Electromagnetic Waves: theory

CONCEPT

Nature of EM Waves & Their Direction

In one line
A changing electric field makes a changing magnetic field, and a changing magnetic field makes a changing electric field. Together they travel as a transverse wave.

NATURE

32.1.1 What is an EM wave?

E and B oscillate with time and keep creating each other.
They need no material medium; they travel through vacuum at speed c.
They are transverse; polarization proves this.
They carry energy and momentum.
They carry no charge, so E and B fields do not deflect them.
The electric vector causes the optical effects, so it is called the light vector.

E and B oscillate in phase, at right angles to each other and to c

PHASE

32.1.2 E and B together

E and B are always in phase: both reach maximum and zero together.
E ⊥ B, and both are ⊥ to the direction of travel.

DIRECTION

32.1.3 Direction of propagation

direction of travel = E × B

E, B and c form a right-handed set

E along	B along	Wave travels along
+x	+y	+z
+y	+z	+x
+z	+x	+y

Use î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ.
Reversing the order gives a minus sign.
Unit vector along travel: k̂ = Ê × B̂.

FINDING B

32.1.4 B from k and E

B = k × Eω

k is the propagation vector, along the direction of travel.
A wave along y can have only x and z components of E and B.
For valid E and B directions: E · B = 0, and E × B is along the travel.

Practise PYQs on this topicNature of EM Waves & Wave Equation · 93 questions with solutions

CONCEPT

Wave Equation, E₀ = cB₀ & Force on a Charge

In one line
The two fields follow the same sine wave. The ratio of their amplitudes is always the wave speed: E₀ = cB₀.

WAVE EQUATION

32.2.1 Wave along +x

E_y = E₀ sin(kx − ωt)

B_z = B₀ sin(kx − ωt)

E₀, B₀: amplitudes of the fields.
Other forms: sin(ωt − kx), or sin ω(t − xc).

k = 2πλ · ω = 2πf

c = fλ = ωk

READING IT

32.2.2 What the equation tells you

In the phase	Meaning
kx − ωt	travels along +x
kx + ωt	travels along −x
number with x	k, and λ = 2πk
number with t	ω, and f = ω2π
number in front	amplitude

AMPLITUDES

32.2.3 E₀ = cB₀

E₀B₀ = EB = c

True at every point and every instant, not only at the peaks.
So BE = 1c.
The frequency is not needed to find one field from the other.

E₀ = √2 E_rms · B₀ = √2 B_rms

WRITING THE OTHER FIELD

32.2.4 B from E, or E from B

Step 1: amplitude from E₀ = cB₀.
Step 2: keep the same phase: same sin or cos, same bracket.
Step 3: direction from E × B = direction of travel.

FORCE ON A CHARGE

32.2.5 Lorentz force

F = qE + q(v × B)

Electric force: F_E = qE.
Magnetic force, v ⊥ B: F_B = qvB.
Since E = cB:

F_EF_B = cv

Practise PYQs on this topicNature of EM Waves & Wave Equation · 93 questions with solutions

CONCEPT

Displacement Current, Maxwell’s Equations & Production

In one line
Maxwell found that a changing electric field also makes a magnetic field. He called this term the displacement current.

AMPERE’S LAW

32.3.1 Where it fails

∮B · dl = μ₀ i

Holds only for steady current.
Take a capacitor being charged by a battery.

Loop l₁ is around the wire; loop l₂ is between the plates

Loop l₁: ∮B · dl = μ₀i.
Loop l₂: no current passes, so the law gives 0.
But a magnetic field is found between the plates. So the law fails.

DISPLACEMENT CURRENT

32.3.2 Ampere-Maxwell law

i_d = ε₀ dφ_Edt

∮B · dl = μ₀(i_c + i_d)

i_c: conduction current, from flow of charges.
i_d: from the changing electric field. It has the dimension of current.
Between the plates, i_d = current in the connecting wires.

CAPACITOR RESULTS

32.3.3 Working forms

i_d = C dVdt = ε₀Ad dVdt

AC: I₀ = V₀ωC · I_rms = V_rmsωC

E is uniform, so a part of area A₀ carries i_d × A₀A.
Leaky capacitor: conduction and displacement currents cancel, so B = 0 between the plates.

CONDUCTING MEDIUM

32.3.4 Comparing the two currents

J_c = σE · J_d = ε dEdt

J_{c, max}J_{d, max} = σωε

MAXWELL’S EQUATIONS

32.3.5 The four laws

Law	Equation
Gauss (electrostatics)	∮E · ds = qε₀
Gauss (magnetism)	∮B · ds = 0
Faraday (EMI)	∮E · dl = −dφ_Bdt
Ampere-Maxwell	∮B · dl = μ₀(i_c + ε₀dφ_Edt)

For static fields ∮E · dl = 0; Faraday’s form is needed when B changes.

PRODUCTION

32.3.6 What makes an EM wave?

Stationary charge: only an E field.
Uniform velocity: steady E and B, no wave.
Accelerated charge: radiates an EM wave.
A charge oscillating harmonically gives a wave of the same frequency.
An LC oscillator gives waves of a chosen frequency:

ν = 12π√(LC)

LC oscillator feeding an antenna

HISTORY

32.3.7 Who did what

Name	Work
Maxwell	first predicted EM waves
Hertz	produced and detected them, λ about 6 m
J. C. Bose	produced λ from 5 mm to 25 mm
Marconi	transmitted them over a few km

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Hertz’s experiment

Hertz’s set-up

Plates P₁ and P₂ act as a capacitor.
Wires joining spheres S₁, S₂ to the plates give a low inductance.
High voltage makes a spark across the gap; the spark sets up oscillations.
The oscillations send out EM waves, picked up by the receiver ring.

ν = 12π√(LC)

Practise PYQs on this topicDisplacement Current, Maxwell's Equations & Production · 15 questions with solutions

CONCEPT

Speed of EM Waves in a Medium

In one line
In vacuum every EM wave travels at c. In a medium it slows down by the factor n = √(μ_rε_r).

VACUUM

32.4.1 Speed in free space

c = 1√μ₀ε₀ = E₀B₀ = 3 × 10⁸ m/s

Same in all directions.
Does not depend on the motion of the source.
Same for all wavelengths in vacuum.

MEDIUM

32.4.2 Speed in a medium

v = 1√(με) = c√μ_rε_r

μ = μ_rμ₀, ε = ε_rε₀.
Dielectric constant k is the same as ε_r.
Speed in a medium depends on wavelength (dispersion), not on intensity.

REFRACTIVE INDEX

32.4.3 Link with n

n = cv = √μ_rε_r

Non-magnetic medium, μ_r = 1:

n = √ε_r · ε_r = n²

n = √μεμ₀ε₀

FROM THE EQUATION

32.4.4 Reading v from a wave

v = ωk = E₀B₀

Find v from the two numbers in the phase.
Then n = cv, and ε_r = n² (non-magnetic).
Time for an echo: t = 2dv.

ENTERING A MEDIUM

32.4.5 What changes

Quantity	In the medium
frequency	unchanged
speed	v = cn
wavelength	λ_m = λn
wave number	k_m = nk

cv = sin isin r

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Wave impedance

Hindrance a medium offers to the wave.
H is the magnetic intensity: H = Bμ.

Z = EH = √με = √μ_rε_r √μ₀ε₀

free space: Z = √μ₀ε₀ = 376.6 Ω ≈ 120π Ω

Practise PYQs on this topicSpeed of EM Waves in a Medium · 28 questions with solutions

CONCEPT

Energy Density

In one line
The energy of an EM wave is shared equally by its electric and magnetic fields.

AT ANY INSTANT

32.5.1 Energy per unit volume

u_E = 12ε₀E²

u_B = B²2μ₀

Using E = cB, these two are always equal.
Same result in a medium, with ε and μ.

u = u_E + u_B = ε₀E² = B²μ₀

AVERAGE VALUES

32.5.2 Over one cycle

Quantity	Value
average electric	14ε₀E₀²
average magnetic	B₀²4μ₀
total average	12ε₀E₀² = B₀²2μ₀

The average of sin²ωt is one half.
Ratio of average electric to average magnetic = 1.
Ratio of average electric to total average = 12.

u_av = ε₀E_rms²

ENERGY IN A VOLUME

32.5.3 Total energy

U = u_av × V

Same energy in a smaller volume needs a larger E₀: E₀² ∝ 1V.
Half the diameter of a cylinder: V becomes one fourth, so E₀ doubles.

HOW IT VARIES

32.5.4 Oscillation of energy

u ∝ E² ∝ sin²ωt.
So the energy in a small volume oscillates at twice the frequency of the wave.

Practise PYQs on this topicEnergy Density & Intensity · 36 questions with solutions

CONCEPT

Intensity

In one line
Intensity is the energy crossing unit area per unit time, on a surface perpendicular to the direction of travel.

DEFINITION

32.6.1 Energy flow

I = energyarea × time = u_av × c

All the energy in a column of length cΔt crosses area A in time Δt

Unit: W/m². Also called energy flux.
E and B contribute equally: ratio 1 : 1.

FROM AMPLITUDES

32.6.2 Working formula

I = 12ε₀E₀²c = cB₀²2μ₀

E₀ = √2Iε₀c · B₀ = E₀c

ε₀c = 2.655 × 10⁻³ in SI: a handy number.
√2Iε₀c has the unit of electric field, N/C.
In terms of rms: I = ε₀E_rms²c.

LASER BEAM

32.6.3 Power and area

I = PA

IN A MEDIUM

32.6.4 Refractive index n

I = 12nε₀cE₀²

With a fraction reflected: I_transmitted = (1 − r)I.

POINT SOURCE

32.6.5 Bulb, lamp, LED

Power spreads over a sphere

I = P_rad4πr²

P_rad = efficiency × rated power.
I ∝ 1r², and E₀ ∝ 1r.
All points at the same r have the same I, whatever the angle.
Intensity due to the electric field alone = I2.

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Poynting vector

Rate of energy flow per unit area; unit W/m².
Points along the direction of travel.

S = 1μ₀ (E × B)

S_max = E₀B₀μ₀

S_av = E₀B₀2μ₀ = 12ε₀E₀²c = cB₀²2μ₀

Its direction does not oscillate; its size swings between 0 and S_max.
S_av is the intensity I.

Practise PYQs on this topicEnergy Density & Intensity · 36 questions with solutions

CONCEPT

Momentum & Radiation Pressure

In one line
EM waves have no rest mass, yet they carry momentum, so they push on any surface they strike.

MOMENTUM

32.7.1 Carried by the wave

p = Uc

Momentum given: absorbing Uc, reflecting 2Uc

Surface	Momentum delivered
fully absorbing	Uc
fully reflecting	2Uc
fraction r reflected	(1 + r)Uc

FORCE

32.7.2 On a surface of area A

Energy per second on the surface = IA.
Force = momentum delivered per second.

Surface	Force F
absorbing	IAc
reflecting	2IAc
fraction r reflected	(1 + r)IAc

The time of exposure does not change the force or the pressure.
It only changes the total momentum: p = F × t.

RADIATION PRESSURE

32.7.3 Force per unit area

P_absorbing = Ic

P_reflecting = 2Ic

Check the units: change W/cm² to W/m² when A is in m².
I can be found first from the amplitude of E.

IN A MEDIUM

32.7.4 Wave speed v

P_absorbing = Iv = nIc

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Force on a reflecting hemisphere

Point source of power P at the centre, inner surface fully reflecting.
Sideways parts of the force cancel; only the axial part adds up over projected area πR².

F = 2Ic × πR² = P2c

Practise PYQs on this topicMomentum & Radiation Pressure · 14 questions with solutions

CONCEPT

Electromagnetic Spectrum: Order & Ranges

In one line
The whole ordered range of frequencies and wavelengths of EM waves is the EM spectrum.

ORDER

32.8.1 Shortest to longest wavelength

The EM spectrum

γ < X < UV < visible < IR < micro < radio

Wavelength rises left to right; frequency and photon energy fall.
All of them travel at the same speed c in vacuum.

RANGES

32.8.2 Wavelength of each part

Wave	Wavelength
Radio	above 0.1 m
Microwave	0.1 m to 1 mm
Infrared	1 mm to 700 nm
Visible	700 nm to 400 nm
Ultraviolet	400 nm to 1 nm
X-rays	1 nm to 10⁻³ nm
Gamma rays	below 10⁻³ nm

VISIBLE LIGHT

32.8.3 VIBGYOR

Violet has the shortest wavelength (about 400 nm).
Red has the longest wavelength (about 700 nm).
Blue photon has more energy than yellow.

PHOTON ENERGY

32.8.4 Finding the region

E = hf = hcλ

λ(nm) ≈ 1240E(eV)

A photon of about 10 keV gives λ near 0.1 nm: X-ray region.

NOT EM WAVES

32.8.5 Watch out

α-rays and β-rays are charged particles.
Cathode rays are electrons.
Sound is a mechanical wave.
Heat rays are infrared, so they are EM waves.

Practise PYQs on this topicElectromagnetic Spectrum & Atmosphere · 47 questions with solutions

CONCEPT

Production, Uses & the Atmosphere

In one line
Each part of the spectrum has its own source and its own uses. The atmosphere decides which of them reach us.

PRODUCTION

32.9.1 Source of each wave

Wave	Produced by
Radio	rapid acceleration of electrons in aerials
Microwave	klystron valve, magnetron
Infrared	vibration of atoms and molecules
Visible	outer electrons changing levels
Ultraviolet	inner electrons, arc lamps
X-rays	inner shell electrons, X-ray tube
Gamma rays	radioactive decay of the nucleus

USES

32.9.2 What each wave does

Wave	Uses
Radio	AM and FM, TV, communication
Microwave	radar, aircraft navigation, oven
Infrared	physiotherapy, muscular strain, night and fog photography, remote control
Visible	to see objects
Ultraviolet	sterilizing, water purification, food preservation, Lasik surgery, finger prints
X-rays	medical diagnosis, crystal structure
Gamma rays	nuclear structure, cancer treatment

WORTH REMEMBERING

32.9.3 Common questions

Microwave oven: its frequency matches the resonant frequency of water molecules.
Human body radiates all the time, in the infrared.
X-rays study crystals: their λ is close to the atomic spacing.
To observe a particle, choose λ of the order of its size.
Radio waves diffract more than microwaves: longer λ.
Object much smaller than λ: the wave is scattered.
Optical waves have shorter λ than microwaves.

GREENHOUSE EFFECT

32.9.4 Warming of the earth

Infrared from the earth is sent back

Infrared from the earth is reflected back by low clouds and CO₂.
So the cause is infrared rays.

OZONE LAYER

32.9.5 The UV shield

Absorbs most of the sun’s ultraviolet rays.
Blocks wavelengths below about 3 × 10⁻⁷ m.
This protects living cells.

RADIO PROPAGATION

32.9.6 How signals travel

Ground wave and sky wave

Ground wave: follows the surface of the earth.
Sky wave: reflected back by the ionosphere.
AM band: frequency up to 30 MHz. Amplitude of the carrier follows the signal.
FM band: 80 MHz to 200 MHz. Frequency of the carrier follows the signal.

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Layers of the atmosphere

Layers above the earth

78% N₂, 21% O₂, traces of other gases.
Troposphere: temperature falls from 290 K to 220 K with height.
Stratosphere: temperature 220 K to 200 K.
Mesosphere: temperature falls to 180 K.
Ionosphere: ions, electrons and neutral molecules.
Kennelly-Heaviside layer: about 110 km up, very high electron density.

Radio bands

Band	Frequency
VLF	10 kHz to 30 kHz
LF	30 kHz to 300 kHz
MF (MW)	300 kHz to 3000 kHz
HF (SW)	3 MHz to 30 MHz
VHF	30 MHz to 300 MHz
UHF	300 MHz to 3000 MHz
SHF (micro)	3000 MHz to 300000 MHz

TV signals and antenna height

Range of a tall antenna

TV signals are frequency modulated.
h = antenna height, R = radius of the earth.

d = √(2hR)

area covered A = πd² = 2πhR

Population covered = area × population density.

Practise PYQs on this topicElectromagnetic Spectrum & Atmosphere · 47 questions with solutions

EXAM-DAY CARD

Just see this before the exam

E₀B₀ = c

wave

i_d = ε₀dφ_Edt

displacement current

n = √μ_rε_r

medium

I = 12ε₀E₀²c

intensity

P = Ic or 2Ic

radiation pressure

γ X UV V IR M R

spectrum

FORMULA SHEET

Whole chapter on one page

Wave

travel along E × B

E_y = E₀ sin(kx − ωt)

k = 2πλ, c = ωk = fλ

E₀ = cB₀

Maxwell

i_d = ε₀dφ_Edt = CdVdt

∮B·dl = μ₀(i_c + i_d)

J_c to J_d = σ to ωε

ν = 12π√(LC)

Medium

c = 1√μ₀ε₀

v = c√μ_rε_r

n = cv = √μ_rε_r

f same, λ_m = λn

Energy

u_E = 12ε₀E², u_B = B²2μ₀

u_E = u_B

u_av = 12ε₀E₀²

U = u_avV

Intensity

I = u_avc = 12ε₀E₀²c

I = cB₀²2μ₀

point source: I = P4πr²

laser: I = PA

Pressure

absorbing: p = Uc, P = Ic

reflecting: p = 2Uc, P = 2Ic

F = (1 + r)IAc

λ(nm) ≈ 1240E(eV)

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

Direction of travel from B × E

Using c in a medium

Total u_av = ¼ε₀E₀²

Reflecting surface: F = IAc

CORRECT ✓

Always use E × B

v = cn, and frequency stays the same

¼ is electric alone; total is ½ε₀E₀²

Reflecting: F = 2IAc

Electromagnetic Waves

Nature of EM Waves & Their Direction

32.1.1 What is an EM wave?

32.1.2 E and B together

32.1.3 Direction of propagation

32.1.4 B from k and E

Wave Equation, E0 = cB0 & Force on a Charge

32.2.1 Wave along +x

32.2.2 What the equation tells you

32.2.3 E0 = cB0

32.2.4 B from E, or E from B

32.2.5 Lorentz force

Displacement Current, Maxwell’s Equations & Production

32.3.1 Where it fails

32.3.2 Ampere-Maxwell law

32.3.3 Working forms

32.3.4 Comparing the two currents

32.3.5 The four laws

32.3.6 What makes an EM wave?

32.3.7 Who did what

Corner

Speed of EM Waves in a Medium

32.4.1 Speed in free space

32.4.2 Speed in a medium

32.4.3 Link with n

32.4.4 Reading v from a wave

32.4.5 What changes

Corner

Energy Density

32.5.1 Energy per unit volume

32.5.2 Over one cycle

32.5.3 Total energy

32.5.4 Oscillation of energy

Intensity

32.6.1 Energy flow

32.6.2 Working formula

32.6.3 Power and area

32.6.4 Refractive index n

32.6.5 Bulb, lamp, LED

Corner

Momentum & Radiation Pressure

32.7.1 Carried by the wave

32.7.2 On a surface of area A

32.7.3 Force per unit area

32.7.4 Wave speed v

Corner

Electromagnetic Spectrum: Order & Ranges

32.8.1 Shortest to longest wavelength

32.8.2 Wavelength of each part

32.8.3 VIBGYOR

32.8.4 Finding the region

32.8.5 Watch out

Production, Uses & the Atmosphere

32.9.1 Source of each wave

32.9.2 What each wave does

32.9.3 Common questions

32.9.4 Warming of the earth

32.9.5 The UV shield

32.9.6 How signals travel

Corner

Just see this before the exam

Whole chapter on one page

Avoid these = easy marks

Wave Equation, E₀ = cB₀ & Force on a Charge

32.2.3 E₀ = cB₀