Electromagnetic Waves

8.1 Introduction — what Maxwell did

• Chapter 4: a current produces a magnetic field. Chapter 6: a changing magnetic field produces an electric field. Maxwell asked the converse — does a changing electric field produce a magnetic field? — and answered yes.
• Applying Ampere's circuital law to a point outside a capacitor being charged by a time-varying current, Maxwell found an inconsistency. To remove it he introduced an extra current, the displacement current, due to the time-varying electric field.
• Maxwell's equations, together with the Lorentz force formula, express all the basic laws of electromagnetism.
• Their most important prediction is the existence of electromagnetic waves — coupled, time-varying electric and magnetic fields that propagate through space.
• The speed the equations gave, $c = 1/\sqrt{\mu_0\varepsilon_0} \approx 3\times10^8$ m s$^{-1}$, matched the optically measured speed of light, so light is an electromagnetic wave. Maxwell thus unified electricity, magnetism and optics.
• Hertz demonstrated em waves experimentally in 1887 (NCERT also mentions 1885 in the introduction); J. C. Bose in Calcutta produced much shorter waves ($25$ mm to $5$ mm); Marconi transmitted them over many kilometres, starting radio communication.

8.2 Displacement current

• Definition: the current that appears, in addition to conduction current, wherever the electric field (and hence the electric flux) in a region is changing with time. $$I_D = \varepsilon_0\dfrac{d\phi_E}{dt}$$
• The inconsistency it removes: for a circular Amperian loop of radius $r$ round the wire feeding a capacitor, the flat surface pierced by the wire gives $\oint\vec B\cdot d\vec l = \mu_0 I$, but a pot-shaped or tiffin-shaped surface with the same rim, whose bottom lies between the plates, is pierced by no conduction current and gives zero. Two answers for the same loop integral — impossible.
• Maxwell's fix: the flat surface between the plates does have something crossing it — the electric field. Adding $\varepsilon_0\, d\phi_E/dt$ to the enclosed current makes the answer the same for every surface bounded by the loop.
• For a parallel-plate capacitor, $E = Q/(A\varepsilon_0)$, so $\phi_E = Q/\varepsilon_0$ and $I_D = \varepsilon_0\, d\phi_E/dt = dQ/dt = I_C$. The displacement current between the plates exactly equals the conduction current in the wire, so total current is continuous round the whole circuit.
• Outside the plates: only conduction current ($I_C = I$, $I_D = 0$). Between the plates: only displacement current ($I_C = 0$, $I_D = I$).
• Displacement current has all the same physical effects as conduction current; in particular it produces a magnetic field. The field measured just inside the gap (point M) equals the field just outside (point P).
• In a dc circuit at steady state the capacitor is fully charged, the field between the plates is constant, so $I_D = 0$, and the insulating gap carries no conduction current — hence no steady current flows in a capacitor branch. During charging or discharging, $Q$ changes, so a momentary current does flow.

Maxwell's equations in vacuum

• The last two make the laws almost symmetric: a changing $\vec B$ makes $\vec E$, and a changing $\vec E$ makes $\vec B$. They are not perfectly symmetric because there are no magnetic monopoles.
• That mutual regeneration is exactly what allows a self-sustaining electromagnetic wave.

8.3.1 Sources of electromagnetic waves

• Stationary charges give only an electrostatic field. Charges in uniform motion (steady current) give a magnetic field that does not change with time. Neither radiates.
• Accelerated charges radiate electromagnetic waves. An oscillating charge is an accelerating charge, so it radiates; an oscillating electric dipole is the basic source of em waves.
• The wave frequency equals the frequency of oscillation of the source charge. The energy carried away comes from the energy of the oscillating charge.
• Yellow light needs about $6\times10^{14}$ Hz, but electronic circuits reach only about $10^{11}$ Hz — which is why the first experimental demonstration (Hertz) was in the radio region.

8.3.2 Nature of electromagnetic waves

• $\vec E$ and $\vec B$ are perpendicular to each other and both perpendicular to the direction of propagation — em waves are transverse.
• $\vec E$ and $\vec B$ oscillate in the same phase: they reach their maxima, zeros and minima at the same instants.
• $\vec E \times \vec B$ points along the direction of propagation; $\vec E$, $\vec B$ and the propagation direction form a right-handed set.
• Standard forms: $E_x = E_0\sin(kz-\omega t)$ and $B_y = B_0\sin(kz-\omega t)$ for a wave travelling along $+z$, with $k = 2\pi/\lambda$, $\omega = 2\pi\nu$ and speed $\omega/k$.
• Amplitudes are related by $E_0 = cB_0$, so $B_0$ is smaller than $E_0$ by a factor of $3\times10^8$.
• They need no material medium; they are self-sustaining oscillations of the fields in vacuum. This is what makes them different from sound and from waves on a string.
• In vacuum every em wave travels at the same speed $c$, independent of wavelength — the ratio of the speeds of two light waves of different wavelengths in vacuum is $1:1$.
• The constancy of $c$ is so well established that it is used to define the standard of length.
• In a medium the speed falls to $v = 1/\sqrt{\mu\varepsilon} = c/\sqrt{\mu_r\varepsilon_r}$; the refractive index is $n = c/v$.
• Em waves carry energy and momentum, and are electrically neutral, so they are not deflected by electric or magnetic fields.
• Energy is shared equally between the electric and magnetic fields: $\bar u_E = \bar u_B = \dfrac14\varepsilon_0E_0^2$.

Standard diagram in words — linearly polarised plane wave

• Draw the propagation direction (say the $z$-axis) horizontally across the page.
• Draw the electric field as a sine curve in the $x$–$z$ plane, its arrows along $\pm x$, drawn as a function of $z$ at a fixed instant.
• Draw the magnetic field as a sine curve in the $y$–$z$ plane, its arrows along $\pm y$, with the same wavelength.
• Draw the two curves in phase — both crossing zero at the same values of $z$ and peaking at the same values of $z$ — and check that $\vec E\times\vec B$ points along $+z$.
• Label $E_0$, $B_0$, one wavelength $\lambda$, and the axes. For a wave along $+x$ with $\vec E$ along $+y$, the same picture rotates so that $\vec B$ is along $+z$.

Reading a wave equation quickly

8.4 Electromagnetic spectrum

• The classification by frequency (or wavelength) is the electromagnetic spectrum. There are no sharp boundaries; the classification is based roughly on how the waves are produced and detected.
• In order of increasing wavelength: gamma rays, X-rays, ultraviolet, visible, infrared, microwaves, radio waves — from about $10^{-12}$ m to about $10^{6}$ m.
• All bands travel at the same speed $c$ in vacuum; they differ only in wavelength/frequency, and therefore in how they interact with matter.
• Radio waves — accelerated electrons in aerials; radio, TV, cellular communication.
• Microwaves — klystrons, magnetrons, Gunn diodes; radar and aircraft navigation, speed guns, microwave ovens (frequency matched to the resonant frequency of water molecules).
• Infrared — hot bodies and vibrating atoms and molecules; heat waves, physical therapy, remote switches, greenhouse effect, satellite crop monitoring, seeing through fog and haze.
• Visible light — electrons in atoms dropping to a lower energy level; the only band the human eye detects, $700$ nm to $400$ nm.
• Ultraviolet — special lamps, very hot bodies, the Sun, welding arcs; water purifiers, LASIK eye surgery, food preservation. Absorbed by ordinary glass and by the ozone layer.
• X-rays — bombarding a metal target with high-energy electrons; medical diagnosis, cancer treatment, crystal structure.
• Gamma rays — radioactive decay of nuclei; radiotherapy for cancer, sterilisation.
• Photon energy $E = h\nu$ increases from radio to gamma; the more energetic the band, the more energetic the process producing it (circuit currents, molecular vibrations, outer electrons, inner-shell electrons, nuclei).

Comparisons CBSE asks for

Results worth memorising

• $I_D = I_C$ for a capacitor being charged; if the charging current is $0.25$ A, the displacement current across the plates is $0.25$ A.
• $c = 1/\sqrt{\mu_0\varepsilon_0} = E_0/B_0 = 3\times10^{8}$ m s$^{-1}$.
• $v = c/\sqrt{\mu_r\varepsilon_r}$ in a medium; $n = \sqrt{\mu_r\varepsilon_r}$.
• $\bar u_E = \bar u_B = \dfrac14\varepsilon_0E_0^2$, total $\bar u = \dfrac12\varepsilon_0E_0^2$.
• $p = U/c$ and radiation pressure $I/c$ (absorbed) or $2I/c$ (reflected).
• Between the plates of a circular parallel-plate capacitor, $B = \dfrac{\mu_0 r I_D}{2\pi R^2}$ for $r < R$.
• A charge oscillating at $10^{9}$ Hz radiates em waves of frequency $10^{9}$ Hz — the frequencies are identical.
• $21$ cm hydrogen line: microwave region.
• $5\times10^{19}$ Hz: gamma-ray region. $10^{16}$ to $10^{20}$ Hz: X-rays.
• If the Earth had no atmosphere its average surface temperature would be lower, because greenhouse gases trap re-radiated infrared.

Traps that cost marks

• Displacement current is not a flow of charge. Never say "electrons jump across the gap"; say the changing electric flux acts as a source of magnetic field.
• $I_D = \varepsilon_0\, d\phi_E/dt$ — do not drop the $\varepsilon_0$, and do not write $d\phi_B/dt$, which belongs to Faraday's law.
• In the Ampere-Maxwell law the $\mu_0$ multiplies both terms: $\mu_0(I_C + I_D)$, so the second term carries $\mu_0\varepsilon_0$.
• $\vec E$ and $\vec B$ in an em wave are in the same phase, not out of phase by $90^\circ$. That confusion is a standard MCQ distractor.
• $\vec E$ and $\vec B$ are perpendicular to each other, never parallel — the other standard distractor.
• Em waves are transverse, not longitudinal, and need no medium.
• $E_0 = cB_0$, so the magnetic amplitude is the smaller number. Multiplying instead of dividing (or the reverse) is the commonest numerical slip.
• $\sin(kz-\omega t)$ travels along $+z$; $\sin(\omega t + kz)$ travels along $-z$. Check the relative signs before naming a direction.
• When asked for the direction of $\vec B$, use $\vec E\times\vec B$ along the propagation direction — and remember to include the sign, e.g. along $-y$, not just "along $y$".
• All em waves have the same speed in vacuum: the ratio of the speeds of $4000$ Å and $8000$ Å light in vacuum is $1:1$, not $1:2$.
• Frequency does not change when a wave enters a medium; the wavelength and the speed do.
• Ultraviolet, not X-rays, is what a welder's goggles block, what purifies water and what is used in LASIK eye surgery.
• Infrared is the heat wave, and it lies between visible (shorter $\lambda$ side) and microwaves (longer $\lambda$ side).
• Microwaves for radar; do not answer "radio waves". The reason is short wavelength giving a narrow, well-directed beam and good resolution.
• Gamma rays and X-rays overlap in wavelength; separate them by origin (nucleus versus inner-shell electrons), which is what the marking scheme wants.
• Write units: displacement current in ampere, energy density in J m$^{-3}$, $\varepsilon_0$ in C$^2$ N$^{-1}$ m$^{-2}$, $\mu_0$ in T m A$^{-1}$.
• When a band is asked for "with its range", give the range — a band name alone loses the second mark.