Physics
Electromagnetic Waves
Electromagnetic Waves Formula Sheet
All key Electromagnetic Waves formulas: displacement current, Maxwell's equations, E-B relations, energy density, intensity, radiation pressure & the EM spectrum for JEE.
Every must-know formula, relation, and constant for Electromagnetic Waves in one scannable sheet — built for last-minute JEE Main and Advanced revision.
Displacement Current
The headline formula Maxwell added to fix Ampere’s law for time-varying fields:
$$\boxed{I_d = \varepsilon_0 \frac{d\Phi_E}{dt}}$$| Quantity | Formula | Notes |
|---|---|---|
| Displacement current | $I_d = \varepsilon_0 \dfrac{d\Phi_E}{dt}$ | $\Phi_E = \int \vec{E} \cdot d\vec{A}$ = electric flux |
| Displacement current (uniform field) | $I_d = \varepsilon_0 A \dfrac{dE}{dt}$ | $A$ = area of surface |
| Displacement current density | $J_d = \varepsilon_0 \dfrac{dE}{dt}$ | Units A/m² |
| Conduction current | $I_c = \dfrac{dq}{dt}$ | Real charge flow |
| Charging capacitor | $I_d = I_c$ | Always equal in a charging/discharging capacitor |
In a parallel-plate capacitor, the field is $E = \dfrac{\sigma}{\varepsilon_0} = \dfrac{q}{\varepsilon_0 A}$, giving $\dfrac{dE}{dt} = \dfrac{I_c}{\varepsilon_0 A}$, which forces $I_d = I_c$.
Ampere–Maxwell Law
$$\oint \vec{B} \cdot d\vec{l} = \mu_0(I_c + I_d) = \mu_0 I_c + \mu_0\varepsilon_0\frac{d\Phi_E}{dt}$$Differential form:
$$\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$$Magnetic Field Inside a Charging Capacitor
For circular plates of radius $R$, at distance $r < R$ from the axis:
$$\boxed{B = \frac{\mu_0 \varepsilon_0\, r}{2}\frac{dE}{dt}}$$- $B \propto r$ (inside the capacitor); $B = 0$ at the centre.
High-Yield
For a square-plate capacitor (side $a$) carrying conduction current $I_0$, the displacement current through a small circular area of radius $r$ between the plates is $I_d = \dfrac{\pi r^2 I_0}{a^2}$ — it scales with the area fraction.
Maxwell’s Equations (Integral Form)
| # | Law | Equation |
|---|---|---|
| 1 | Gauss’s law | $\oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{enc}}{\varepsilon_0}$ |
| 2 | Gauss’s law (magnetism) | $\oint \vec{B} \cdot d\vec{A} = 0$ |
| 3 | Faraday’s law | $\oint \vec{E} \cdot d\vec{l} = -\dfrac{d\Phi_B}{dt}$ |
| 4 | Ampere–Maxwell law | $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_c + \mu_0 \varepsilon_0 \dfrac{d\Phi_E}{dt}$ |
Key symmetry: changing $\vec{B} \rightarrow \vec{E}$ (Faraday); changing $\vec{E} \rightarrow \vec{B}$ (Maxwell). This symmetry is what produces self-propagating EM waves.
Speed of EM Waves
$$\boxed{c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3 \times 10^8 \text{ m/s}}$$| Quantity | Formula | Notes |
|---|---|---|
| Speed in vacuum | $c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$ | Universal for all EM waves |
| Speed in medium | $v = \dfrac{1}{\sqrt{\mu\varepsilon}} = \dfrac{c}{n}$ | $n$ = refractive index |
| Refractive index | $n = \sqrt{\mu_r \varepsilon_r}$ | Non-magnetic media: $n \approx \sqrt{\varepsilon_r}$ |
| Wave equation | $\dfrac{\partial^2 \vec{E}}{\partial t^2} = c^2 \nabla^2 \vec{E}$ | Same form for $\vec{B}$ |
Sample medium speeds: water ($n \approx 1.33$) → $v \approx 2.25 \times 10^8$ m/s; glass ($n \approx 1.5$) → $v \approx 2 \times 10^8$ m/s.
Wave Structure: E and B Fields
For a plane wave travelling in the $+x$ direction:
$$\vec{E} = E_0 \sin(kx - \omega t)\,\hat{j}, \qquad \vec{B} = B_0 \sin(kx - \omega t)\,\hat{k}$$Direction of propagation: $\hat{j} \times \hat{k} = \hat{i}$.
| Relation | Formula | Notes |
|---|---|---|
| Field ratio | $\dfrac{E}{B} = \dfrac{E_0}{B_0} = c$ | $E$ and $B$ in phase |
| Wave number | $k = \dfrac{2\pi}{\lambda}$ | |
| Angular frequency | $\omega = 2\pi f$ | |
| Dispersion relation | $\omega = ck$ | $c = \dfrac{\omega}{k}$ |
| Wave relation | $c = f\lambda$ |
Transverse & In-Phase
EM waves are transverse: $\vec{E} \perp \vec{B} \perp$ direction of propagation, and $\vec{E} \times \vec{B}$ points along the propagation direction. $\vec{E}$ and $\vec{B}$ reach their maxima and zeros at the same instant. Polarization is described by the $\vec{E}$-field direction. EM waves need no medium — they travel through vacuum at $c$.
Energy in EM Waves
Energy Density
| Quantity | Formula |
|---|---|
| Electric energy density | $u_E = \dfrac{1}{2}\varepsilon_0 E^2$ |
| Magnetic energy density | $u_B = \dfrac{B^2}{2\mu_0}$ |
| Total energy density | $u = u_E + u_B = \varepsilon_0 E^2 = \dfrac{B^2}{\mu_0} = \dfrac{EB}{\mu_0 c}$ |
Energy is split equally between the electric and magnetic fields.
Intensity and Poynting Vector
| Quantity | Formula | Notes |
|---|---|---|
| Poynting vector | $\vec{S} = \dfrac{1}{\mu_0}(\vec{E} \times \vec{B})$ | Energy flow per unit area per unit time |
| Instantaneous intensity | $I = uc = \varepsilon_0 c E^2 = \dfrac{cB^2}{\mu_0} = \dfrac{EB}{\mu_0}$ | Units W/m² |
| Average intensity | $\langle I \rangle = \dfrac{1}{2}\varepsilon_0 c E_0^2 = \dfrac{c B_0^2}{2\mu_0} = \dfrac{E_0 B_0}{2\mu_0}$ | $E_0, B_0$ = amplitudes |
Don't Drop the ½
For sinusoidal waves the time-average introduces a factor of $\tfrac{1}{2}$: use $\langle I \rangle = \tfrac{1}{2}\varepsilon_0 c E_0^2$ with amplitudes, not the instantaneous $I = \varepsilon_0 c E^2$.
Inverse Square Law (Point Source)
$$I = \frac{P}{4\pi r^2}$$Solar constant at Earth ($r = 1.5 \times 10^{11}$ m) ≈ 1350 W/m².
Momentum and Radiation Pressure
| Quantity | Formula | Notes |
|---|---|---|
| Momentum density | $\vec{p} = \dfrac{\vec{S}}{c^2} = \dfrac{u}{c}\hat{n}$ | |
| Photon momentum | $p = \dfrac{E}{c} = \dfrac{h}{\lambda}$ | EM waves carry momentum despite zero mass |
| Radiation pressure (absorption) | $P_{abs} = \dfrac{I}{c} = u$ | Surface absorbs the wave |
| Radiation pressure (reflection) | $P_{ref} = \dfrac{2I}{c} = 2u$ | Factor of 2 from momentum reversal |
| Force on surface | $F = P \times A$ | $F = \dfrac{IA}{c}$ (abs.), $\dfrac{2IA}{c}$ (refl.) |
Common Trap
The factor of 2 in $P_{ref} = \dfrac{2I}{c}$ comes from the momentum change $\Delta p = p - (-p) = 2p$ on reflection. Forgetting it is the most common error in radiation-pressure problems.
Doppler Effect for EM Waves
$$f' = f\sqrt{\frac{c + v_r}{c - v_r}}$$$v_r$ = radial velocity (positive if approaching). Used for astronomical redshift and radar speed detection.
Electromagnetic Spectrum
$$\boxed{c = f\lambda}, \qquad f = \frac{c}{\lambda}, \quad \lambda = \frac{c}{f}$$Order from longest wavelength / lowest energy to shortest / highest:
graph LR
A[Radio] --> B[Microwave] --> C[Infrared] --> D[Visible] --> E[Ultraviolet] --> F[X-rays] --> G[Gamma]| Type | Wavelength | Frequency | Source |
|---|---|---|---|
| Radio | > 0.1 m | < 3 GHz | Oscillating circuits, antennas |
| Microwave | 1 mm – 0.1 m | 3 GHz – 300 GHz | Magnetron, klystron, Gunn diode |
| Infrared | 700 nm – 1 mm | 300 GHz – 4×10¹⁴ Hz | Hot objects, molecular vibrations |
| Visible | 400 – 700 nm | 4×10¹⁴ – 7.5×10¹⁴ Hz | Atomic transitions (outer e⁻) |
| Ultraviolet | 10 – 400 nm | 7.5×10¹⁴ – 3×10¹⁶ Hz | Sun, mercury lamps |
| X-rays | 0.01 – 10 nm | 3×10¹⁶ – 3×10¹⁹ Hz | Inner-shell e⁻, X-ray tube |
| Gamma rays | < 0.01 nm | > 3×10¹⁹ Hz | Nuclei, cosmic sources |
Full span: frequency 10⁴ Hz (radio) → 10²² Hz (gamma); wavelength 10⁴ m → 10⁻¹⁴ m. All travel at $c$ in vacuum.
Order Mnemonics
Spectrum order: Radio → Microwave → Infrared → Visible → Ultraviolet → X-rays → Gamma (“Real Men In Vegas Usually X-ray Gorillas”). Visible: VIBGYOR — Violet (shortest λ, highest f) to Red (longest λ, lowest f). Ionizing: UV (partially), X-rays, and Gamma.
Visible Spectrum
| Color | Wavelength (nm) |
|---|---|
| Violet | 400–450 (highest frequency) |
| Indigo | 450–475 |
| Blue | 475–495 |
| Green | 495–570 |
| Yellow | 570–590 |
| Orange | 590–620 |
| Red | 620–700 (lowest frequency) |
Photon Energy and Momentum
| Quantity | Formula | Notes |
|---|---|---|
| Photon energy | $E = hf = \dfrac{hc}{\lambda}$ | $h = 6.626 \times 10^{-34}$ J·s |
| Energy in eV | $E(\text{eV}) = \dfrac{1240}{\lambda(\text{nm})}$ | Shortcut for visible/UV |
| Photon momentum | $p = \dfrac{E}{c} = \dfrac{h}{\lambda}$ |
Quick values: red (620 nm) ≈ 2 eV; blue (450 nm) ≈ 2.76 eV; UV (300 nm) ≈ 4.13 eV; X-ray (1 nm) ≈ 1.24 keV.
Key Constants
| Constant | Value | Notes |
|---|---|---|
| Permittivity of free space | $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m | Also C²/(N·m²) |
| Permeability of free space | $\mu_0 = 4\pi \times 10^{-7}$ T·m/A | $\approx 1.26 \times 10^{-6}$; also H/m |
| Speed of light | $c = 3 \times 10^8$ m/s | $c = 1/\sqrt{\mu_0\varepsilon_0}$ |
| Planck’s constant | $h = 6.626 \times 10^{-34}$ J·s | |
| Electron volt | $1 \text{ eV} = 1.6 \times 10^{-19}$ J |
Unit Gotchas
$1$ nm $= 10\,$Å (X-ray $1$ Å $= 0.1$ nm, not 1 nm). $B$ is numerically much smaller than $E$ since $B = E/c$. Intensity in W/m², energy density in J/m³, radiation pressure in Pa — don’t confuse them. Keep the exponent sign right: $\varepsilon_0 = 10^{-12}$, not $10^{12}$.