Electromagnetic Waves: JEE & NEET Physics Notes, Class 12

In this chapter: displacement current and the Ampere-Maxwell law, Maxwell's equations in words, sources of EM waves, nature of EM waves (transverse, E and B in phase, E₀/B₀ = c), speed in vacuum and in a medium, energy and momentum carried by the waves, the electromagnetic spectrum with the production and uses of each band.

Displacement current

Apply Ampere's law to a capacitor being charged. Take a small circle around the wire just outside the capacitor. If the surface bounded by the circle is a flat disc cut by the wire, the enclosed current is i and ∮B · dl = μ₀i. If instead we use a pot-shaped surface with the same rim that passes between the plates, no conduction current crosses it, and we get zero. The same rim cannot give two answers, so Ampere's law as it stands is incomplete.

Maxwell resolved this. Between the plates the electric flux Φ_E = EA = q/ε₀ changes as the capacitor charges, and ε₀ dΦ_E/dt = dq/dt = i. He called this term the displacement current:

i_d = ε₀ dΦ_Edt∮ B · dl = μ₀(i_c + i_d)the Ampere-Maxwell law; i_c is the conduction current

Outside the capacitor there is only conduction current; between the plates there is only displacement current; the total is continuous. The displacement current is not a flow of charges, but it produces a magnetic field exactly as a conduction current does. So a changing electric field produces a magnetic field, just as Faraday's law says a changing magnetic field produces an electric field. This symmetry is what makes electromagnetic waves possible.

Maxwell's equations in words

Gauss's law for electricity: ∮E · dA = Q/ε₀.
Gauss's law for magnetism: ∮B · dA = 0.
Faraday's law: ∮E · dl = −dΦ_B/dt.
Ampere-Maxwell law: ∮B · dl = μ₀i_c + μ₀ε₀ dΦ_E/dt.

Sources of electromagnetic waves

A charge at rest produces only an electric field; a charge moving uniformly also produces a steady magnetic field. Neither radiates. An accelerated charge produces a changing magnetic field, which produces a changing electric field, and so on; the fields regenerate each other and travel outward. A charge oscillating with frequency ν produces an EM wave of the same frequency. This is how an antenna works. Hertz first produced and detected such waves in 1887, and Jagadish Chandra Bose later produced waves of much shorter wavelength (25 mm to 5 mm) in the laboratory.

Nature of electromagnetic waves

For a plane wave travelling along x, the fields are

E_y = E₀ sin(kx − ωt), B_z = B₀ sin(kx − ωt)k = 2π/λ, ω = ck, E₀B₀ = c

In a plane EM wave travelling along x, E oscillates along y and B along z. They reach their maxima together, and the direction of E × B gives the direction of travel.

E and B are perpendicular to each other and to the direction of propagation, so EM waves are transverse.
E and B oscillate in phase, and the direction of travel is along E × B.
They need no material medium. In vacuum all EM waves travel at c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m s⁻¹, whatever their wavelength.
In a medium, v = 1/√(με), which is less than c.
They carry energy and momentum. In a region with fields, energy is stored in both E and B. If a wave transfers energy U to a surface and is completely absorbed, the momentum delivered is p = U/c, so light exerts radiation pressure. It is tiny for sunlight, which is why we do not feel it.

Worked example: A plane EM wave of frequency 25 MHz travels in vacuum along the x direction. At a certain point and instant, E = 6.3 V m⁻¹ along +y. Find the wavelength and B at that point.
Solution: λ = c/ν = (3 × 10⁸)/(25 × 10⁶) = 12 m. B = E/c = 6.3/(3 × 10⁸) = 2.1 × 10⁻⁸ T. For E × B to point along +x with E along +y, B must be along +z (since ŷ × ẑ = x̂).

The electromagnetic spectrum

EM waves of all wavelengths are the same kind of wave; they differ in frequency and in how they are produced and detected. The bands overlap and their boundaries are not sharp.

The electromagnetic spectrum, from gamma rays with the shortest wavelengths to radio waves with the longest. Wavelength ranges follow the NCERT table.

Type	Wavelength range	Produced by	Uses and notes
Radio	> 0.1 m	Accelerated motion of charges in antennas (aerials)	AM band 530 kHz to 1710 kHz; TV 54 MHz to 890 MHz; FM 88 MHz to 108 MHz; mobile phones use ultra high frequencies
Microwaves	0.1 m to 1 mm	Klystron, magnetron, Gunn diode	Radar in aircraft navigation, speed guns; microwave ovens, where the frequency matches a resonant frequency of water molecules so energy goes into the water in food
Infrared	1 mm to 700 nm	Hot bodies and vibrating molecules	Heat waves; remote controls; physical therapy; infrared detectors in the military and on earth satellites; responsible for the greenhouse effect
Visible	700 nm to 400 nm	Electrons in atoms changing energy levels	Vision; the eye is sensitive to this range
Ultraviolet	400 nm to 1 nm	Special lamps, very hot bodies (the sun)	Mostly absorbed by the ozone layer; kills germs in water purifiers; LASIK eye surgery; glass absorbs much of it, so welders wear glass goggles
X-rays	1 nm to 10⁻³ nm	Bombarding a metal target with high-energy electrons	Medical diagnosis and treatment of cancer; overexposure damages living tissue
Gamma rays	< 10⁻³ nm	Nuclear reactions, radioactive nuclei	Used in medicine to destroy cancer cells

A handy order to memorise, from high frequency to low: gamma, X-ray, UV, visible (violet to red), infrared, microwave, radio. Within visible light, violet has the shortest wavelength (about 400 nm) and red the longest (about 700 nm).

Common mistakes: (1) Thinking displacement current is a flow of charge; it is a changing electric flux. (2) Saying E and B are 90° out of phase in an EM wave; they are in phase and only perpendicular in direction. (3) Believing different EM waves travel at different speeds in vacuum; they all travel at c. (4) Writing B₀ = cE₀; it is E₀ = cB₀. (5) Mixing up the order of UV and infrared in the spectrum.

JEE and NEET focus

Displacement current in a charging capacitor and why it equals the conduction current in the wires.
Relations E₀ = cB₀, c = 1/√(μ₀ε₀) and v = 1/√(με).
Finding the direction of B or of propagation from E × B.
Order of the spectrum by wavelength, frequency and photon energy.
Sources and uses of each band, especially microwaves, infrared, UV and X-rays.

Practice questions

Displacement current is produced by:

a steady electric field
a changing electric flux
moving electrons in a wire
a steady magnetic field

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B. i_d = ε₀ dΦ_E/dt.

The peak electric field of an EM wave in vacuum is 30 V m⁻¹. Its peak magnetic field is:

10⁻⁷ T
10⁻⁸ T
9 × 10⁹ T
3 × 10⁻⁷ T

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A. B₀ = E₀/c = 30/(3 × 10⁸).

The ratio of the speed of gamma rays to that of radio waves in vacuum is:

greater than 1
less than 1
1
depends on the source

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C. All EM waves travel at c in vacuum.

Which of these is in order of increasing wavelength?

X-rays, UV, infrared, microwaves
UV, X-rays, microwaves, infrared
microwaves, infrared, UV, X-rays
infrared, X-rays, UV, microwaves

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A. Wavelength increases from X-rays to microwaves.

The ozone layer in the atmosphere mainly absorbs:

infrared
visible light
ultraviolet
microwaves

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C. It protects us from most of the sun's UV.

In an EM wave, E is along +x and B is along +y. The wave travels along:

+z
−z
+x
−y

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A. E × B ∝ x̂ × ŷ = ẑ.

The speed of EM waves in a non-magnetic medium of relative permittivity 4 is:

3 × 10⁸ m s⁻¹
1.5 × 10⁸ m s⁻¹
0.75 × 10⁸ m s⁻¹
6 × 10⁸ m s⁻¹

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B. v = c/√(μ_rε_r) = c/2.

Microwave ovens heat food mainly because microwaves:

ionise the food molecules
have a frequency matching a resonant frequency of water molecules
are reflected by food
are absorbed by the container

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B. Energy is transferred to the water in the food, which heats up.