Electromagnetic Waves notes: CBSE Class 12 Physics

1. Displacement current

The puzzle: for a loop round the wire of a charging capacitor, Ampere’s law $\oint \vec B\cdot d\vec l = \mu_0 I$ gives $\mu_0 I$ for a flat surface pierced by the wire but zero for a surface passing through the gap, where no charge flows.

Maxwell’s fix: a changing electric flux acts like a current, the displacement current $I_d = \varepsilon_0\dfrac{d\Phi_E}{dt}$ . In the gap $\Phi_E = q/\varepsilon_0$ , so $I_d = dq/dt = I$ : the total current is continuous.

Ampere–Maxwell law: $\oint \vec B\cdot d\vec l = \mu_0 I_c + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$ . A changing electric field produces a magnetic field (the counterpart of Faraday’s law).

Formulas

Displacement current
$I_d = \varepsilon_0\dfrac{d\Phi_E}{dt}$
Ampere–Maxwell law
$\oint \vec B\cdot d\vec l = \mu_0\left(I_c + \varepsilon_0\dfrac{d\Phi_E}{dt}\right)$

Key terms

Displacement current: The term ε₀ dΦE/dt that acts like a current wherever the electric flux is changing.

2. Electromagnetic waves and their characteristics

Source: accelerated (e.g. oscillating) charges. A charge at rest gives only E; steady motion gives E and steady B; acceleration radiates. The wave’s frequency equals the frequency of oscillation of the charge.
Self-sustaining: a changing E makes B and a changing B makes E, so the disturbance travels; no medium is needed.
Plane wave along x: $E_y = E_0\sin(kx - \omega t)$ , $B_z = B_0\sin(kx - \omega t)$ : E and B are perpendicular to each other and to the direction of travel (transverse), and in phase; $\vec E \times \vec B$ is along the direction of travel.
Speed: $c = 1/\sqrt{\mu_0\varepsilon_0} \approx 3.0 \times 10^8$ m s⁻¹ in vacuum, the same for all frequencies; in a medium $v = 1/\sqrt{\mu\varepsilon} < c$ .
$E_0/B_0 = c$ ; $c = f\lambda$ ; $k = 2\pi/\lambda$ , $\omega = 2\pi f$ .
EM waves carry energy and momentum (they exert radiation pressure).

E × B points along the direction of travel (checked).

E / E₀
B / B₀ (same phase)

E₀ = cB₀; the fields reach their peaks at the same places.

Formulas

EM wave relations
$c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}},\qquad E_0 = cB_0,\qquad c = f\lambda$

Key terms

Transverse wave: A wave whose oscillations are perpendicular to its direction of travel.

3. The electromagnetic spectrum

In order of decreasing wavelength (increasing frequency and photon energy); the edges are approximate:

Band	Wavelength	Produced by	Uses
Radio	> 0.1 m	oscillating currents in aerials	AM/FM radio, TV, mobile phones
Microwave	0.1 m – 1 mm	magnetron, klystron	radar, ovens (2.45 GHz), satellite links, Wi-Fi
Infrared	1 mm – 700 nm	hot bodies, molecular vibrations	remote controls, night vision, physiotherapy; greenhouse effect
Visible	700 – 400 nm	electron transitions in atoms	vision, photography
Ultraviolet	400 nm – 1 nm	Sun, arc lamps	sterilisation, LASIK; absorbed by the ozone layer
X-rays	1 nm – about 1 pm	fast electrons hitting a metal target	medical imaging, crystal structure
Gamma rays	below about 1 pm	radioactive nuclei	cancer therapy, sterilising equipment

Band edges are approximate; the bands overlap slightly in practice.

Key terms

Electromagnetic spectrum: The full range of EM waves arranged by wavelength or frequency.
Ozone layer: Upper-atmosphere layer that absorbs most of the Sun’s ultraviolet.

Exam tips

Displacement current: describe the two surfaces with one boundary, then $I_d = \varepsilon_0\,d\Phi_E/dt$ .
Transverse nature: E ⟂ B ⟂ direction of travel; E and B in phase; E × B gives the direction.
$c = 1/\sqrt{\mu_0\varepsilon_0}$ , $E_0 = cB_0$ , $c = f\lambda$ .
Spectrum: order, one source and two uses for each band; convert f to λ before naming a band.

1. Displacement current

Formulas

Key terms

2. Electromagnetic waves and their characteristics

Formulas

Key terms

3. The electromagnetic spectrum

Key terms

Exam tips

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