Electromagnetic Waves

1. Check this before you revise anything

The "Additional Exercises" section has been removed from this chapter, as from all 14 chapters of the current Class 12 Physics book. The questions run contiguously from 8.1 to 8.10 with no gaps. At 14 pages this is one of the shortest chapters in the book.

Energy density is required by an exercise but never defined in the chapter. Exercise 8.10(c) asks you to "show that the average energy density of the $E$ field equals the average energy density of the $B$ field."

Searching the whole chapter for "energy density" returns exactly two hits — both inside that exercise. Searching for "intensity", "momentum", "radiation pressure" and "Poynting" returns zero hits each. Older editions carried a passage on electromagnetic waves transporting energy and momentum, including radiation pressure; only the qualitative sentence that these waves "carry energy" survives.

You can still answer it, using results from two earlier chapters:

• $u_E=\frac{1}{2}{\epsilon }_0{E}^2$ comes from the energy stored in a capacitor, Chapter 2.
• $u_B=\frac{B^2}{2{\mu }_0}$ comes from Example 6.9, where the magnetic energy of a solenoid is rewritten in terms of $B$, $A$ and $l$.

With $E_0=c{B}_0$ and $c^2=1/{\mu }_0{\epsilon }_0$, the two averages come out equal. Our solution shows the substitution in full.

A symbol trap if you work from an extracted PDF. As in Chapters 1 to 7, the micro symbol extracts as a plain "m", which affects ${\mu }_0$ and any microfarad or micrometre value.

2. Displacement Current (Textbook 8.2)

Maxwell noticed that Ampere's circuital law, as it stood, contradicts itself — and fixing the contradiction predicted electromagnetic waves.

The inconsistency. Consider a capacitor being charged. Take an Amperian loop around the connecting wire, and apply Ampere's law using two different surfaces bounded by that same loop:

• A flat surface cutting the wire encloses the conduction current $I$, giving $\oint \vec{B}\cdot d\vec{l}={\mu }_{0}I$.
• A bulging surface passing between the capacitor plates encloses no conduction current, giving zero.

Same loop, same field, two different answers. Ampere's law in its original form is therefore incomplete.

Maxwell's resolution. Between the plates there is no charge flow, but there is a changing electric field. Maxwell proposed that a changing electric flux is itself a source of magnetic field, equivalent to a current:

$i_d={\epsilon }_0\frac{d{\Phi }_E}{dt}$

The Ampere-Maxwell law then reads:

$\oint \vec{B}\cdot d\vec{l}={\mu }_0(i_c+i_d)$

The displacement current is exactly equal to the conduction current at every instant during charging, which is what makes the two surfaces agree. Exercise 8.2(b) asks you to confirm this, and Exercise 8.1 works it out numerically for a 12 cm capacitor.

The word "current" here is a name, not a claim: nothing flows between the plates. What produces the magnetic field is the changing electric field.

Maxwell's four equations are Gauss's law for electricity, Gauss's law for magnetism, Faraday's law of induction, and this corrected Ampere-Maxwell law. Together they predicted waves travelling at $3\times 10^8$ m s${}^{-1}$ — the measured speed of light — which is how light was identified as an electromagnetic wave. Hertz produced and detected such waves experimentally in 1887.

3. The Nature of Electromagnetic Waves (Textbook 8.3)

Sources (8.3.1). An electromagnetic wave is radiated by an accelerating charge. A stationary charge has a static field and radiates nothing; a charge in uniform motion produces a steady current and a static magnetic field, and still radiates nothing.

A charge oscillating at frequency $\nu$ radiates a wave of the same frequency $\nu$. That single sentence answers Exercise 8.6 completely.

Nature (8.3.2). Electromagnetic waves are transverse. The oscillating electric and magnetic fields are:

• perpendicular to each other,
• both perpendicular to the direction of propagation,
• and in phase, reaching their maxima and zeros together.

The direction of travel is along $\vec{E}\times \vec{B}$. For a wave travelling along $z$, both $\vec{E}$ and $\vec{B}$ lie in the $x$-$y$ plane, which is what Exercise 8.4 asks you to state.

The amplitude relation:

$\frac{E_0}{B_0}=c\text{so}{E}_0=c{B}_0$

Because $c$ is so large, $B_0$ is always numerically tiny beside $E_0$. Exercises 8.7 and 8.8 use this in both directions.

The speed. Maxwell's equations give the speed from two electrostatic and magnetostatic constants alone:

$c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=3\times 10^8{\text{m s}}^{-1}$

In a medium the speed falls to $v=1/\sqrt{\mu \epsilon }$, which is less than $c$.

No medium is required. Unlike sound, electromagnetic waves propagate through vacuum — the fields sustain each other, a changing $\vec{E}$ generating $\vec{B}$ and a changing $\vec{B}$ generating $\vec{E}$.

All electromagnetic waves travel at the same speed in vacuum, whatever their frequency. That is the whole of Exercise 8.3: X-rays, red light and radio waves differ in wavelength and frequency, but share $c$ exactly.

The usual wave relations apply throughout:

$c=\nu \lambda ,\omega =2\pi \nu ,k=\frac{\omega }{c}=\frac{2\pi }{\lambda }$

4. The Electromagnetic Spectrum (Textbook 8.4)

The bands differ only in frequency. They are continuous and overlapping, with no sharp boundaries, and are named by how they are produced rather than by any physical difference.

Frequency, energy and penetration all rise together as wavelength falls, since $E=h\nu$. This is the pattern Exercise 8.9 asks you to bring out: photon energies run from around $10^{-9}$ eV for radio waves to about $10^5$ eV for gamma rays, and the source of each band matches its energy scale — aerial electrons for the weakest, atomic transitions in the middle, nuclear processes at the top.

Two atmospheric effects the chapter highlights:

• The greenhouse effect. The atmosphere passes visible sunlight but absorbs the infrared re-radiated by the warmed Earth, trapping heat.
• The ozone layer absorbs most solar ultraviolet, which would otherwise damage living tissue.

Microwave ovens work because the microwave frequency is matched to the rotational frequency of water molecules, so energy is absorbed efficiently by the water in the food and heats it throughout rather than only at the surface.

Summary

• Ampere's original law gives two different answers for two surfaces sharing one loop when a capacitor charges — that inconsistency is what displacement current fixes.
• $i_d={\epsilon }_{0}d{\Phi }_E/dt$: a changing electric field produces a magnetic field just as a current does.
• Ampere-Maxwell law: $\oint \vec{B}\cdot d\vec{l}={\mu }_0(i_c+i_d)$, with $i_d=i_c$ at every instant during charging.
• Nothing physically flows between the plates; "current" here is a name for a changing flux.
• Maxwell's four equations predicted waves at $3\times 10^8$ m s${}^{-1}$, identifying light as electromagnetic; Hertz confirmed it in 1887.
• Only an accelerating charge radiates; a stationary or uniformly moving charge does not.
• A charge oscillating at frequency $\nu$ radiates at the same frequency $\nu$.
• Electromagnetic waves are transverse, with $\vec{E}\perp \vec{B}$, both perpendicular to the propagation direction, and in phase; travel is along $\vec{E}\times \vec{B}$.
• $E_0=c{B}_0$, so the magnetic amplitude is always numerically far smaller.
• $c=1/\sqrt{{\mu }_0{\epsilon }_0}$; in a medium $v=1/\sqrt{\mu \epsilon }<c$.
• No medium is needed — the two fields sustain each other.
• All electromagnetic waves share the same speed in vacuum, whatever their frequency.
• $c=\nu \lambda$, $\omega =2\pi \nu$, $k=\omega /c=2\pi /\lambda$.
• The spectrum runs radio, microwave, infrared, visible, ultraviolet, X-ray, gamma — continuous, overlapping, named by how each is produced.
• Frequency, photon energy $E=h\nu$ and penetrating power all rise as wavelength falls.
• The greenhouse effect traps re-radiated infrared; the ozone layer absorbs solar ultraviolet.
• Microwave ovens match the rotational frequency of water molecules.
• Energy density, intensity, momentum and radiation pressure have been removed from this chapter, though Exercise 8.10(c) still requires energy density — take $u_E=\frac{1}{2}{\epsilon }_0{E}^2$ from Chapter 2 and $u_B=B^2/2{\mu }_0$ from Example 6.9.