PhysicsCBSE Class 12 · Electromagnetic Waves

Nature of Electromagnetic Waves Class 12: Properties and Wave Equation

Updated 15 October 2026 · Fact-checked

Electromagnetic waves are produced by accelerated charges and consist of electric and magnetic fields oscillating in phase, perpendicular to each other and to the direction of travel, so they are transverse. They need no medium, travel in vacuum at c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s, have E₀/B₀ = c, and carry energy and momentum.

Understand Nature of Electromagnetic Waves

Light from the Sun crosses 150 million km of empty space; radio signals reach spacecraft beyond the planets. No medium is needed, because an electromagnetic (EM) wave is made of fields that keep regenerating each other.

Source. A charge at rest has only an electric field; a charge moving at constant velocity also has a steady magnetic field. Neither radiates. An accelerated charge produces changing electric and magnetic fields that spread out as an EM wave. A charge oscillating at frequency ν radiates a wave of the same frequency ν. Hertz first produced and detected such waves (of a few metres wavelength) with an oscillating spark circuit, confirming Maxwell's prediction.

Structure of a plane EM wave. For a wave travelling along +z:
$E_{x} = E_{0} sin (k z - ω t)$ and $B_{y} = B_{0} sin (k z - ω t)$ ,
where $k = \frac{2 π}{λ}$ and $ω = 2 π ν$ . Key features (a qualitative idea of the transverse nature is what the syllabus needs):

$E$ and $B$ are perpendicular to each other and to the direction of propagation: EM waves are transverse.
The direction of travel is the direction of $E \times B$ .
$E$ and $B$ oscillate in phase: both reach their maxima at the same place and time.
The amplitudes are related by $\frac{E _{0}}{B _{0}} = c$ , so the magnetic field amplitude is tiny (E₀ of 3 V/m goes with B₀ of only 10⁻⁸ T).
Speed: $c = \frac{ω}{k} = ν λ$ .

Speed from μ₀ and ε₀. Maxwell's equations give the speed of EM waves in vacuum as $c = \frac{1}{μ _{0} ε _{0}}$ . Putting in $μ_{0} = 4 π \times 1 0^{- 7}$ T m/A and $ε_{0} = 8.854 \times 1 0^{- 12}$ C² N⁻¹ m⁻² gives about $3 \times 1 0^{8}$ m/s, the measured speed of light. This showed that light is an electromagnetic wave. All EM waves, from radio to gamma rays, have this same speed in vacuum.

In a medium. In a material of permeability μ and permittivity ε, $v = \frac{1}{μ ε}$ , slower than c. The refractive index is $n = \frac{c}{v} = μ_{r} ε_{r}$ (≈ $ε_{r}$ for non-magnetic materials). When a wave passes from one medium into another, its frequency stays the same, while its speed and wavelength change: $λ_{m e d i u m} = \frac{λ _{0}}{n}$ .

Energy and momentum. EM waves carry energy, shared equally between the electric and magnetic fields, and also momentum. When a wave carrying energy U is completely absorbed by a surface, it delivers momentum $p = \frac{U}{c}$ , which is why light exerts a (very small) pressure. Sunlight warming your skin is energy delivered by EM waves.

Nature of Electromagnetic Waves: formulas and key facts

Plane wave along +z: $E_{x} = E_{0} sin (k z - ω t), B_{y} = B_{0} sin (k z - ω t)$; k = 2π/λ, ω = 2πν; E, B in phase.
Amplitude relation: $\frac{E _{0}}{B _{0}} = c$; In a medium, E₀/B₀ = v.
Speed in vacuum: $c = \frac{1}{μ _{0} ε _{0}} = ν λ = \frac{ω}{k}$; ≈ 3 × 10⁸ m/s for all EM waves.
Speed and refractive index in a medium: $v = \frac{1}{μ ε}, n = \frac{c}{v} = μ_{r} ε_{r}$; Frequency unchanged; λ_medium = λ₀/n.
Direction of propagation: $\overset{v}{^} ∥ E \times B$; Use to find a missing field direction.
Momentum delivered on absorption: $p = \frac{U}{c}$; For complete absorption of energy U.

How to solve Nature of Electromagnetic Waves questions

Use this method for questions on EM wave properties and wave equations.

1From a wave equation, read the amplitude, k and ω: $λ = \frac{2 π}{k}$ , $ν = \frac{ω}{2 π}$ , $v = \frac{ω}{k}$ .
2The sign between kx and ωt gives direction: $(k x - ω t)$ travels along +x; $(k x + ω t)$ along −x.
3Find the other amplitude with $E_{0} = c B_{0}$ (or $v B_{0}$ in a medium).
4Find the direction of the missing field so that $E \times B$ points along the direction of travel.
5For media, use $n = μ_{r} ε_{r}$ , $v = \frac{c}{n}$ and $λ = \frac{λ _{0}}{n}$ ; keep the frequency unchanged.
6Check that $v = ν λ$ gives 3 × 10⁸ m/s in vacuum.

Nature of Electromagnetic Waves in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Wavelength of radio waves

Suppose an FM radio station broadcasts at 100 MHz, and a simple car antenna works best when its length is a quarter of the wavelength.

(a) Find the wavelength of the station's waves. (b) Find the ideal antenna length. (c) What is the frequency of oscillation of the electrons in the transmitting antenna?

Try it first, then show the solutionSolution

$λ = \frac{c}{ν} = \frac{3 \times 1 0 ^{8}}{1 0 ^{8}} = 3$ m.
Quarter-wave antenna: $\frac{3}{4} = 0.75$ m.
The radiated wave has the same frequency as the oscillating charges: 100 MHz.

Answer: (a) 3 m; (b) 0.75 m; (c) 100 MHz.

Memory hook: The wave copies the frequency of the charges that make it.

Magnetic part of an EM wave is tiny

Imagine a microwave link whose signal at a receiving dish has an electric field amplitude of 0.6 V/m.

(a) Find the magnetic field amplitude. (b) Why are receivers usually designed to respond to the electric field?

Try it first, then show the solutionSolution

$B_{0} = \frac{E _{0}}{c} = \frac{0.6}{3 \times 1 0 ^{8}} = 2 \times 1 0^{- 9}$ T.
Because E₀/B₀ = c, the magnetic field is numerically tiny; the force it exerts on charges ( $q v B$ ) is far smaller than qE for slowly moving charges, so the electric field drives the antenna current.

Answer: (a) $2 \times 1 0^{- 9}$ T; (b) the electric field has the larger effect on the charges in the antenna.

Memory hook: E is the strong partner; B = E/c is the quiet one.

Tricks to remember Nature of Electromagnetic Waves

Short memory aids. Say each one aloud once, then use it on a question the same day.

E cross B points the way
Fingers along E, curl towards B, thumb shows the direction of travel.
Use it when: Direction questions.
Accelerate to radiate
Charge at rest: E only; steady motion: E and B but no wave; acceleration: EM wave.
Use it when: Source of EM waves.
E over B is c
E₀/B₀ = c, just like distance over time.
Use it when: Finding one amplitude from the other.
Frequency is loyal
Crossing into a new medium, frequency stays; speed and wavelength change.
Use it when: Waves in media.

Nature of Electromagnetic Waves: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Amplitudes and wavelength · 2 marks · Very short answerBoard level

A 50 MHz EM wave travels in vacuum with an electric field amplitude of 30 V/m. Find (a) the magnetic field amplitude and (b) the wavelength.

Try it first, then show the solutionSolution

(a) $B_{0} = \frac{E _{0}}{c} = \frac{30}{3 \times 1 0 ^{8}} = 1 0^{- 7}$ T. (1)
(b) $λ = \frac{c}{ν} = \frac{3 \times 1 0 ^{8}}{5 \times 1 0 ^{7}} = 6$ m. (1)

Answer: (a) $1 0^{- 7}$ T; (b) 6 m.

Example 2: Properties of EM waves · 3 marks · Short answerBoard level

Give the source of electromagnetic waves and list four characteristics. Sketch the variation of E and B along the direction of propagation.

Try it first, then show the solutionSolution

Source: accelerated (for example, oscillating) charges; the wave frequency equals the frequency of oscillation. (1)
Characteristics: transverse (E ⊥ B ⊥ direction of travel); E and B in phase with E₀/B₀ = c; speed $\frac{1}{μ _{0} ε _{0}}$ in vacuum; need no medium and carry energy and momentum. (1)
Sketch: two sine curves along z, E in the x-z plane and B in the y-z plane, crests at the same points; arrow along z marked as the direction of E × B. (1)

Answer: Produced by accelerated charges; transverse, E and B in phase with E₀/B₀ = c, speed c in vacuum, no medium needed, carry energy and momentum.

Example 3: Reading a wave equation · 3 marks · Short answerHarder board level

The magnetic field of an EM wave in vacuum is $B_{z} = 3 \times 1 0^{- 8} sin (2 \times 1 0^{3} x - 6 \times 1 0^{11} t)$ T. Find (a) the wavelength and frequency, (b) the speed, and (c) the amplitude and direction of the electric field. Write the expression for E.

Try it first, then show the solutionSolution

(a) $λ = \frac{2 π}{k} = \frac{2 π}{2 \times 1 0 ^{3}} \approx 3.14 \times 1 0^{- 3}$ m; $ν = \frac{ω}{2 π} = \frac{6 \times 1 0 ^{11}}{2 π} \approx 9.55 \times 1 0^{10}$ Hz. (1)
(b) $v = \frac{ω}{k} = 3 \times 1 0^{8}$ m/s, along +x. (1)
(c) $E_{0} = c B_{0} = 9$ V/m. For travel along +x with B along z, E must be along y ( $\overset{y}{^} \times \overset{z}{^} = \overset{x}{^}$ ): $E_{y} = 9 sin (2 \times 1 0^{3} x - 6 \times 1 0^{11} t)$ V/m. (1)

Answer: (a) about 3.14 mm and 9.55 × 10¹⁰ Hz; (b) 3 × 10⁸ m/s along +x; (c) 9 V/m along y, E_y = 9 sin(2 × 10³x − 6 × 10¹¹t) V/m.

Example 4: Speed in a dielectric · 3 marks · Short answerHarder board level

A non-magnetic material has a relative permittivity of 1.69. Find (a) its refractive index and (b) the speed of EM waves in it. (c) Show that $\frac{1}{μ _{0} ε _{0}}$ gives the speed of light in vacuum.

Try it first, then show the solutionSolution

(a) $n = μ_{r} ε_{r} = 1 \times 1.69 = 1.3$ . (1)
(b) $v = \frac{c}{n} = \frac{3 \times 1 0 ^{8}}{1.3} \approx 2.31 \times 1 0^{8}$ m/s. (1)
(c) $\frac{1}{4 π \times 1 0 ^{- 7} \times 8.854 \times 1 0 ^{- 12}} \approx 3.0 \times 1 0^{8}$ m/s. (1)

Answer: (a) 1.3; (b) about $2.31 \times 1 0^{8}$ m/s; (c) about $3.0 \times 1 0^{8}$ m/s.

Example 5: Finding field directions · 5 marks · Long answerHOTS / competency

(a) An EM wave travels along −y. At some instant, its electric field at a point is along +z. Find the direction of the magnetic field there. (2)
(b) An electron oscillates at $1 0^{9}$ Hz. Find the wavelength of the wave it radiates and name the region of the spectrum. (2)
(c) Why does an electron moving with constant velocity not radiate? (1)

Try it first, then show the solutionSolution

(a) Need $E \times B$ along −y with $E$ along +z. Try $B$ along −x: $\overset{z}{^} \times (- \overset{x}{^}) = - \overset{y}{^}$ . (1)
So B is along −x. (1)
(b) $λ = \frac{c}{ν} = \frac{3 \times 1 0 ^{8}}{1 0 ^{9}} = 0.3$ m. (1)
This is in the radio-wave region (wavelengths above 0.1 m), close to the boundary with microwaves. (1)
(c) Radiation needs changing fields spreading out; a uniformly moving charge carries steady fields along with it, so no wave is emitted. Only acceleration produces EM waves. (1)

Answer: (a) along −x; (b) 0.3 m, radio waves (near the microwave boundary); (c) a uniformly moving charge has no accelerating changes to radiate.

Example 6: Light entering glass · 4 marks · Case-basedHarder board level

Read the passage and answer.

Light of frequency $5 \times 1 0^{14}$ Hz passes from vacuum into glass of relative permittivity 2.56 (non-magnetic).
(a) Find its wavelength in vacuum. (1)
(b) Find its speed and wavelength in the glass. (1)
(c) Does the frequency change in glass? Why? (1)
(d) If the electric field amplitude in glass is 6 V/m, find the magnetic field amplitude there. (1)

Try it first, then show the solutionSolution

(a) $λ_{0} = \frac{3 \times 1 0 ^{8}}{5 \times 1 0 ^{14}} = 6 \times 1 0^{- 7}$ m = 600 nm. (1)
(b) n = √2.56 = 1.6: $v = \frac{3 \times 1 0 ^{8}}{1.6} = 1.875 \times 1 0^{8}$ m/s and $λ = \frac{600}{1.6} = 375$ nm. (1)
(c) No: the frequency is set by the source (the oscillating charges), and the waves crossing the boundary must stay in step on both sides. (1)
(d) $B_{0} = \frac{E _{0}}{v} = \frac{6}{1.875 \times 1 0 ^{8}} = 3.2 \times 1 0^{- 8}$ T. (1)

Answer: (a) 600 nm; (b) 1.875 × 10⁸ m/s and 375 nm; (c) no, frequency is fixed by the source; (d) 3.2 × 10⁻⁸ T.

Where students lose marks in Nature of Electromagnetic Waves

The slips examiners see most often, and what to write instead.

Mistake: Saying E and B are 90° out of phase in an EM wave.
Confusing with AC circuits.
Do this: In a travelling EM wave, E and B are in phase.
Usually costs: 1 mark.
Mistake: Saying any moving charge radiates.
Missing the role of acceleration.
Do this: Only accelerated charges radiate EM waves.
Usually costs: 1 mark.
Mistake: Changing the frequency when a wave enters a medium.
Thinking all wave properties change.
Do this: Frequency stays the same; speed and wavelength change.
Usually costs: 1 mark.
Mistake: Getting the direction of travel from B × E.
Reversing the order.
Do this: Direction of travel is along E × B.
Usually costs: 1 mark.
Mistake: Discussing polarisation of EM waves.
It appears in older notes.
Do this: For 2026-27 the transverse nature is a qualitative idea only, and polarisation is not in the syllabus.
Usually costs: Time.

Exam tips

List properties as short points: source, transverse, in phase, E₀/B₀ = c, speed formula, no medium, energy and momentum.
In equation questions, write k, ω, λ, ν and the direction first, then the missing field.
Questions on the nature and propagation of EM waves appeared in 2025 (55/7/1 Q15) and 2026 (55/1/1 Q5 and 55/5/1 Q6).
Draw the standard sketch of E and B sine curves at right angles whenever a figure is asked.
Keep answers on the transverse nature qualitative, as the syllabus specifies.

Practice questions on Nature of Electromagnetic Waves

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
In an electromagnetic wave, the electric and magnetic fields are
(a) parallel to each other and to the direction of travel (b) perpendicular to each other and to the direction of travel (c) perpendicular to each other but parallel to the direction of travel (d) out of phase by 90°
Hint
Transverse wave.
Answer
(b).
Q2 · 1 mark · Very short answerBoard level
What is the source of electromagnetic waves? What determines their frequency?
Hint
What must charges be doing?
Answer
Accelerated charges; the frequency equals the frequency with which the charges oscillate.
Q3 · 1 mark · Very short answerHarder board level
The magnetic field amplitude of an EM wave in vacuum is $4 \times 1 0^{- 8}$ T. Find the electric field amplitude.
Hint
E₀ = cB₀.
Answer
$E_{0} = 3 \times 1 0^{8} \times 4 \times 1 0^{- 8} = 12$ V/m.
Q4 · 1 mark · Very short answerHarder board level
Find the speed of EM waves in a non-magnetic medium of relative permittivity 4, and its refractive index.
Hint
n = √ε_r.
Answer
n = 2; v = 1.5 × 10⁸ m/s.
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Electromagnetic waves can travel through vacuum.
Reason (R): They consist of oscillating electric and magnetic fields that sustain each other and do not need material particles to vibrate.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
What vibrates in an EM wave?
Answer
(a).
Q6 · 2 marks · Very short answerHOTS / competency
At a point, the electric field of an EM wave points along +x and the magnetic field along +y. In which direction does the wave travel? If it then enters a medium, what happens to its direction relative to E and B?
Hint
x̂ × ŷ.
Answer
$\overset{x}{^} \times \overset{y}{^} = \overset{z}{^}$ : the wave travels along +z. In any medium the wave stays transverse: E, B and the direction of travel remain mutually perpendicular, with travel along E × B.
Q7 · 2 marks · Very short answerHOTS / competency
Show that $\frac{1}{μ _{0} ε _{0}}$ has the units of speed and evaluate it.
Hint
Use T m/A and C² N⁻¹ m⁻².
Answer
μ₀ε₀ has units (N/A²)(C²/(N m²)) = C²/(A² m²) = s²/m², so 1/√(μ₀ε₀) has units m/s. Its value is $\frac{1}{4 π \times 1 0 ^{- 7} \times 8.854 \times 1 0 ^{- 12}} \approx 3.0 \times 1 0^{8}$ m/s.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Source of EM waves?Show answer
Accelerated charges.
2.Relation between E₀ and B₀?Show answer
E₀/B₀ = c.
3.Speed of EM waves in vacuum in terms of μ₀ and ε₀?Show answer
c = 1/√(μ₀ε₀).
4.Phase relation between E and B in an EM wave?Show answer
In phase.
5.Direction of propagation?Show answer
Along E × B.
6.Refractive index of a medium in terms of μ_r and ε_r?Show answer
n = √(μ_rε_r).

Nature of Electromagnetic Waves: frequently asked questions

Why are electromagnetic waves called transverse?

Their electric and magnetic fields oscillate perpendicular to the direction in which the wave travels, just as a wave on a string moves sideways while travelling along the string.

Do electromagnetic waves need a medium?

No. The changing electric and magnetic fields regenerate each other, so the wave can travel through vacuum. Light from the Sun reaches us through empty space.

Do all EM waves travel at the same speed?

In vacuum, yes: all EM waves, from radio waves to gamma rays, travel at c ≈ 3 × 10⁸ m/s. In a material their speed is less and can depend on frequency.

Why is the magnetic field of an EM wave so small?

Because E₀/B₀ = c, a very large number. A wave with E₀ of a few volts per metre has B₀ of only about 10⁻⁸ T.

Do EM waves carry momentum?

Yes. A wave carrying energy U delivers momentum U/c when it is completely absorbed. This produces radiation pressure, which is tiny for everyday light.