Electromagnetic Waves notes: ISC Class 12 Physics

1. Displacement current

The problem: Ampère’s law $\oint \vec B \cdot d\vec l = \mu_0 i$ allows any surface bounded by the loop. For a loop round the lead of a charging capacitor, a flat disc encloses the current $i$ , but a surface bulging through the gap encloses no conduction current: two answers for one loop.
Maxwell’s fix: a changing electric flux acts as a current, the displacement current $i_d = \varepsilon_0\,\dfrac{d\Phi_E}{dt}$ . Ampère–Maxwell law: $\oint \vec B \cdot d\vec l = \mu_0(i + i_d)$ .
For a parallel-plate capacitor: $\Phi_E = EA = q/\varepsilon_0$ , so $i_d = dq/dt = i$ : the current is continuous through the gap. Also $i_d = C\,dV/dt = \varepsilon_0A\,dE/dt$ .
The displacement current produces a magnetic field (circles round the axis of the plates) exactly like a conduction current. It exists only while E changes: zero once the capacitor is fully charged, so steady DC is blocked.
Conduction current: moving charges in a conductor. Displacement current: changing E, possible even in vacuum. ISC: basic idea and formula only.

charging: slope 3.0 μC per ms
steady: slope zero
discharging: slope −6.0 μC per ms

The displacement current in the gap is the slope: 3.0 mA, then zero, then 6.0 mA in the opposite sense.

Formulas

Displacement current
$i_d = \varepsilon_0\,\dfrac{d\Phi_E}{dt}$
Ampère–Maxwell law
$\oint \vec B \cdot d\vec l = \mu_0(i + i_d)$

Key terms

Displacement current: ε₀ dΦE/dt: the current-like effect of a changing electric flux, which produces a magnetic field.
Continuity of current: Conduction current in the leads plus displacement current in the gap form one unbroken current round a circuit with a capacitor.

2. Nature of electromagnetic waves

A changing B produces E (Faraday); a changing E produces B (Maxwell). Each regenerates the other, so the disturbance travels through empty space.
Source: accelerated charges, for example charges oscillating in an aerial; the wave has the frequency of the oscillation. A charge at rest or in uniform motion does not radiate.
Transverse: in a plane wave along x, E oscillates along y and B along z: $\vec E \perp \vec B$ , both perpendicular to the direction of travel, which is the direction of $\vec E \times \vec B$ . E and B are in phase.
Sketch: a sine curve of E in one plane and of B in the perpendicular plane, both on the same axis of travel, peaking together; label E, B, the direction of travel and λ.

Key terms

Electromagnetic wave: Linked oscillating electric and magnetic fields, at right angles to each other and to the direction of travel, moving at c in vacuum.
Transverse wave: A wave whose oscillations are at right angles to its direction of travel.

3. Speed and properties of EM waves

Speed in vacuum: $c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.0 \times 10^{8}$ m s⁻¹. It matched the measured speed of light, so light is an EM wave; Hertz produced and detected radio waves to confirm the theory.
Peak fields: $\dfrac{E_0}{B_0} = c$ . For every EM wave in vacuum: $c = f\lambda$ . In a material medium the speed is less than c (qualitative only).
Common properties: transverse; same speed c in vacuum; no medium needed; E and B in phase; carry energy and momentum; uncharged, so not deflected by electric or magnetic fields; show reflection, refraction, interference, diffraction and polarisation.
The bands differ only in frequency (and so wavelength).

Formulas

Speed; peak fields
$c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}},\qquad \dfrac{E_0}{B_0} = c$
Wave relation
$c = f\lambda$

Key terms

Speed of light in vacuum: c = 1/√(μ₀ε₀) = 3.0 × 10⁸ m s⁻¹, the same for every EM wave.

4. The electromagnetic spectrum: order and ranges

Increasing frequency (decreasing wavelength): radio, microwaves, infrared, visible, ultraviolet, X-rays, gamma rays.
Approximate wavelengths: radio longer than 30 cm; microwaves 1 mm to 30 cm (1–300 GHz); infrared 700 nm to 1 mm; visible 400–700 nm; ultraviolet 1–400 nm; X-rays 1 pm to 1 nm; gamma rays below about 10 pm. Boundaries are not sharp.
X-rays and gamma rays overlap; they are named by origin (electrons hitting a target, or nuclei).
Place a wave in its band by converting with $c = f\lambda$ . From ultraviolet upwards the radiation is ionising.

Key terms

Electromagnetic spectrum: The whole range of EM waves arranged by frequency or wavelength, from radio waves to gamma rays.
Ionising radiation: Radiation able to knock electrons out of atoms: ultraviolet, X-rays and gamma rays.

5. Radio waves, microwaves and infrared

Radio: source — oscillating currents in aerials; detection — aerial with a tuned circuit; uses — AM and FM broadcasting, TV, two-way radio, radio astronomy. Medium waves are reflected by the ionosphere.
Microwaves: source — magnetron, klystron, Gunn diode and transistor oscillators; detection — dish or horn with a crystal diode; uses — radar (distance = ct/2), microwave ovens (2.45 GHz, absorbed by water in food), mobile phones, Wi-Fi, satellite links (they pass through the ionosphere).
Infrared: source — hot and warm bodies, IR LEDs; detection — thermopile, bolometer, IR photodiode, thermal camera, the skin; uses — remote controls, non-contact thermometers, night vision, physiotherapy, optical fibres, greenhouses (absorbed by water vapour and CO₂).

λ = c/f

Doubling the frequency halves the wavelength: λ (cm) = 30 ÷ f (GHz).

Formulas

Radar ranging
$d = \dfrac{ct}{2}$

Key terms

Magnetron: Valve that generates the microwaves in a microwave oven and in some radars.
Radar: Finding the distance of an object from the echo time t of a microwave pulse: d = ct/2.

6. Visible light, ultraviolet, X-rays and gamma rays

Visible (400–700 nm): source — Sun, flames, lamps, LEDs, lasers; detection — eye, photocell, camera sensor, film; uses — vision, photography, photosynthesis.
Ultraviolet: source — Sun, mercury-vapour lamps, welding arcs; detection — photocells, fluorescent materials, film; uses — sterilising water and surfaces, detecting forged notes, vitamin D. Mostly absorbed by the ozone layer; excess causes sunburn, eye damage, skin cancer.
X-rays: source — fast electrons striking a metal target in an X-ray tube; detection — film, fluorescent screen, digital detector, GM counter; uses — radiography, CT scans, baggage checks, crystal structure (wavelengths near atomic spacing). Ionising: limit exposure, shield with lead.
Gamma rays: source — radioactive nuclei (e.g. cobalt-60); detection — GM and scintillation counters; uses — radiotherapy, sterilising medical items, food irradiation, checking welds. Most penetrating.

Key terms

Fluorescence: Absorbing ultraviolet and giving out visible light; used to detect UV and to check banknotes.
X-ray tube: Evacuated tube in which fast electrons strike a metal target and produce X-rays.

Exam tips

Continuity of current: one loop, two surfaces; the gap surface needs the displacement current $\varepsilon_0\,d\Phi_E/dt$ .
For a capacitor, $i_d = C\,dV/dt$ ; it is zero once the capacitor is fully charged.
Sketch E, B and the direction of travel mutually perpendicular, E and B in phase; $E_0 = cB_0$ , $c = f\lambda$ .
Spectrum: order both ways, approximate ranges, and one source, detector and use for each band.

1. Displacement current

Formulas

Key terms

2. Nature of electromagnetic waves

Key terms

3. Speed and properties of EM waves

Formulas

Key terms

4. The electromagnetic spectrum: order and ranges

Key terms

5. Radio waves, microwaves and infrared

Formulas

Key terms

6. Visible light, ultraviolet, X-rays and gamma rays

Key terms

Exam tips

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