuits.

Chapter

Parallel Plate Capacitance

C = \frac{\varepsilon_0 A}{d}, \quad C_{\text{dielectric}} = \frac{\varepsilon_0 A}{d - t + \frac{t}{K}}

Capacitance with vacuum and with dielectric slab of thickness t and constant K.

Filling entire gap with dielectric increases capacitance to K*C₀.

Stored Energy & Energy Density

U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV, \quad u_E = \frac{1}{2}\varepsilon_0 E^2

Electrostatic potential energy stored in the electric field between capacitor plates.

Energy density u_E has units of Joules per cubic meter (J/m³).

Combinations of Capacitors

C_{\text{parallel}} = \sum C_i, \quad \frac{1}{C_{\text{series}}} = \sum \frac{1}{C_i}

Series combination has same charge on each capacitor; parallel has same potential difference.

Notice rules are opposite to resistor combinations.

Charge Sharing & Heat Loss

V_{\text{common}} = \frac{C_1 V_1 + C_2 V_2}{C_1 + C_2}, \quad \Delta H = \frac{1}{2}\frac{C_1 C_2}{C_1 + C_2}(V_1 - V_2)^2

Common potential and heat dissipated when two charged capacitors are connected.

Heat loss ΔH is always positive (energy is lost, never gained).

Transient RC Circuit (Charging & Discharging)

q(t) = Q_0(1 - e^{-t/RC}) \; (\text{Charging}), \quad q(t) = Q_0 e^{-t/RC} \; (\text{Discharging}), \quad \tau = RC

Exponential charging and discharging with capacitive time constant τ = RC.

At t = τ, capacitor charges to ~63.2% of Q₀; at t = 5τ, it is ~99.3% charged.

Attractive Force Between Plates & Electrostatic Pressure

F = \frac{Q^2}{2\varepsilon_0 A} = \frac{1}{2}QE, \quad P_{\text{elec}} = \frac{F}{A} = \frac{\sigma^2}{2\varepsilon_0} = \frac{1}{2}\varepsilon_0 E^2

Mutual electrostatic attraction between oppositely charged plates, and electrostatic outward pulling pressure.

The factor of 1/2 arises because the charge on one plate experiences the electric field E/2 produced solely by the other plate.

Spherical & Cylindrical Capacitor Geometries

C_{\text{spherical}} = 4\pi\varepsilon_0 \left(\frac{ab}{b - a}\right), \quad C_{\text{isolated sphere}} = 4\pi\varepsilon_0 R, \quad \frac{C_{\text{cylindrical}}}{L} = \frac{2\pi\varepsilon_0}{\ln(b/a)}

Capacitance formulas for concentric spherical shells and coaxial cylindrical conductors of radii a and b (b > a).

For Earth (R ≈ 6400 km), isolated spherical capacitance is C ≈ 711 μF.

Dielectric Slab Partially Inserted (Series & Parallel)

\text{Slab (thickness } t\text{, full area):}\; C = \frac{\varepsilon_0 A}{d - t + t/K} \quad \text{(series stacking)}\\[4pt] \text{Slab (partial area } A_1\text{, rest } A_2\text{):}\; C = \frac{\varepsilon_0 K A_1}{d} + \frac{\varepsilon_0 A_2}{d} = \frac{\varepsilon_0}{d}(K A_1 + A_2)

When dielectric occupies only part of the gap thickness (series): effective gap is reduced by t(1 - 1/K). When dielectric covers part of the plate area (parallel): two sub-capacitors add directly.

Battery connected while inserting: V = const → Q increases, C increases, E between plates unchanged. Battery disconnected: Q = const → V decreases, C increases, E decreases. Energy is LOST when battery connected, REDUCED when disconnected.

ElectromagnetismTop Priority (3-4 Qs in JEE)902 PYQs

Current Electricity

Ohm's law, drift velocity, resistivity, Kirchhoff's laws, Wheatstone bridge, potentiometer, and electrical power.

Chapter

Drift Velocity & Current Density

I = n e A v_d, \quad v_d = \frac{e E \tau}{m}, \quad j = \sigma E = \frac{E}{\rho}, \quad \rho = \frac{m}{n e^2 \tau}

Microscopic electron transport relations.

Relaxation time τ decreases with temperature in metals, increasing resistivity ρ.

Resistance & Temperature Coefficient

R = \rho \frac{L}{A}, \quad R_T = R_0(1 + \alpha \Delta T)

Geometric resistance formula and thermal resistance variation.

Stretching a wire to n times its length increases resistance by a factor of n² (volume constant).

Kirchhoff's Laws (KCL & KVL)

\sum I_{\text{node}} = 0 \; (\text{Charge Conservation}), \quad \sum \Delta V_{\text{loop}} = 0 \; (\text{Energy Conservation})

Fundamental nodal current and loop voltage laws for circuit analysis.

Always maintain consistent sign conventions across resistors and battery terminals.

Grouping of Cells (EMF & Internal Resistance)

E_{eq} = \frac{\sum E_i / r_i}{\sum 1 / r_i} \; (\text{Parallel}), \quad \frac{1}{r_{eq}} = \sum \frac{1}{r_i}, \quad E_{eq} = \sum E_i \; (\text{Series})

Equivalent battery combination formulas.

Maximum power transfer occurs when external load resistance equals internal resistance (R = r).

Wheatstone Bridge & Meter Bridge

\frac{P}{Q} = \frac{R}{S} \implies I_G = 0, \quad \text{Meter Bridge: } \frac{R}{S} = \frac{l}{100 - l}

Null deflection condition for measuring unknown resistances.

Sensitivity is maximum when all four branch resistances are of comparable magnitude.

Potentiometer Principle

\frac{E_1}{E_2} = \frac{l_1}{l_2}, \quad r = R\left(\frac{l_1}{l_2} - 1\right)

Compares EMFs and determines internal resistance without drawing current from the test cell.

Potentiometer is superior to voltmeter because it draws zero current at balance point.

Maximum Power Transfer Theorem & Heating

P_{\text{load}} = \frac{E^2 R}{(R + r)^2} \implies P_{\text{max}} = \frac{E^2}{4r} \quad (\text{when } R = r), \quad H = I^2 R t

Maximum electrical power is transferred to load when load resistance equals the internal resistance of the power source.

Efficiency of power transfer at maximum power output is exactly 50% (half energy dissipated in internal resistance r).

Seebeck Effect & Thermoelectric Inversion

E = \alpha T + \frac{1}{2}\beta T^2, \quad T_n = -\frac{\alpha}{\beta}, \quad T_i - T_n = T_n - T_0 \implies T_i = 2T_n - T_0

Thermoelectric EMF generated across thermocouple junctions at temperatures T and T₀ (T_n is neutral temperature, T_i is inversion temperature).

At neutral temperature T_n, thermoelectric power (dE/dT) is zero and EMF reaches its peak maximum.

Joule's Heating Law & Electrical Power

P = VI = I^2 R = \frac{V^2}{R}, \quad H = Pt = I^2 R t = \frac{V^2 t}{R}, \quad \text{kWh} = 1 \text{ unit} = 3.6 \times 10^6 \text{ J}

Heat produced by current I through resistance R in time t. All three forms of power (P = VI, I²R, V²/R) are equivalent by Ohm's law.

For resistors in series, same I → more heat in larger R. For parallel, same V → more heat in smaller R. Domestic electrical energy billing unit 'one unit' = 1 kWh.

ElectromagnetismHigh Yield (2-3 Qs in JEE)480 PYQs

Magnetic Effects of Current

Biot-Savart law, Ampere's circuital law, magnetic field of straight wire, circular loop, solenoid, and Lorentz magnetic force.

Chapter

Lorentz Magnetic Force

\vec{F}_m = q(\vec{v} \times \vec{B}), \quad r = \frac{m v_\perp}{qB}, \quad T = \frac{2\pi m}{qB}

Force on moving charge in magnetic field and helical circular radius.

Time period and cyclotron frequency are completely independent of particle speed v.

Biot-Savart Law

d\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{l} \times \hat{r})}{r^2}, \quad \frac{\mu_0}{4\pi} = 10^{-7} \text{ T m/A}

Fundamental differential magnetic field generated by current element.

Direction is given by right-hand palm rule.

Magnetic Field of Standard Geometries

B_{\text{wire}} = \frac{\mu_0 I}{2\pi d}, \quad B_{\text{ring, center}} = \frac{\mu_0 I}{2R}, \quad B_{\text{ring, axis}} = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}

Field formulas derived from Biot-Savart integration.

For ideal long solenoid, B = μ₀ n I inside, and 0 outside.

Force Between Two Parallel Current Wires

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}

Attractive if currents flow in same direction; repulsive if in opposite directions.

Used to define the international Ampere standard.

Magnetic Dipole Moment & Torque on Loop

\vec{M} = N I \vec{A}, \quad \vec{\tau} = \vec{M} \times \vec{B}, \quad U = -\vec{M} \cdot \vec{B}

Magnetic dipole behavior of planar current loop.

Gyromagnetic ratio for revolving electron: M / L = e / (2m).

Ampère's Circuital Law, Solenoid & Thick Wire

\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}, \quad B_{\text{solenoid}} = \mu_0 n I, \quad B_{\text{thick wire}}(r \le R) = \frac{\mu_0 I r}{2\pi R^2}

Circuital law line integral and internal fields for infinite solenoid, toroid, and solid cylindrical conductors.

Field at the exact ends of a semi-infinite solenoid is exactly half the interior field: B_end = (1/2)μ₀ n I.

Moving Coil Galvanometer, Ammeter & Voltmeter

\theta = \left(\frac{NAB}{C}\right)I, \quad \text{Shunt: } S = \frac{I_g R_g}{I - I_g}, \quad \text{Multiplier: } R = \frac{V}{I_g} - R_g

Torsional suspension equilibrium and conversion into ammeter (low parallel shunt S) and voltmeter (high series resistance R).

Ideal ammeter has zero resistance; ideal voltmeter has infinite resistance.

Bohr Magneton & Revolving Electron

\mu_l = \left(\frac{e}{2m}\right)L = n\mu_B, \quad \mu_B = \frac{e\hbar}{2m} = \frac{eh}{4\pi m} \approx 9.27 \times 10^{-24}\,\text{A}\cdot\text{m}^2

Magnetic moment of an electron orbiting a nucleus in the n-th Bohr orbit.

Gyromagnetic ratio e/(2m) is a universal constant independent of orbit radius and velocity.

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Electromagnetic Induction (EMI)

Faraday's law, Lenz's law, motional EMF, self & mutual inductance, and LR growth/decay circuits.

Chapter

Faraday's & Lenz's Law

\varepsilon = -\frac{d\Phi_B}{dt}, \quad \Phi_B = \vec{B} \cdot \vec{A} = B A \cos\theta

Induced electromotive force opposing changing magnetic flux.

Total induced charge q = ΔΦ / R is independent of the rate of change of flux.

Motional EMF

\varepsilon = (\vec{v} \times \vec{B}) \cdot \vec{l} = B v l \; (\text{Translational}), \quad \varepsilon = \frac{1}{2}B \omega L^2 \; (\text{Rotating Rod})

Voltage induced across conductors moving through magnetic field.

Induced current direction given by Fleming's Right-Hand Rule.

Self & Mutual Inductance

\Phi = L I \implies \varepsilon = -L \frac{dI}{dt}, \quad L_{\text{solenoid}} = \mu_0 n^2 A l = \frac{\mu_0 N^2 A}{l}, \quad M = k\sqrt{L_1 L_2}

Inertia of electric current in coils and coupling coefficient k.

Stored magnetic energy density: u_B = B² / (2μ₀).

Transient LR Circuit (Current Growth & Decay)

I(t) = \frac{V}{R}(1 - e^{-t / \tau}) \; (\text{Growth}), \quad I(t) = I_0 e^{-t / \tau} \; (\text{Decay}), \quad \tau = \frac{L}{R}

Inductive time constant governing current evolution.

At t = 0, inductor acts as open circuit; at t -> ∞, acts as ideal short wire.

Non-Conservative Induced Electric Field

\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} \implies E(r) = \begin{cases} \frac{r}{2}\frac{dB}{dt} & (r \le R) \\ \frac{R^2}{2r}\frac{dB}{dt} & (r > R) \end{cases}

Electric field produced by time-varying magnetic flux inside and outside a cylindrical field region of radius R.

Unlike electrostatic fields, induced electric field lines form closed continuous loops and work done along a closed path is non-zero.

Magnetic Force & Power Dissipated in Moving Loop

F_{\text{magnetic}} = \frac{B^2 L^2 v}{R}, \quad P_{\text{mechanical}} = P_{\text{Joule}} = \frac{B^2 L^2 v^2}{R}, \quad v(t) = v_0 e^{-t / \tau}

Retarding Lorentz braking force on a conducting rod sliding across uniform field rails with load resistance R.

Mechanical power input equals rate of Joule heat dissipation in the circuit, verifying energy conservation.

Inductance Combinations & Stored Magnetic Energy

U_B = \frac{1}{2} L I^2, \quad u_B = \frac{B^2}{2\mu_0}, \quad L_{\text{series}} = L_1 + L_2 \pm 2M, \quad L_{\text{parallel}} = \frac{L_1 L_2 - M^2}{L_1 + L_2 \mp 2M}

Total magnetic energy stored in current-carrying inductor and equivalent inductance of coupled coils.

Use +2M when magnetic fluxes aid each other; use -2M when fluxes oppose each other.

AC Generator (Rotating Coil) EMF

\varepsilon = NBA\omega\sin(\omega t) = \varepsilon_0\sin(\omega t), \quad \varepsilon_0 = NBA\omega, \quad \varepsilon_{\text{rms}} = \frac{\varepsilon_0}{\sqrt{2}} = \frac{NBA\omega}{\sqrt{2}}

EMF produced by N-turn coil of area A rotating with angular velocity ω in uniform magnetic field B. Peak EMF ε₀ = NBAω. Coil plane is parallel to B at maximum EMF; perpendicular to B at zero EMF.

Maximum EMF occurs when coil plane is parallel to B (flux = 0, rate of change of flux is maximum). Zero EMF when coil is perpendicular to B (flux maximum, rate of change = 0). Phase difference between flux and EMF is 90°.

Eddy Currents — Power Loss & Applications

P_{\text{eddy}} \propto B^2 f^2 t^2 / \rho_{\text{resistivity}}, \quad \text{Skin depth: } \delta = \sqrt{\frac{2\rho}{\omega \mu}}

Eddy currents are induced in bulk conductors by changing B. Loss is reduced by laminating the core (thin silicon steel sheets increase resistivity path). Power loss ∝ (frequency)² × (thickness)² × (B)².

Applications: electromagnetic braking (train brakes), induction furnace (heating by eddy currents), galvanometer damping, speedometers. Silicon steel (high resistivity) is used in transformer cores to minimise eddy current losses.

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Alternating Current (AC)

RMS and peak values, reactance, impedance of LCR circuits, resonance, Q-factor, and power factor.

Chapter

RMS & Average Values of AC

I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0, \quad I_{\text{avg}} = \frac{2I_0}{\pi} \approx 0.637 I_0 \; (\text{half cycle})

Root-mean-square effective heating current and half-wave rectification average.

Domestic AC supply of 220 V means V_rms = 220 V (peak V₀ ≈ 311 V).

Inductive & Capacitive Reactance

X_L = \omega L = 2\pi f L, \quad X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}

Frequency-dependent opposition to current.

Inductor blocks high frequencies (X_L ∝ f); capacitor blocks DC (X_C -> ∞ at f = 0).

Series LCR Impedance & Phase Angle

Z = \sqrt{R^2 + (X_L - X_C)^2}, \quad \tan\phi = \frac{X_L - X_C}{R}

Total impedance vector and phase angle between voltage and current.

Current leads voltage if X_C > X_L (capacitive); lags if X_L > X_C (inductive).

Resonance & Quality Factor in LCR

\omega_0 = \frac{1}{\sqrt{LC}} \implies f_0 = \frac{1}{2\pi\sqrt{LC}}, \quad Z_{\text{min}} = R, \quad Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}

Condition when inductive and capacitive reactances cancel out.

At resonance, current is maximum, impedance is purely resistive, and power factor cos φ = 1.

Average Power & Power Factor

P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi, \quad \text{Power Factor } \cos\phi = \frac{R}{Z}

Real electrical power dissipated in AC circuit.

Wattless current component is I_rms * sin φ.

Transformer Equation & Efficiency

\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}, \quad \eta = \frac{P_{\text{out}}}{P_{\text{in}}} = \frac{V_s I_s}{V_p I_p}

Voltage transformation ratio in mutual induction transformer.

Step-up transformer increases voltage but decreases current.

LC Oscillations & Total Stored Energy

q(t) = Q_0 \cos(\omega_0 t), \quad I(t) = -\omega_0 Q_0 \sin(\omega_0 t), \quad E_{\text{total}} = \frac{q^2}{2C} + \frac{1}{2}LI^2 = \frac{Q_0^2}{2C}

Harmonic energy exchange between electrical field in capacitor and magnetic field in inductor.

Analogy with mechanical SHM: mass m ↔ inductance L, spring constant k ↔ 1/C, velocity v ↔ current I.

Bandwidth & Half-Power Frequencies in LCR

\Delta\omega = \omega_2 - \omega_1 = \frac{R}{L}, \quad Q = \frac{\omega_0}{\Delta\omega} = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}

Frequency width between half-power points where current drops to I_max / √2. Smaller bandwidth implies sharper resonance.

High Q circuits are required in radio receivers for tuning sharply to a single carrier frequency.

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Electromagnetic Waves (EM Waves)

Maxwell equations, displacement current, wave speed in vacuum, Poynting vector, and electromagnetic spectrum.

Chapter

Wave Speed & Field Amplitude Ratio

c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = \frac{E_0}{B_0} = \frac{E_{\text{rms}}}{B_{\text{rms}}} \approx 3 \times 10^8 \text{ m/s}

Fundamental connection between permittivity, permeability, and optical field amplitudes.

In a medium of refractive index n: v = c / n = 1 / √(μ ε).

Energy Density & Intensity of EM Waves

u_{\text{avg}} = \frac{1}{2}\varepsilon_0 E_0^2 = \frac{B_0^2}{2\mu_0}, \quad I = u_{\text{avg}} c = \frac{1}{2}\varepsilon_0 c E_0^2

Energy is divided equally between electric and magnetic fields.

Poynting vector S = (1/μ₀)(E × B) represents directional power flux.

Displacement Current

I_d = \varepsilon_0 \frac{d\Phi_E}{dt} = C \frac{dV}{dt}

Maxwell's addition to Ampere's circuital law inside capacitor dielectric gap.

Displacement current between capacitor plates equals conduction current in lead wires.

Poynting Vector (Directional Energy Flux)

\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B}), \quad I = \langle S \rangle = \frac{E_0 B_0}{2\mu_0} = \frac{E_0^2}{2\mu_0 c} = \frac{1}{2}\varepsilon_0 c E_0^2

Instantaneous directional energy flow per unit area carried by propagating electromagnetic waves.

Wave propagates strictly in the direction of vector cross product (E × B).

Radiation Pressure & Momentum

p = \frac{U}{c}, \quad P_{\text{rad}} = \begin{cases} \frac{I}{c} & (\text{Complete Absorption}) \\ \frac{2I}{c} & (\text{Complete Reflection}) \end{cases}

Mechanical force per unit area exerted by electromagnetic radiation on physical surfaces.

For oblique incidence at angle θ: P_rad = (I/c) cos²θ (absorption); P_rad = (2I/c) cos²θ (reflection).

Free Space Impedance & EM Spectrum Hierarchy

Z_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \frac{E_0}{H_0} \approx 376.7 \,\Omega \approx 120\pi \,\Omega

Intrinsic wave impedance of vacuum. Frequency order: Radio < Micro < IR < Visible < UV < X-ray < Gamma.

Memory mnemonic for increasing frequency / decreasing wavelength: R-M-I-V-U-X-G (Radio, Microwave, IR, Visible, UV, X-ray, Gamma).

Maxwell's Four Fundamental Equations

\oint \vec{E}\cdot d\vec{A} = \frac{Q}{\varepsilon_0}, \; \oint \vec{B}\cdot d\vec{A} = 0, \; \oint \vec{E}\cdot d\vec{l} = -\frac{d\Phi_B}{dt}, \; \oint \vec{B}\cdot d\vec{l} = \mu_0\left(I_c + \varepsilon_0\frac{d\Phi_E}{dt}\right)

Unification of electricity and magnetism describing generation of fields by charges, currents, and time-varying opposite fields.

Gauss's law for magnetism ∮ B · dA = 0 implies that magnetic monopoles do not exist in nature.

Wave Speed in Material Media & Refractive Index

v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{\sqrt{\mu_r \varepsilon_r}} = \frac{c}{n} \implies n \approx \sqrt{\varepsilon_r} = \sqrt{K} \quad (\text{for } \mu_r \approx 1)

Maxwell's relation linking optical refractive index n to static dielectric constant K.

Both E and B amplitudes decrease by factor n in dielectric medium, while wave frequency f remains constant.