Electromagnetic Waves Class 12: Notes, Spectrum and Solved Questions

What Electromagnetic Waves is about

Weightage: This chapter is part of Units V and VI, Electromagnetic Waves; Optics, which carry 18 of the 70 theory marks together with Ray Optics and Optical Instruments and Wave Optics.Source: CBSE Physics syllabus 2026-27, Units V and VI Electromagnetic Waves; Optics

In real board papers: asked in 8 of the 8 board papers we hold (2023 to 2026), on average 3.1 marks per paper. See the questions

Key concepts

Displacement current and the Ampere-Maxwell law

NCERT section 8.2

While a capacitor charges, a conduction current i flows in the wires but no charge crosses the gap between the plates. Apply Ampere's law round a small loop around the wire: one surface bounded by the loop is pierced by the wire (enclosed current i), while another surface with the same boundary can pass between the plates (enclosed current 0). Ampere's law gives two different answers for the same loop, so something is missing.

Between the plates the electric field is changing: $E=\frac{q}{{\epsilon }_{0}A}$, so the electric flux is ${\Phi }_E=EA=\frac{q}{{\epsilon }_0}$ and ${\epsilon }_0\frac{d{\Phi }_E}{dt}=\frac{dq}{dt}=i$. Maxwell called $i_d={\epsilon }_0\frac{d{\Phi }_E}{dt}$ the displacement current and corrected Ampere's law:

$\oint \vec{B}\cdot d\vec{l}={\mu }_0(i_c+i_d)={\mu }_0{i}_c+{\mu }_0{\epsilon }_0\frac{d{\Phi }_E}{dt}$

The total current $i_c+i_d$ is continuous across the gap. Displacement current is not a flow of charge, but it produces a magnetic field just as a conduction current does. It is significant only where electric fields change rapidly; for steady fields it is zero.

Worked example

A parallel plate capacitor has plates of area $0.02m^2$ separated by 2 mm in air. The voltage across it is changing at $1\times 10^{6}V{s}^{-1}$. Find the displacement current between the plates. (${\epsilon }_0=8.85\times 10^{-12}{C}^2{N}^{-1}{m}^{-2}$) [2 marks]

Try it first, then show the solutionSolution

1. $i_d={\epsilon }_0\frac{d{\Phi }_E}{dt}={\epsilon }_{0}A\frac{dE}{dt}=\frac{{\epsilon }_{0}A}{d}\frac{dV}{dt}=C\frac{dV}{dt}$
2. $C=\frac{8.85\times 10^{-12}\times 0.02}{2\times 10^{-3}}=8.85\times 10^{-11}F$
3. $i_d=8.85\times 10^{-11}\times 10^6=8.85\times 10^{-5}A$

Answer: About $8.9\times 10^{-5}A$ (89 µA), equal to the conduction current in the leads

Sources of electromagnetic waves

NCERT section 8.3.1

A charge at rest produces only an electric field. A charge moving steadily (a steady current) also produces a steady magnetic field. Neither radiates. An accelerated charge produces changing electric and magnetic fields that regenerate each other and travel outward: an electromagnetic wave.

A charge oscillating at frequency ν (and so constantly accelerating) produces an oscillating electric field, which produces an oscillating magnetic field, and so on: the wave has the same frequency ν as the oscillating charge. Radio antennas work this way. For visible light, the oscillating charges are electrons in atoms, and the wavelength is close to atomic sizes. In general, the wavelength of radiation is often related to the size of the system that emits it, for example gamma rays come from nuclei (about $10^{-14}m$) and long radio waves from large aerials.

Worked example

An electric charge oscillates about its mean position with a frequency of $5\times 10^{8}Hz$. What is the frequency and wavelength of the electromagnetic waves it produces? [1 mark]

Try it first, then show the solutionSolution

1. The wave has the same frequency as the oscillating charge: $\nu =5\times 10^{8}Hz$
2. $\lambda =\frac{c}{\nu }=\frac{3\times 10^8}{5\times 10^8}=0.6m$

Answer: $5\times 10^{8}Hz$; 0.6 m (radio waves)

Nature and properties of electromagnetic waves

NCERT section 8.3.2

For a wave travelling along z, the fields are $E_x=E_0\sin (kz-\omega t)$ and $B_y=B_0\sin (kz-\omega t)$, with $k=\frac{2\pi }{\lambda }$ and $\omega =2\pi \nu$.

• Transverse: $\vec{E}$ and $\vec{B}$ are perpendicular to each other and to the direction of travel, which is along $\vec{E}\times \vec{B}$. (Qualitative idea only in the syllabus.)
• E and B oscillate in phase, reaching their maxima together.
• Speed in vacuum: $c=\frac{\omega }{k}=\nu \lambda =\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}\approx 3\times 10^{8}m{s}^{-1}$, the same for all wavelengths.
• Ratio of amplitudes: $\frac{E_0}{B_0}=c$.
• In a medium, $v=\frac{1}{\sqrt{\mu \epsilon }}$, which is less than c; the frequency stays the same and the wavelength shortens.
• They need no medium: they travel through vacuum.
• They carry energy and momentum, so they exert pressure on surfaces they strike (radiation pressure).
• They are not deflected by electric or magnetic fields (they are uncharged).

Worked example

A plane EM wave of frequency 50 MHz travels in vacuum along the x-axis. At a point the magnetic field has amplitude $2\times 10^{-8}T$ along z. Find (a) the wavelength, (b) the amplitude and direction of the electric field. [2 marks]

Try it first, then show the solutionSolution

1. (a) $\lambda =\frac{c}{\nu }=\frac{3\times 10^8}{5\times 10^7}=6m$
2. (b) $E_0=c{B}_0=3\times 10^8\times 2\times 10^{-8}=6V{m}^{-1}$
3. Direction: $\vec{E}\times \vec{B}$ must be along $+\hat{i}$. With $\vec{B}$ along $\hat{k}$, $\hat{j}\times \hat{k}=\hat{i}$, so $\vec{E}$ is along y.

Answer: (a) 6 m (b) $6V{m}^{-1}$, along the y-axis

Reading the wave equation

NCERT section 8.3.2

Given a field such as $E_y=E_0\sin (kx-\omega t)$ or $B_z=B_0\sin (kx+\omega t)$, you can read off everything:

• Direction of travel: the variable inside the sine. $(kx-\omega t)$ means travel along $+x$; $(kx+\omega t)$ means along $-x$.
• Wavelength: $\lambda =\frac{2\pi }{k}$. Frequency: $\nu =\frac{\omega }{2\pi }$. Speed: $v=\frac{\omega }{k}$.
• The other field: perpendicular to both the given field and the direction of travel, with amplitude $B_0=\frac{E_0}{v}$ (or $E_0=v{B}_0$), chosen so that $\vec{E}\times \vec{B}$ points along the direction of travel.

If $v=\frac{\omega }{k}$ comes out less than c, the wave is in a medium, with refractive index $n=\frac{c}{v}$.

Worked example

The electric field of an EM wave is $E_y=60\sin (2\pi \times 10^{2}x-6\pi \times 10^{10}t)$ in SI units. Find the wavelength, frequency, speed and the amplitude of the magnetic field. Is the wave in vacuum? [3 marks]

Try it first, then show the solutionSolution

1. $k=2\pi \times 10^2{m}^{-1}$, so $\lambda =\frac{2\pi }{k}=10^{-2}m=1cm$
2. $\omega =6\pi \times 10^{10}rad{s}^{-1}$, so $\nu =3\times 10^{10}Hz$
3. $v=\frac{\omega }{k}=\frac{6\pi \times 10^{10}}{2\pi \times 10^2}=3\times 10^{8}m{s}^{-1}$: equal to c, so it is in vacuum
4. $B_0=\frac{E_0}{c}=\frac{60}{3\times 10^8}=2\times 10^{-7}T$, along z (travel is along +x and $\hat{j}\times \hat{k}=\hat{i}$)

Answer: λ = 1 cm; ν = $3\times 10^{10}Hz$ (microwave); v = $3\times 10^{8}m{s}^{-1}$ (vacuum); $B_0=2\times 10^{-7}T$ along z

The spectrum: radio waves and microwaves

NCERT section 8.4.1 and 8.4.2

The electromagnetic spectrum is the full range of EM waves arranged by wavelength or frequency. All travel at c in vacuum; they differ in how they are produced, detected and used.

Radio waves (wavelength above about 0.1 m; frequency below about $10^{9}Hz$): produced by accelerated motion of charges in conducting wires (oscillating circuits, aerials). Used in radio and television communication; AM band about 530 kHz to 1710 kHz, higher frequencies for short-wave and TV, very high frequencies for mobile phones.

Microwaves (about 0.1 m to 1 mm; short-wavelength radio waves of frequency in the GHz range): produced by special vacuum tubes (klystrons, magnetrons, Gunn diodes). Used in radar for aircraft navigation and speed guns, and in microwave ovens, where the frequency is chosen to match the resonant frequency of water molecules so that food containing water heats up.

Worked example

The FM radio band runs from 88 MHz to 108 MHz. What is the corresponding wavelength band, and which end has the longer wavelength? [1 mark]

Try it first, then show the solutionSolution

1. $\lambda =\frac{c}{\nu }$: at 88 MHz, $\lambda =\frac{3\times 10^8}{88\times 10^6}\approx 3.41m$
2. At 108 MHz, $\lambda =\frac{3\times 10^8}{108\times 10^6}\approx 2.78m$; the lower-frequency end has the longer wavelength

Answer: About 2.78 m to 3.41 m; the 88 MHz end is longer

The spectrum: infrared, visible and ultraviolet

NCERT section 8.4.3 to 8.4.5

Infrared (about 1 mm to 700 nm): produced by hot bodies and molecules; often called heat waves, because water and other molecules absorb them readily and warm up. Uses: physical therapy, infrared lamps, remote controls of TVs and other electronic devices, infrared detectors in earth satellites. Infrared also keeps the earth warm through the greenhouse effect.

Visible light (about 700 nm to 400 nm): emitted when electrons in atoms move between energy levels; detected by our eyes. It is the part of the spectrum our eyes respond to.

Ultraviolet (about 400 nm down to 0.6 nm): produced by special lamps and very hot bodies; the sun is an important source. Large doses are harmful (UV can cause skin damage); ordinary glass absorbs much of it, and the ozone layer in the atmosphere absorbs most of the sun's UV. UV lamps are used to kill germs in water purifiers, and UV lasers have medical uses such as eye surgery.

Worked example

Name the part of the electromagnetic spectrum that (a) is absorbed by the ozone layer, (b) is used in the remote control of a TV, (c) is used to kill germs in water purifiers. [2 marks]

Try it first, then show the solutionSolution

1. (a) Ultraviolet: the ozone layer absorbs most of the sun's UV.
2. (b) Infrared: remote controls use infrared signals.
3. (c) Ultraviolet: UV lamps kill germs.

Answer: (a) ultraviolet (b) infrared (c) ultraviolet

The spectrum: X-rays and gamma rays

NCERT section 8.4.6 and 8.4.7

X-rays (about $10^{-8}m$ down to $10^{-13}m$): produced by bombarding a metal target with high-energy electrons. Used in medicine as a diagnostic tool (X-ray images) and to treat some cancers. Overexposure damages living tissue.

Gamma rays (below about $10^{-10}m$): produced in nuclear reactions and emitted by radioactive nuclei. Used in medicine to destroy cancer cells.

The divisions between regions are not sharp; neighbouring regions overlap, and the names reflect how the radiation is produced (for example, X-rays and gamma rays of the same wavelength differ only in their origin). Moving from radio to gamma rays, wavelength decreases and frequency, and the energy carried by each photon (Chapter 11), increases.

Worked example

Arrange in increasing order of frequency: X-rays, microwaves, visible light, radio waves, gamma rays, infrared, ultraviolet. Which has the shortest wavelength? [1 mark]

Try it first, then show the solutionSolution

1. Frequency rises as wavelength falls: radio, microwave, infrared, visible, ultraviolet, X-rays, gamma rays.
2. Highest frequency means shortest wavelength: gamma rays.

Answer: Radio < microwave < infrared < visible < ultraviolet < X-rays < gamma rays; gamma rays have the shortest wavelength.

Physics in real life: Electromagnetic Waves

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.

Topics in this chapter

• Displacement Current
• Nature of Electromagnetic Waves
• Electromagnetic Spectrum

Electromagnetic Waves formula sheet

Displacement current

$i_d={\epsilon }_0\frac{d{\Phi }_E}{dt}$

For a capacitor $i_d=CdV/dt$.

Ampere-Maxwell law

$\oint \vec{B}\cdot d\vec{l}={\mu }_0{i}_c+{\mu }_0{\epsilon }_0\frac{d{\Phi }_E}{dt}$

Speed of EM waves in vacuum

$c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}\approx 3\times 10^{8}m{s}^{-1}$

Speed in a medium

$v=\frac{1}{\sqrt{\mu \epsilon }}=\frac{c}{\sqrt{{\mu }_r{\epsilon }_r}},n=\frac{c}{v}$

Ratio of field amplitudes

$\frac{E_0}{B_0}=c$

Wave relations

$c=\nu \lambda =\frac{\omega }{k},k=\frac{2\pi }{\lambda },\omega =2\pi \nu$

Plane wave

$E_x=E_0\sin (kz-\omega t),B_y=B_0\sin (kz-\omega t)$

Travels along $\vec{E}\times \vec{B}$ (here +z).

Electromagnetic Waves: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the board asks Electromagnetic Waves

The question patterns to expect, and the method that works for each.

Tricks to remember Electromagnetic Waves

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

• Raging Martians Invaded Venus Using X-ray Guns

  Radio, Microwave, Infrared, Visible, Ultraviolet, X-rays, Gamma rays: increasing frequency, decreasing wavelength.

  Use it when: Ordering the spectrum.

• E is c times B

  Big E, tiny B: $E_0=c{B}_0$, so B in tesla is E in V/m divided by $3\times 10^8$.

  Use it when: Finding one field amplitude from the other.

• E cross B points the way

  Point your fingers along E, curl them to B: the thumb shows the direction of travel.

  Use it when: Direction questions.

• Minus inside, moving plus

  $\sin (kx-\omega t)$ moves along $+x$; $\sin (kx+\omega t)$ along $-x$.

  Use it when: Reading wave equations.

• Uses in one line each

  Radio: broadcast. Micro: radar, oven. IR: remote, heat therapy. Visible: seeing. UV: germ-killing, absorbed by ozone. X: diagnosis. Gamma: cancer treatment.

  Use it when: One-mark use questions.

Electromagnetic Waves: board-level solved questions

Ordered from board level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section D style)

Read the situation, then answer every part. In the board paper this is a 4-mark question split into 1 + 1 + 1 + 1 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Electromagnetic Waves

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

• ✗Mistake: Calling displacement current a flow of charge

  The word 'current' misleads.

  ✓Do this: Write: it is not a flow of charge; it is ${\epsilon }_0$ times the rate of change of electric flux, and it produces a magnetic field like a conduction current.

  Usually costs: 1 mark.

• ✗Mistake: Getting the direction of the second field wrong

  Not checking $\vec{E}\times \vec{B}$.

  ✓Do this: After writing the other field's axis, check that $\vec{E}\times \vec{B}$ points along the direction of travel.

  Usually costs: ½-1 mark.

• ✗Mistake: Assuming the speed is c in every equation

  Not computing ω/k.

  ✓Do this: Always compute $v=\omega /k$; if it is less than c, the wave is in a medium and $B_0=E_0/v$.

  Usually costs: 1 mark.

• ✗Mistake: Mixing up the order or uses of UV and IR

  Both are 'invisible light'.

  ✓Do this: IR: longer than red, heat, remote controls. UV: shorter than violet, germ-killing, absorbed by ozone.

  Usually costs: 1 mark per item.

• ✗Mistake: Saying frequency changes when light enters glass

  Confusing with wavelength.

  ✓Do this: Frequency is fixed by the source; speed and wavelength fall in a denser medium.

  Usually costs: 1 mark in MCQs.

NCERT exercise guide: Electromagnetic Waves

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Electromagnetic Waves: previous year questions

• Asked in 8 of the 8 board papers we hold (2023 to 2026), on average 3.1 marks per paper; 16 questions in all, counting OR alternatives. In 4 of 4 sample papers.

Counted from the real papers on our previous year papers page, not estimated. Average marks are over every board paper we hold, with an OR pair counted once. Chapter tags are ours.

Solved board questions from Electromagnetic Waves

Real questions from recent papers. Try each one first, then open our step-by-step solution with marks per step.

Every Electromagnetic Waves question, by year

Practice questions on Electromagnetic Waves

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

1. Q1 · 1 mark · MCQBoard level

   The waves used for purifying water in water purifiers are:
   (a) infrared
   (b) ultraviolet
   (c) microwaves
   (d) radio waves

   Hint

   Germ-killing radiation.

   Answer

   (b) ultraviolet

2. Q2 · 1 mark · MCQBoard level

   The ratio of the amplitudes of the electric and magnetic fields in an EM wave in vacuum is:
   (a) 1
   (b) c
   (c) 1/c
   (d) c²

   Hint

   $E_0=c{B}_0$.

   Answer

   (b) c

3. Q3 · 1 mark · Very short answerBoard level

   Name the electromagnetic waves used in the remote control of a television.

   Hint

   Just beyond red.

   Answer

   Infrared

4. Q4 · 1 mark · Assertion-reasonHarder board level

   Assertion (A): A charge moving with uniform velocity produces electromagnetic waves.
   Reason (R): A charge moving with uniform velocity produces a magnetic field.
   (a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

   Hint

   What kind of motion radiates?

   Answer

   (d). Only accelerated charges radiate; a uniformly moving charge produces a steady magnetic field but no wave.

5. Q5 · 2 marks · Very short answerHarder board level

   Show that the displacement current between the plates of a capacitor equals $C\frac{dV}{dt}$. Find it for a 10 nF capacitor whose voltage changes at 1000 V s⁻¹.

   Hint

   ${\Phi }_E=EA=\frac{VA}{d}$.

   Answer

   $i_d={\epsilon }_0\frac{d}{dt}\left(\frac{VA}{d}\right)=\frac{{\epsilon }_{0}A}{d}\frac{dV}{dt}=C\frac{dV}{dt}$. Value: $10^{-8}\times 1000=10^{-5}A$.

6. Q6 · 3 marks · Short answerHarder board level

   The magnetic field in a plane EM wave in vacuum is $B_y=3\times 10^{-7}\sin (1.0\times 10^{3}x+3\times 10^{11}t)$ T. Find (a) the wavelength, (b) the frequency, (c) the direction of travel, (d) the amplitude of the electric field.

   Hint

   Note the + sign.

   Answer

   (a) $\lambda =\frac{2\pi }{10^3}\approx 6.28\times 10^{-3}m$ (b) $\nu =\frac{3\times 10^{11}}{2\pi }\approx 4.8\times 10^{10}Hz$ (c) along −x (d) $E_0=c{B}_0=90V{m}^{-1}$ (check: $\omega /k=3\times 10^8$, so vacuum).

7. Q7 · 2 marks · Very short answerHarder board level

   Arrange the following in increasing order of wavelength and give one use of each: gamma rays, microwaves, ultraviolet, infrared.

   Hint

   Gamma is shortest.

   Answer

   Gamma rays (cancer treatment) < ultraviolet (germ-killing) < infrared (remote controls, heat therapy) < microwaves (radar, ovens).

8. Q8 · 2 marks · Very short answerHarder board level

   A mobile phone network uses 900 MHz and a Wi-Fi router uses 2.4 GHz. (a) Find the wavelength of each. (b) Name the region of the spectrum each lies in. (c) A simple aerial is often made a quarter of a wavelength long. How long should each aerial be?

   Hint

   $\lambda =c/\nu$; radio waves lie below about $10^{9}Hz$, microwaves in the GHz range.

   Answer

   (a) 900 MHz: $\lambda =\frac{3\times 10^8}{9\times 10^8}\approx 0.33m$; 2.4 GHz: $\lambda =\frac{3\times 10^8}{2.4\times 10^9}=0.125m$. (b) 900 MHz is at the top of the radio-wave region; 2.4 GHz is a microwave. (c) Quarter wavelengths: about 8.3 cm and about 3.1 cm.

9. Q9 · 2 marks · Very short answerHOTS / competency

   In a region the electric field is $E_x=E_0\sin (kz-\omega t)$ in one place and $E_x=E_0$ (constant) in another. In which region can there be an electromagnetic wave, and why?

   Hint

   A wave needs changing fields.

   Answer

   Only where $E_x=E_0\sin (kz-\omega t)$: the field changes in space and time and travels along z. A constant field does not change, so it produces no magnetic field and no wave.

10. Q10 · 3 marks · Short answerHOTS / competency

    Suppose a satellite detects radiation of wavelength 0.1 nm from a distant star. (a) Which region of the spectrum is it? (b) A student suggests building a telescope for it on a high mountain instead. Would that work? (c) Explain why thinning of the ozone layer increases the risk of skin damage on earth.

    Hint

    Think of which bands the atmosphere lets through, and what ozone absorbs.

    Answer

    (a) $0.1nm=10^{-10}m$: X-rays. (b) No. The atmosphere absorbs X-rays, so even on a mountain top they would not reach the telescope; X-ray telescopes must be above the atmosphere, on satellites. (c) The ozone layer absorbs most of the sun's ultraviolet radiation. If it thins, more UV reaches the ground, and large doses of UV damage the skin.

11. Q11 · 2 marks · Very short answerHarder board level

    An EM wave in vacuum has an electric field amplitude of $120V{m}^{-1}$ and frequency $5\times 10^{14}Hz$. Find the magnetic field amplitude and the wavelength. Which region is it in?

    Hint

    $B_0=E_0/c$.

    Answer

    $B_0=4\times 10^{-7}T$; $\lambda =6\times 10^{-7}m=600nm$; visible light (orange).

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. 1.Write the expression for displacement current.Show answer

   $i_d={\epsilon }_{0}d{\Phi }_E/dt$.

2. 2.State the Ampere-Maxwell law.Show answer

   $\oint \vec{B}\cdot d\vec{l}={\mu }_0(i_c+{\epsilon }_{0}d{\Phi }_E/dt)$.

3. 3.What produces electromagnetic waves?Show answer

   Accelerated (for example oscillating) charges.

4. 4.Write the speed of EM waves in terms of ${\mu }_0$ and ${\epsilon }_0$.Show answer

   $c=1/\sqrt{{\mu }_0{\epsilon }_0}$.

5. 5.What is the ratio $E_0/B_0$ in vacuum?Show answer

   c.

6. 6.Why are EM waves called transverse?Show answer

   E and B oscillate perpendicular to the direction in which the wave travels.

7. 7.Which EM waves are used in radar?Show answer

   Microwaves.

8. 8.Find the wavelength of a 300 MHz wave.Show answer

   1 m.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you explain why Ampere's law needed correcting for a charging capacitor?
• ☐Can you show that ${\epsilon }_{0}d{\Phi }_E/dt$ equals the conduction current for a capacitor?
• ☐Can you state one similarity and one difference between conduction and displacement current?
• ☐Can you say why an accelerated charge radiates but a uniformly moving one does not?
• ☐Can you list five properties of EM waves?
• ☐Can you find $B_0$ from $E_0$ and the direction of B from the direction of travel?
• ☐Can you read λ, ν and v from a wave equation?
• ☐Can you arrange the spectrum by wavelength and frequency?
• ☐Can you name the source and two uses of each type of wave?
• ☐Can you identify a wave from its wavelength range?

Electromagnetic Waves: frequently asked questions