A capacitor is being charged through a resistor. The displacement current between its plates is:

Equal to the conduction current in the connecting wires

Zero, because no charge crosses the gap

Larger than the conduction current by the factor ε₀

Equal to the conduction current only when the capacitor is fully charged

Which of the following lists is in order of increasing frequency?

Microwaves, infrared, ultraviolet, X-rays

X-rays, ultraviolet, infrared, microwaves

Infrared, microwaves, X-rays, ultraviolet

Ultraviolet, X-rays, microwaves, infrared

X-rays and gamma rays can have the same wavelength. They are distinguished by:

Their origin — outside the nucleus for X-rays, inside it for gamma rays

The greenhouse effect arises because atmospheric gases such as CO₂:

Transmit incoming visible light but absorb outgoing infrared

Absorb incoming visible light and re-emit it as ultraviolet

Reflect all incoming solar radiation back to space

Absorb ultraviolet radiation in the stratosphere

Electromagnetic Waves — Displacement Current, Maxwell’s Synthesis and the Spectrum

This is a short chapter with an unusually high return: it is almost always one or two straightforward questions, and every one of them comes from a small set of facts. The physics has a clean storyline. Ampère's law, as first written, quietly fails for a charging capacitor; Maxwell repaired it by inventing displacement current; and the repaired equations then predicted, with no further assumption, that a changing field pair must travel through empty space at exactly the measured speed of light. That single prediction converted optics into a branch of electromagnetism. Learn the story and the numbers attach themselves to it.

Displacement Current — the Gap Maxwell Closed

Ampère's circuital law says ∮B·dl = μ₀I. Apply it to a charging capacitor and it contradicts itself: choose a surface pierced by the wire and you get μ₀I, but slide the same loop's surface into the gap between the plates, where no charge crosses, and you get zero. Two answers for one loop means the law is incomplete. Maxwell's fix was to notice that although no charge crosses the gap, the electric flux there is changing, and to add a corresponding term.

🎯 Why the displacement current is exactly equal to the conduction current

Maxwell defined I_d = ε₀ dΦ_E/dt. For a parallel-plate capacitor the field between the plates is E = q/(ε₀A), so Φ_E = EA = q/ε₀ and therefore I_d = ε₀ · (1/ε₀) · dq/dt = dq/dt = I, the very current flowing in the wire. That equality is not a coincidence engineered to make the law work; it is what makes current continuous through the circuit. The charge stops at the plate, but the current — now carried by a changing electric field rather than by moving charge — continues across the gap without interruption, so Kirchhoff's laws survive and the magnetic field around the capacitor gap is exactly what it is around the wire. Two exam consequences: the displacement current between the plates equals the conduction current in the leads at every instant, and it becomes zero once the capacitor is fully charged, because then dΦ_E/dt = 0. The generalised law is ∮B·dl = μ₀(I_c + I_d).

Maxwell’s four equations, in words
Equation	What it says	Consequence
Gauss’s law for electricity	Electric flux out of a closed surface = q_enclosed/ε₀	Electric field lines begin and end on charges
Gauss’s law for magnetism	Net magnetic flux out of any closed surface is zero	No magnetic monopoles — field lines are closed loops
Faraday’s law of induction	A changing magnetic flux creates a circulating electric field	Half of the wave mechanism
Ampère–Maxwell law	A current or a changing electric flux creates a circulating magnetic field	The other half — together they self-sustain a wave

📖 The moment optics became electromagnetism

Combine the last two equations and they feed each other: a changing E makes a circulating B, that changing B makes a circulating E, and the pair propagates without any medium. The algebra delivers a speed c = 1/√(μ₀ε₀), built entirely out of two constants measured in laboratory experiments on capacitors and current-carrying wires — nothing about light went into them. Putting in μ₀ = 4π × 10⁻⁷ and ε₀ = 8.85 × 10⁻¹² gives 3 × 10⁸ m/s, which was already known as the speed of light from astronomical and optical measurements. Maxwell drew the only reasonable conclusion in 1865: light is an electromagnetic wave. Hertz produced and detected radio waves in the laboratory in 1887 and confirmed it. This is worth remembering as the standard example of theory predicting an entire new domain — the radio spectrum was discovered on paper before anyone built a transmitter.

The Nature of an Electromagnetic Wave

Properties every question draws on
Property	Statement	Why it matters
Transverse and mutually perpendicular	E ⊥ B, and both ⊥ to the direction of travel; the direction is along E × B	Makes EM waves polarisable, unlike sound
Ratio of amplitudes	E₀/B₀ = c, and E and B are in phase	B is numerically tiny — for E₀ = 3 V/m, B₀ = 10⁻⁸ T
Needs no medium	Travels through vacuum at c = 1/√(μ₀ε₀); in a medium v = c/n = 1/√(μ ε)	Sunlight reaches us across empty space; sound could not
Energy is shared equally	u_E = ½ε₀E² equals u_B = B²/2μ₀ on average	Total u_av = ½ε₀E₀²; intensity I = u_av·c
Carries momentum	p = U/c for energy U absorbed; radiation pressure = I/c	Doubles to 2I/c on a perfect reflector
Frequency is fixed by the source	Entering a medium changes λ and v, never ν	The standard "what changes on refraction" question

🧠 Getting the direction right with one right-hand rule

The propagation direction is along E × B — point the fingers along E, curl them towards B, and the thumb gives the direction of travel. Questions almost always give two of the three and ask for the third, so it is worth practising in reverse too: if a wave travels along +z with E along +x, then B must be along +y, since x̂ × ŷ = ẑ. A useful check is that the answer must be perpendicular to both given vectors; if your answer shares a component with E or with B, you have used the wrong rule. And be alert to the trap of a wave described as having E along the direction of propagation — that is impossible for an electromagnetic wave, and such an option is there to be eliminated on sight.

⚠️ Radiation pressure: absorbing versus reflecting, and the factor of two

A perfectly absorbing (black) surface takes the wave's momentum and stops, so the pressure is I/c. A perfectly reflecting (mirror) surface sends the momentum back the way it came, so the momentum change is twice as large and the pressure is 2I/c — the same reasoning as a ball bouncing off a wall versus a ball of putty sticking to it. Students who remember only "p = U/c" apply it to a mirror and lose the mark. The numbers are small but real: sunlight at the Earth delivers about 1.4 kW/m², giving roughly 5 μPa of pressure on a black surface and 10 μPa on a mirror. Small as that is, it is why comet tails point away from the Sun and why a solar sail can accelerate a spacecraft continuously with no fuel, and why designers of solar sails make them reflective rather than black.

The Electromagnetic Spectrum

The spectrum is one continuous family with no gaps and no upper or lower bound; the seven named bands are conveniences, and they are named after how we make and detect them, not after any difference in their physics. All of them travel at c in vacuum, all are transverse, all carry energy and momentum. What changes across the spectrum is the frequency, and with it the photon energy E = hν — which is why the high-frequency end is dangerous and the low-frequency end is not.

The seven bands: wavelength, production and use
Band	Wavelength	Produced by / used for
Radio waves	> 0.1 m	Oscillating circuits and antennas; broadcasting, mobile telephony, MRI
Microwaves	0.1 m – 1 mm	Klystron and magnetron; radar, satellite links, microwave ovens
Infrared	1 mm – 700 nm	Hot bodies and molecular vibration; thermal imaging, remote controls, physiotherapy
Visible light	700 nm – 400 nm	Atomic outer-shell transitions; vision, photosynthesis, optical fibre
Ultraviolet	400 nm – 1 nm	Arcs and the Sun; sterilisation, LASIK, detecting forgery — absorbed by ozone
X-rays	1 nm – 10⁻³ nm	Fast electrons striking a metal target; radiography, crystallography, cancer therapy
Gamma rays	< 10⁻³ nm	Radioactive nuclei; sterilising equipment, killing tumour cells

🧠 Fixing the order, and the one distinction that is about origin not wavelength

Read the table downwards for increasing frequency and photon energy, decreasing wavelength: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma. Questions usually ask for one comparison only, so all you need is the order and which way it runs. Note that X-rays and gamma rays overlap in wavelength, so they cannot be told apart by that; the distinction is where they come from — X-rays from electron transitions and decelerating electrons outside the nucleus, gamma rays from inside the nucleus. That is a favourite one-line question. Within visible light, remember VIBGYOR from violet (shortest λ, highest energy) to red (longest λ), which also sets which colour bends most on refraction and which scatters most in the sky.

ℹ️ Two atmospheric windows, and why they are in the news

The atmosphere is transparent to visible light and to most radio waves, and largely opaque to everything else — which is why optical and radio telescopes work from the ground while X-ray and infrared astronomy needs satellites. Two absorption bands carry consequences worth knowing. Ozone in the stratosphere absorbs most solar ultraviolet, and its depletion by chlorofluorocarbons therefore raises the incidence of skin cancer and cataract; this is the physics behind the Montreal Protocol. Separately, CO₂, methane and water vapour are transparent to incoming visible light but absorb outgoing infrared from the warmed surface, trapping heat — the greenhouse effect, which keeps the Earth about 33 °C warmer than it would otherwise be and, when strengthened, drives global warming. Both are standard cross-links between this chapter and environmental questions in general studies papers.

Key takeaways

Displacement current I_d = ε₀ dΦ_E/dt equals the conduction current, keeping current continuous across a capacitor gap.
c = 1/√(μ₀ε₀) — light’s speed came out of two electrical constants, which is how optics joined electromagnetism.
EM waves are transverse with E ⊥ B, in phase, E₀/B₀ = c, and travel along E × B.
Radiation pressure is I/c on a black surface but 2I/c on a mirror.
Order of increasing frequency: radio, microwave, infrared, visible, UV, X-ray, gamma.
X-rays and gamma rays overlap in wavelength; they differ by origin — outside versus inside the nucleus.
On entering a medium the frequency stays fixed while wavelength and speed both fall.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

A capacitor is being charged through a resistor. The displacement current between its plates is:
1. Equal to the conduction current in the connecting wires
2. Zero, because no charge crosses the gap
3. Larger than the conduction current by the factor ε₀
4. Equal to the conduction current only when the capacitor is fully charged
Show answer
Answer: A — Equal to the conduction current in the connecting wires
Since Φ_E = q/ε₀ between the plates, I_d = ε₀ dΦ_E/dt = dq/dt, exactly the conduction current — which is what keeps current continuous round the circuit. The tempting "zero" confuses charge flow with current: a changing electric field carries the current across the gap. The last option inverts the truth, since once fully charged dΦ_E/dt = 0 and the displacement current vanishes.
An electromagnetic wave travels along the +z direction with its electric field along +x. Its magnetic field is along:
1. +y
2. +z
3. −x
4. It has no fixed direction
Show answer
Answer: A — +y
Propagation is along E × B, and x̂ × ŷ = ẑ, so B must lie along +y. The option +z is the classic error of putting a field along the direction of travel, which is impossible for a transverse wave; −x would make E and B parallel, so their cross product — and hence the propagation direction — would vanish.
The electric field amplitude of a plane electromagnetic wave in vacuum is 6 V/m. The amplitude of its magnetic field is:
1. 2 × 10⁻⁸ T
2. 1.8 × 10⁹ T
3. 6 T
4. 3 × 10⁻⁴ T
Show answer
Answer: A — 2 × 10⁻⁸ T
E₀/B₀ = c, so B₀ = 6/(3 × 10⁸) = 2 × 10⁻⁸ T. Multiplying by c instead of dividing gives the absurd 1.8 × 10⁹ T, which should be rejected immediately — no laboratory magnet exceeds about 100 T, and this is a reliable sanity check on every question of this type.
Light of intensity I falls normally on a perfectly reflecting mirror. The radiation pressure exerted on it is:
1. 2I/c
2. I/c
3. I/2c
4. Ic
Show answer
Answer: A — 2I/c
A mirror reverses the wave’s momentum instead of absorbing it, so the momentum change per unit time is doubled and the pressure is 2I/c. The distractor I/c is the correct answer for a perfectly black, absorbing surface — the question is testing exactly that distinction, and the word "reflecting" is the whole clue.
Which of the following lists is in order of increasing frequency?
1. Microwaves, infrared, ultraviolet, X-rays
2. X-rays, ultraviolet, infrared, microwaves
3. Infrared, microwaves, X-rays, ultraviolet
4. Ultraviolet, X-rays, microwaves, infrared
Show answer
Answer: A — Microwaves, infrared, ultraviolet, X-rays
Frequency increases as radio, microwave, infrared, visible, ultraviolet, X-ray, gamma, so the first list is correct. The second is the same sequence reversed, which is the order of increasing wavelength — the commonest way to lose this mark is to read the question as asking for wavelength when it asks for frequency.
X-rays and gamma rays can have the same wavelength. They are distinguished by:
1. Their origin — outside the nucleus for X-rays, inside it for gamma rays
2. Their speed in vacuum
3. Whether they are transverse
4. Their charge
Show answer
Answer: A — Their origin — outside the nucleus for X-rays, inside it for gamma rays
Where the wavelengths overlap the only real distinction is the source: X-rays come from electron transitions and decelerating electrons outside the nucleus, gamma rays from nuclear decay within it. Speed cannot distinguish them since all electromagnetic waves travel at c in vacuum, and charge is a non-starter — electromagnetic waves are uncharged.
A monochromatic electromagnetic wave passes from vacuum into glass. Which quantity is unchanged?
1. Frequency
2. Wavelength
3. Speed
4. Both wavelength and speed
Show answer
Answer: A — Frequency
Frequency is set by the source and the boundary cannot change how many oscillations arrive per second, so ν is fixed; since v = νλ and the speed falls to c/n, the wavelength must fall by the same factor n. Choosing wavelength is the standard error, and it comes from picturing the wave as a fixed pattern in space rather than a fixed rhythm in time.
The greenhouse effect arises because atmospheric gases such as CO₂:
1. Transmit incoming visible light but absorb outgoing infrared
2. Absorb incoming visible light and re-emit it as ultraviolet
3. Reflect all incoming solar radiation back to space
4. Absorb ultraviolet radiation in the stratosphere
Show answer
Answer: A — Transmit incoming visible light but absorb outgoing infrared
Sunlight arrives mainly as visible light, which these gases let through; the warmed surface re-radiates in the infrared, which they absorb, trapping the energy. The last option describes ozone and ultraviolet, which is a different mechanism in a different layer — confusing ozone depletion with global warming is the standard mix-up this question is set to catch.