Chapter 8: Electromagnetic Waves

In 1864 James Clerk Maxwell unified electricity, magnetism and light. By adding a single correction term to Ampere's law (the displacement current) he made the four field equations mathematically consistent and predicted that the fields themselves could propagate as waves through empty space at a speed determined entirely by two electromagnetic constants: $\mu_0$ and $\varepsilon_0$. The predicted speed matched the measured speed of light to within experimental uncertainty — a stunning achievement. Hertz confirmed the existence of these waves in 1887. Today the electromagnetic spectrum, from radio to gamma rays, is the foundation of all modern communication, imaging, and astronomy.

Concept Map

• Displacement current: $i_d = \varepsilon_0\,d\Phi_E/dt$ — fixes the inconsistency of Ampere's law for time-varying fields
• Maxwell's equations (4): Gauss for $E$, Gauss for $B$, Faraday, Ampere–Maxwell
• EM waves: transverse, self-propagating, travel at $c = 1/\sqrt{\mu_0\varepsilon_0} \approx 3\times 10^8\,\text{m/s}$
• In a medium: $v = 1/\sqrt{\mu\varepsilon} = c/n$, with refractive index $n = \sqrt{\mu_r\varepsilon_r}$
• Field structure: $\vec E\perp\vec B\perp\vec k$, in phase, $E_0/B_0 = c$
• Energy density: $u_E = u_B = \tfrac14\varepsilon_0 E_0^2$ (time-averaged); $u = \tfrac12\varepsilon_0 E_0^2$ total
• Intensity: $I = \tfrac12 c\varepsilon_0 E_0^2$, momentum $p = U/c$, radiation pressure $P_{\text{rad}} = I/c$ (absorber) or $2I/c$ (reflector)
• Spectrum: $\gamma$, X-ray, UV, visible, IR, microwave, radio — all the same wave, different wavelengths

8.1 Displacement Current — Maxwell's Correction to Ampere's Law

The problem with Ampere's law

Ampere's law (Ch.4) states

$\oint\vec B\cdot d\vec\ell = \mu_0\,I_{\text{enc}},$

where $I_{\text{enc}}$ is the conduction current passing through any open surface bounded by the loop. But: consider a circuit charging a parallel-plate capacitor. Choose an Amperian loop around the wire, and consider two surfaces both bounded by it: (a) a flat disc cutting the wire — $I_{\text{enc}} = I$, (b) a bulging surface that passes between the plates — $I_{\text{enc}} = 0$ (no conduction current crosses the gap).

The left-hand side of Ampere's law depends only on the loop, not on the surface chosen. We get different values of $I_{\text{enc}}$ from the two surfaces — a contradiction. Something is missing.

Maxwell's fix

Maxwell noted that during charging, the electric field between the plates changes with time — the changing $\vec E$ "carries" the missing current. He defined the displacement current:

$i_d = \varepsilon_0\,\frac{d\Phi_E}{dt},\quad \Phi_E = \int\vec E\cdot d\vec A.$

For a parallel-plate capacitor with area $A$ and charge $q(t)$ on a plate,

$E = q/(\varepsilon_0 A),\quad \Phi_E = EA = q/\varepsilon_0,\quad i_d = \varepsilon_0\,d\Phi_E/dt = dq/dt = i\,\checkmark$

Exactly equal to the conduction current in the wire. Magic.

Modified (Ampere–Maxwell) law

$\boxed{\;\oint\vec B\cdot d\vec\ell = \mu_0\big(I_c + I_d\big) = \mu_0 I_c + \mu_0\varepsilon_0\,\frac{d\Phi_E}{dt}\;}$

Now both surfaces give the same RHS: surface (a) has $I_c = I, I_d = 0$; surface (b) has $I_c = 0, I_d = I$. The contradiction is resolved.

Significance

The displacement current is not a flow of charges — it is the physical fact that a changing electric field itself produces a magnetic field. This symmetry (changing $E$ produces $B$, changing $B$ produces $E$, by Faraday) is the engine of electromagnetic waves.

Worked Example

A parallel-plate capacitor with circular plates of radius $5.0\,\text{cm}$ has the charge on its plates increasing at $dq/dt = 5.0\,\text{A}$. Find the magnetic field at a point $3.0\,\text{cm}$ from the axis, between the plates.

Between the plates, no conduction current; only displacement current. The displacement current density is uniform: $J_d = I/A = 5.0/(\pi\times 0.05^2) = 636.6\,\text{A/m}^2$. By symmetry, apply Ampere–Maxwell on a circle of radius $r = 3\,\text{cm}$ between the plates:

$B\cdot 2\pi r = \mu_0 J_d\cdot\pi r^2\implies B = \frac{\mu_0 J_d r}{2} = \frac{4\pi\times 10^{-7}\times 636.6\times 0.03}{2} = 1.2\times 10^{-5}\,\text{T}.$

Pitfalls

• $i_d = \varepsilon_0\,d\Phi_E/dt$ — the constant is $\varepsilon_0$ (vacuum permittivity).
• Displacement current is not a current in the conventional sense — no charges flow, but it acts like one in producing $B$.
• It is essential for time-varying fields; in steady-state, $i_d = 0$.

8.2 Maxwell's Equations

The four Maxwell equations, in integral form (vacuum, with charges and currents):

Add the Lorentz force $\vec F = q(\vec E + \vec v\times\vec B)$ and the classical theory of electromagnetism is complete.

Symmetry

Equations 3 and 4 are nearly symmetric: $\vec E$ and $\vec B$ each generate the other when changing. The only asymmetry is the source term $I_c$ on the magnetic side and the lack of a "magnetic charge" term — magnetic monopoles, if they existed, would symmetrize them perfectly.

Wave equation from Maxwell

In a source-free region ($q = 0$, $I_c = 0$), taking the curl of Faraday and substituting into Ampere–Maxwell (or vice versa) gives

$\nabla^2\vec E = \mu_0\varepsilon_0\,\frac{\partial^2\vec E}{\partial t^2},$

a classical wave equation with wave speed

$\boxed{\;c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}\;}$

Plugging in $\mu_0 = 4\pi\times 10^{-7}\,\text{T m/A}$ and $\varepsilon_0 = 8.854\times 10^{-12}\,\text{C}^2/(\text{N m}^2)$:

$c = \frac{1}{\sqrt{4\pi\times 10^{-7}\times 8.854\times 10^{-12}}} = 2.998\times 10^8\,\text{m/s}.$

Exactly the speed of light. Light is an electromagnetic wave.

Worked Example (conceptual)

If only Maxwell's equation #4 lacked the displacement-current term ($\mu_0\varepsilon_0\,d\Phi_E/dt$), how would the wave equation change? The wave equation would no longer follow — you cannot derive a self-propagating wave from Ampere without Maxwell's correction. The very existence of EM waves requires the displacement current.

Pitfalls

• Differential vs integral form — both are equivalent, NCERT uses integral.
• Maxwell's equations are partial differential equations; they unify electromagnetic phenomena into a single theory.

8.3 Electromagnetic Waves — Nature and Speed

Properties

EM waves, derived from Maxwell's equations, have the following properties:

1. Transverse: both $\vec E$ and $\vec B$ are perpendicular to the direction of propagation $\hat k$. There is no longitudinal component.
2. $\vec E\perp\vec B$: the two field vectors are also perpendicular to each other.
3. In phase: $\vec E$ and $\vec B$ reach their maxima, zeros and minima at the same point and time.
4. Self-propagating: they need no medium — they travel through vacuum at $c$.
5. Carry energy and momentum — and therefore exert pressure.

Speed in vacuum

$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3.00\times 10^8\,\text{m/s}.$

This is one of the most fundamental constants of nature. The SI second is now defined so that $c$ is exactly $299\,792\,458\,\text{m/s}$.

Speed in a medium

In a non-conducting medium with permeability $\mu$ and permittivity $\varepsilon$:

$v = \frac{1}{\sqrt{\mu\varepsilon}} = \frac{c}{\sqrt{\mu_r\varepsilon_r}} = \frac{c}{n},$

where $n = \sqrt{\mu_r\varepsilon_r}$ is the refractive index of the medium. For non-magnetic media ($\mu_r \approx 1$), $n = \sqrt{\varepsilon_r}$.

For glass ($n \approx 1.5$), light travels at $2\times 10^8\,\text{m/s}$.

Form of the wave

A plane EM wave propagating along $+\hat x$:

$\vec E = E_0\,\hat j\,\sin(kx - \omega t),\quad \vec B = B_0\,\hat k\,\sin(kx - \omega t),$

with $\omega/k = c$ (or $v$ in a medium).

Worked Example

For an EM wave of frequency $f = 10^9\,\text{Hz}$ in vacuum:

$\lambda = c/f = 3\times 10^8/10^9 = 0.30\,\text{m} = 30\,\text{cm},$

$k = 2\pi/\lambda = 20.94\,\text{rad/m},\quad \omega = 2\pi f = 6.28\times 10^9\,\text{rad/s}.$

Pitfalls

• "Transverse" — both $\vec E$ and $\vec B$ are perpendicular to propagation. (Sound waves in air are longitudinal, by contrast.)
• $c = 1/\sqrt{\mu_0\varepsilon_0}$ holds in vacuum; in matter, replace by $\mu, \varepsilon$.
• EM waves don't need a medium — they travel freely through vacuum. (Mechanical waves do.)

8.4 $\vec E$ and $\vec B$ in an EM Wave

Relation $E_0/B_0 = c$

From Maxwell's equations applied to a plane wave $\vec E = E_0\sin(kx - \omega t)\hat j$, $\vec B = B_0\sin(kx - \omega t)\hat k$ (propagating along $\hat i$):

Faraday $\nabla\times\vec E = -\partial\vec B/\partial t$ gives (in 1D)

$-\frac{\partial E_y}{\partial x} = \frac{\partial B_z}{\partial t}\implies -k E_0\cos(kx-\omega t) = -\omega B_0\cos(kx-\omega t),$

so $kE_0 = \omega B_0$, i.e.,

$\boxed{\;\frac{E_0}{B_0} = \frac{\omega}{k} = c\;}$

(or $v$ in a medium). Since $c\gg 1$ in SI units, $E_0$ is numerically much larger than $B_0$ — but neither is "more important", they describe different aspects of the same wave.

Orientation rule

If the wave travels in $+\hat x$, $\vec E$ along $+\hat y$, then $\vec B$ along $+\hat z$ — the cross product $\hat E\times\hat B$ gives the direction of propagation $\hat k$.

Worked Example

A plane EM wave has $E_0 = 60\,\text{V/m}$. Then $B_0 = E_0/c = 60/(3\times 10^8) = 2.0\times 10^{-7}\,\text{T} = 200\,\text{nT}$.

Pitfalls

• The numerical asymmetry $E_0 \gg B_0$ does not mean $\vec E$ is "stronger" — the energy densities of $E$ and $B$ are equal (next section).
• $\hat E\times\hat B$ gives the direction of propagation. Reverse them and you get the wrong direction.

8.5 Energy Density of an EM Wave

Field energy densities

From electrostatics and magnetostatics:

$u_E = \tfrac12\varepsilon_0 E^2,\quad u_B = \frac{B^2}{2\mu_0}.$

For an EM wave with $B = E/c$:

$u_B = \frac{(E/c)^2}{2\mu_0} = \frac{E^2}{2\mu_0 c^2} = \frac{E^2}{2\mu_0}\cdot\mu_0\varepsilon_0 = \tfrac12\varepsilon_0 E^2 = u_E.$

The energy is equally divided between the electric and magnetic fields.

Total instantaneous:

$u = u_E + u_B = \varepsilon_0 E^2 = \frac{B^2}{\mu_0}.$

Time-averaged (using $\langle\sin^2\rangle = 1/2$):

$\boxed{\;\langle u\rangle = \tfrac12\varepsilon_0 E_0^2 = \tfrac12\,\frac{B_0^2}{\mu_0}\;}$

Intensity

The intensity $I$ is the time-averaged power per unit area carried by the wave. The Poynting vector $\vec S = (1/\mu_0)\vec E\times\vec B$ gives the instantaneous energy flux; its time average for a sinusoidal wave is

$\boxed{\;I = \langle S\rangle = \tfrac12 c\varepsilon_0 E_0^2 = \frac{cB_0^2}{2\mu_0}\;}$

For sunlight at the top of Earth's atmosphere, $I \approx 1361\,\text{W/m}^2$ (the solar constant), giving $E_0 \approx 1013\,\text{V/m}$ and $B_0 \approx 3.4\,\mu\text{T}$.

Worked Example

A laser beam of intensity $I = 1.5\,\text{kW/m}^2$ in vacuum. Find $E_0$ and $B_0$.

$E_0 = \sqrt{\frac{2I}{c\varepsilon_0}} = \sqrt{\frac{2\times 1500}{3\times 10^8\times 8.85\times 10^{-12}}} = 1063\,\text{V/m},$

$B_0 = E_0/c = 3.54\times 10^{-6}\,\text{T}.$

Pitfalls

• $u_E = u_B$ — students sometimes think the electric energy dominates because $E_0 \gg B_0$ in SI units. It does not; the factor of $c^2$ in the relation $B = E/c$ compensates.
• Intensity $I$ contains a factor $1/2$ (from $\langle\sin^2\rangle$). Don't forget it.

8.6 Momentum and Radiation Pressure

Momentum of an EM wave

A flux of EM energy carries momentum:

$p = U/c.$

A wave delivering energy $U$ at a surface also delivers momentum $U/c$ in the direction of propagation. The momentum density is $u/c$.

Radiation pressure on an absorber

A wave of intensity $I$ falling on a perfectly absorbing surface delivers energy $I$ per unit area per second and momentum $I/c$ per unit area per second. Pressure (force per unit area) =

$P_{\text{rad}}^{(\text{abs})} = I/c.$

Radiation pressure on a reflector

A perfectly reflecting surface reverses the momentum, so momentum transferred per unit time = $2I/c$:

$P_{\text{rad}}^{(\text{refl})} = 2I/c.$

Worked Example

For sunlight, $I = 1361\,\text{W/m}^2$. Pressure on a black absorber:

$P = I/c = 1361/(3\times 10^8) = 4.54\times 10^{-6}\,\text{Pa}.$

Tiny, but accumulated over the entire Earth's cross-section, it accelerates solar-sail spacecraft. Pressure on a perfect reflector (silver mirror): $9.08\times 10^{-6}\,\text{Pa}$.

Pitfalls

• Reflector pressure is twice that on absorber for the same intensity (factor of 2 from momentum reversal).
• The forces are extremely small for ordinary sources but become significant for stellar radiation, lasers, and the cosmic microwave background.

8.7 The Electromagnetic Spectrum

The full range of EM waves spans some 20 orders of magnitude in wavelength, all of them governed by the same Maxwell equations and all of them traveling at $c$ in vacuum. Here is a table covering all bands of the spectrum.

Spectrum visualization

Going from low frequency to high: radio $\to$ microwave $\to$ IR $\to$ visible $\to$ UV $\to$ X-ray $\to$ gamma. The frequency increases (and wavelength decreases) by ~20 orders of magnitude across the full spectrum.

Boundaries are not sharp

The transitions between regions are conventional, not physical. The same photon at the IR/microwave boundary is just an EM wave — only its production and detection methods differ.

Photon energy

$E = hf = hc/\lambda$. So gamma rays ($\lambda \sim 10^{-13}\,\text{m}$) have photon energies $\sim$ MeV; radio waves ($\lambda \sim 10\,\text{m}$) have photon energies $\sim 10^{-7}\,\text{eV}$. The huge dynamic range of biological and technological effects (warming, browning skin, damaging DNA, ionising tissue) is set by photon energy.

Worked Example

The wavelength of a microwave oven (2.45 GHz):

$\lambda = c/f = 3\times 10^8/(2.45\times 10^9) = 12.2\,\text{cm}.$

The wavelength of green light (550 nm):

$f = c/\lambda = 3\times 10^8/(550\times 10^{-9}) = 5.45\times 10^{14}\,\text{Hz}.$

The photon energy:

$E = hf = 6.626\times 10^{-34}\times 5.45\times 10^{14} = 3.61\times 10^{-19}\,\text{J} = 2.26\,\text{eV}.$

Pitfalls

• Microwaves are part of radio in some classifications; NCERT treats them separately.
• UV is sometimes split into UV-A (long), UV-B, UV-C; the dangerous biological UV is UV-B and UV-C, mostly absorbed by ozone.
• "X-rays from Bremsstrahlung" — high-energy electrons decelerating in a target produce a continuous X-ray spectrum.
• The atmosphere is transparent only to visible and parts of radio/microwave/IR; X-rays and gamma rays from space are observed by space-based telescopes.

Solved Problems

Problem 1 — Displacement current in a capacitor

A capacitor of $C = 100\,\mu\text{F}$ is charged by an AC source with $V(t) = 100\sin(100\pi t)\,\text{V}$. Find the displacement current's RMS value.

$i(t) = C\,dV/dt = 100\times 10^{-6}\times 100\times 100\pi\cos(100\pi t)\,\text{A},$

$I_0 = 3.14\,\text{A},\quad I_{\text{rms}} = I_0/\sqrt 2 = 2.22\,\text{A}.$

The displacement current between the plates equals the conduction current in the wire — consistent with Maxwell's correction.

Problem 2 — Speed of light in glass

For glass with $\varepsilon_r = 4$, $\mu_r = 1$: $n = \sqrt{\varepsilon_r\mu_r} = 2$, $v = c/n = 1.5\times 10^8\,\text{m/s}$.

Problem 3 — Field amplitudes from intensity

A radio wave from a 10 kW transmitter spreads spherically. At distance 10 km the intensity is

$I = \frac{10^4}{4\pi(10^4)^2} = 7.96\times 10^{-6}\,\text{W/m}^2.$

Field amplitudes:

$E_0 = \sqrt{2I/(c\varepsilon_0)} = \sqrt{2\times 7.96\times 10^{-6}/(3\times 10^8\times 8.85\times 10^{-12})} = 0.0774\,\text{V/m},$

$B_0 = E_0/c = 2.58\times 10^{-10}\,\text{T}.$

Problem 4 — Radiation pressure on a reflector

A laser of $1\,\text{W}$ focused on a $1\,\text{mm}^2$ mirror: $I = 10^6\,\text{W/m}^2$. Reflector pressure:

$P = 2I/c = 2\times 10^6/(3\times 10^8) = 6.67\times 10^{-3}\,\text{Pa}.$

Force on mirror = $P\cdot A = 6.67\times 10^{-9}\,\text{N}$. Small but measurable with sensitive torsion balances (Lebedev, 1900).

Problem 5 — Wavelength of an FM radio station

FM station at 100 MHz: $\lambda = c/f = 3\,\text{m}$. The antenna is typically $\lambda/2 = 1.5\,\text{m}$ — this is why FM antennas are about a meter long.

Problem 6 — Photon energy of visible light

A green photon ($\lambda = 532\,\text{nm}$):

$E = hc/\lambda = (6.626\times 10^{-34}\times 3\times 10^8)/(532\times 10^{-9}) = 3.74\times 10^{-19}\,\text{J} = 2.33\,\text{eV}.$

(Just enough to excite electrons in semiconductors with bandgap $\sim 2\,\text{eV}$.)

Problem 7 — X-ray frequency

An X-ray of $\lambda = 0.1\,\text{nm}$: $f = c/\lambda = 3\times 10^{18}\,\text{Hz}$, photon energy $hf = 1.99\times 10^{-15}\,\text{J} = 12.4\,\text{keV}$ — typical of diagnostic X-rays.

JEE/NEET Edge Cases

• Displacement current existence: it exists only when $\vec E$ is changing. In a steady DC circuit with capacitors, $i_d = 0$ at steady state.
• EM waves in conductors: in a conductor, the wave is rapidly attenuated (skin effect). The standard EM-wave equation applies only in non-conducting media.
• Refractive index $n = \sqrt{\mu_r\varepsilon_r}$ in general; for non-magnetic media $n = \sqrt{\varepsilon_r}$, which is itself frequency-dependent (giving rise to dispersion).
• Polarisation: Maxwell's equations allow $\vec E$ to oscillate along any direction perpendicular to $\hat k$. Plane-polarised, circularly polarised, and elliptically polarised waves are all consistent with the theory.
• Standing EM waves (e.g., in a microwave cavity) — the fields are not propagating but oscillating in place. The energy density still equals $\tfrac12\varepsilon_0 E^2$ (instantaneous, then averaged).
• Hertz's experiment (1887): used a spark-gap transmitter and a resonant loop receiver to demonstrate EM waves of $\sim 5\,\text{m}$ wavelength, confirming Maxwell's prediction.
• Why are visible wavelengths special? The atmosphere is transparent in $400$–$700\,\text{nm}$; the Sun's emission peaks there; and our eyes evolved to detect it.
• Trap: $E_0 = B_0\cdot c$, not $B_0 = E_0\cdot c$. Sign of $c$ on the right matters.
• Trap: Intensity has the factor $1/2$ from time-averaging $\sin^2$. The instantaneous magnitude of the Poynting vector is $c\varepsilon_0 E^2(t)$ (no 1/2).
• Trap: Photon momentum $p = h/\lambda = E/c$.
• Trap: "Microwaves heat water" — microwave ovens use $\sim 2.45\,\text{GHz}$ specifically because water molecules absorb at this frequency (rotational resonance) — not because of any deep electromagnetic mechanism.

Quick Recap

• Displacement current $i_d = \varepsilon_0\,d\Phi_E/dt$ fixes Ampere's law for time-varying fields.
• Four Maxwell equations describe all classical electromagnetism.
• EM waves are transverse, travel at $c = 1/\sqrt{\mu_0\varepsilon_0} \approx 3\times 10^8\,\text{m/s}$ in vacuum, $v = c/n$ in a medium.
• $\vec E\perp\vec B\perp\hat k$, in phase, $E_0/B_0 = c$.
• Energy density: $u_E = u_B$, total $u = \varepsilon_0 E^2$, time-averaged $\tfrac12\varepsilon_0 E_0^2$.
• Intensity $I = \tfrac12 c\varepsilon_0 E_0^2$; momentum $p = U/c$; radiation pressure $I/c$ (absorber), $2I/c$ (reflector).
• Spectrum: radio, microwave, IR, visible, UV, X-ray, gamma — increasing frequency, decreasing wavelength.

Formula Sheet