Electromagnetic Waves Notes
Electromagnetic Waves Notes • CBSE • NEET • IIT JEE

Electromagnetic Waves Notes

EM wave theory, wave equation, direction of propagation, vector method, formula sheet, solved questions, PYQs and practice. Common content is kept once, and unique questions are grouped exam-wise.

Section 1: Introduction

Electromagnetic waves are transverse waves consisting of oscillating electric field and magnetic field vectors which are mutually perpendicular and also perpendicular to the direction of propagation.

E ⟂ B
Electric field and magnetic field are always perpendicular in a plane EM wave.
E ⟂ Propagation
The electric field does not oscillate along the direction in which the wave travels.
B ⟂ Propagation
The magnetic field is also transverse and perpendicular to the wave direction.
Propagation +x Electric field E along y-axis Magnetic field B along z-axis Correct rule: E × B = propagation direction. Here y × z = x.

Propagation is along +x, electric field along y-axis and magnetic field along z-axis.

Nature of Electromagnetic Waves

Electromagnetic waves are self-sustaining waves made of time-varying electric and magnetic fields. They can travel through vacuum because they do not need particles of a material medium for propagation.

Produced by accelerated charges
Whenever charge accelerates, it creates changing electric and magnetic fields. These fields detach from the source and travel as an electromagnetic wave.
Travel in vacuum
Unlike sound waves, EM waves do not need air, water or any solid medium. Light from the Sun reaches Earth through space because EM waves can travel in vacuum.
Carry energy and momentum
Electromagnetic waves transport energy through space. In advanced treatment, this energy flow is represented by the Poynting vector, directed along E × B.

Speed from Maxwell's Theory

c = 1 / √(μ0ε0)

Using the free-space constants μ0 and ε0, Maxwell's equations predict the speed of electromagnetic waves.

Speed in Vacuum

c = 3 × 108 m s-1

Radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays all travel with this same speed in vacuum.

Section 2: Derivation of Wave Equation

Start with the general harmonic equation:

y = A sin(ωt + φ)

For a progressive wave, phase difference at position x is:

φ = (2π / λ)x

Substitute phase in the wave equation:

y = A sin(ωt + 2πx / λ)

Use Angular Frequency

ω = 2π / T v = λ / T

Convert Time Term

ωt = (2π / T)t = (2π / λ)vt

Therefore, the two common travelling wave forms are:

Wave Along -x

y = A sin[(2π / λ)(vt + x)]

For constant phase, vt + x = constant. As t increases, x decreases. Hence wave moves along -x.

Wave Along +x

y = A sin[(2π / λ)(vt - x)]

For constant phase, vt - x = constant. As t increases, x increases. Hence wave moves along +x.

Section 3: Electromagnetic Wave Equations

Case 1: Propagation Along +x

Electric field is along y-axis and magnetic field is along z-axis.

Ey = E0 sin[(2π / λ)(vt - x)] Bz = B0 sin[(2π / λ)(vt - x)] Ex = Ez = 0, Bx = By = 0 c = E0 / B0

Case 2: Propagation Along +y

If electric field is along x-axis, magnetic field must be along -z-axis because x × (-z) = +y.

Ex = E0 sin[(2π / λ)(vt - y)] Bz = -B0 sin[(2π / λ)(vt - y)] E × B = Direction of propagation
E B E × B Use right-hand rule: fingers from E to B, thumb gives propagation.

Section 4: Important Formulas

Field Ratioc = E0 / B0
Wave Speedc = νλ
Wave Numberk = 2π / λ
Angular Frequencyω = 2πν
+x Wavey = A sin(ωt - kx)
-x Wavey = A sin(ωt + kx)
DirectionE × B = Propagation
Magnetic AmplitudeB0 = E0 / c

Section 5: Solved Questions

All questions below include a Show Solution toggle, formula, direction logic and final answer.

Q1. By = 3×10-7 sin[(1.5)x + (5×108)t]. Find wavelength, frequency and electric field equation.
  1. Compare with B = B0 sin(kx + ωt). Thus k = 1.5 rad m-1, ω = 5×108 rad s-1.
  2. Wavelength: λ = 2π/k = 2π/1.5 = 4.19 m.
  3. Frequency: f = ω/2π = (5×108)/2π = 7.96×107 Hz.
  4. The plus sign kx + ωt means propagation along -x.
  5. B is along +y. For propagation -x, E × B = -x, so E is along +z.
  6. E0 = cB0 = (3×108)(3×10-7) = 90 V m-1.
Final Answer: λ = 4.19 m, f = 7.96×107 Hz, Ez = 90 sin(1.5x + 5×108t) V m-1
Q2. By = 3.01×10-7 sin(6.28×102x + 2.2×1011t). Find wavelength.
  1. k = 6.28×102 rad m-1.
  2. λ = 2π/k = 6.28/(6.28×102) = 10-2 m.
Final Answer: λ = 0.01 m = 1 cm
Q3. Ey = E0 cos(107t + kx). Find wave number, wavelength, amplitude and direction.
  1. Wave number is the coefficient of x, so it is k rad m-1.
  2. Wavelength: λ = 2π/k.
  3. Amplitude of electric field is E0.
  4. The form ωt + kx represents propagation along -x.
Final Answer: wave number = k, λ = 2π/k, amplitude = E0, direction = -x
Q4. Ey = 2.5×10-5 cos(2π×106t - π×10-2x). Find frequency, wavelength and direction.
  1. Compare with E = E0 cos(ωt - kx).
  2. ω = 2π×106, so f = ω/2π = 106 Hz.
  3. k = π×10-2 rad m-1.
  4. λ = 2π/k = 2π/(π×10-2) = 200 m.
  5. The form ωt - kx represents +x direction.
Final Answer: f = 1.0×106 Hz, λ = 200 m, direction = +x
Q5. Plane EM wave propagates in +x direction. λ = 6.0 mm. Electric field along y, amplitude 33 V/m. Write E and B equations.
  1. λ = 6.0 mm = 6×10-3 m.
  2. k = 2π/λ = 2π/(6×10-3) = 1.047×103 rad m-1.
  3. f = c/λ = (3×108)/(6×10-3) = 5×1010 Hz.
  4. ω = 2πf = 3.14×1011 rad s-1.
  5. B0 = E0/c = 33/(3×108) = 1.1×10-7 T.
  6. For +x propagation with E along +y, B is along +z.
Final Answer: Ey = 33 sin(3.14×1011t - 1.047×103x); Bz = 1.1×10-7 sin(3.14×1011t - 1.047×103x)
Q6. Wave travels along +x direction. E = 9.3 ĵ. Find magnetic field.
  1. E is along +y and propagation is +x.
  2. Need E × B = +x. Since j × k = i, B is along +z.
  3. B = E/c = 9.3/(3×108) = 3.1×10-8 T.
Final Answer: B = 3.1×10-8 k̂ T
Q7. Wave travels along +y direction. Electric field is along +x direction. Find magnetic field equation.
  1. Need E × B = +y.
  2. E is +x. Since i × (-k) = +j, B is along -z.
  3. Generic magnetic field equation uses the same phase as E.
Final Answer: Bz = -B0 sin[(2π/λ)(vt - y)], where B0 = E0/c
Q8. Ex = E0 sin(kz - ωt). Find magnetic field.
  1. The form kz - ωt represents propagation along +z.
  2. E is along +x.
  3. Need E × B = +z. Since i × j = k, B is along +y.
  4. B0 = E0/c.
Final Answer: By = B0 sin(kz - ωt), B0 = E0/c
Q9. Ey = E0 sin(kx - ωt), Bz = B0 sin(kx - ωt). Find relation among E0, B0, k and ω.
  1. The wave speed is v = ω/k.
  2. For an EM wave in vacuum, v = c.
  3. Also c = E0/B0.
Final Answer: E0/B0 = ω/k = c
Q10. Ey = E0 sin(ky - ωt), Bz = B0 sin(ky - ωt). Find relation between E0 and B0.
Important physical check: The phase ky - ωt shows propagation along +y, but Ey is also along y. In a plane EM wave, E must be perpendicular to propagation. Therefore the written directions are physically inconsistent.
  1. If the field direction is corrected so that E is transverse, then the amplitude relation remains E0/B0 = c.
  2. Hence B0 = E0/c.
Final Answer: For a valid EM wave, E0/B0 = c
Q11. Erms = 18 V/m. Find peak magnetic field.
  1. Peak electric field: E0 = √2 Erms = 1.414×18 = 25.46 V/m.
  2. Peak magnetic field: B0 = E0/c.
  3. B0 = 25.46/(3×108) = 8.49×10-8 T.
Final Answer: B0 = 8.49×10-8 T
Q12. E = 103 V/m. Find magnetic field.
  1. Use B = E/c.
  2. B = 103/(3×108) = 3.33×10-6 T.
Final Answer: B = 3.33×10-6 T
Q13. E = 60 cos(1.2x - 3.6×108t). Find magnetic field equation.
  1. The form kx - ωt represents +x propagation.
  2. Assume electric field is along y, so magnetic field is along z.
  3. B0 = E0/c = 60/(3×108) = 2×10-7 T.
  4. Magnetic field has the same phase as electric field.
Final Answer: Bz = 2×10-7 cos(1.2x - 3.6×108t) T
Q14. Bz = 5×10-7 sin(t - x/c). Find maximum electric field.
  1. Magnetic field amplitude B0 = 5×10-7 T.
  2. Use E0 = cB0.
  3. E0 = (3×108)(5×10-7) = 150 V/m.
Final Answer: E0 = 150 V/m
Q15. By = 2×10-7 sin(0.5×103x + 1.5×1011t). Identify EM radiation.
  1. ω = 1.5×1011 rad/s.
  2. f = ω/2π = (1.5×1011)/6.28 = 2.39×1010 Hz.
  3. This frequency is about 23.9 GHz.
  4. GHz range belongs to microwave radiation.
Final Answer: Microwave radiation
Q16. Frequency = 28 MHz. Find wavelength.
  1. f = 28 MHz = 28×106 Hz.
  2. λ = c/f = (3×108)/(28×106) = 10.71 m.
Final Answer: λ = 10.71 m
Q17. By = 3.01×10-7 sin(6.28×102x + 2.2×1011t). Find wavelength.
  1. k = 6.28×102 rad m-1.
  2. λ = 2π/k = 6.28/(6.28×102) = 0.01 m.
Final Answer: λ = 1.0×10-2 m
Q18. y = A sin(ωt - kx). Find direction of propagation.
  1. Write phase as ωt - kx = constant.
  2. As t increases, x must increase to keep phase constant.
  3. Therefore the wave moves along +x.
Final Answer: +x direction
Q19. y = A sin(ωt + kx). Find direction of propagation.
  1. Write phase as ωt + kx = constant.
  2. As t increases, x must decrease to keep phase constant.
  3. Therefore the wave moves along -x.
Final Answer: -x direction
Q20. Ey = E0 sin[(2π/λ)(vt - x)]. Find direction of propagation, electric field and magnetic field.
  1. The term vt - x means propagation along +x.
  2. Ey means electric field is along y-axis.
  3. For +x propagation, E × B = +x. Since y × z = x, magnetic field is along z-axis.
Final Answer: propagation +x, E along y, B along z
Q21. Ex = E0 sin[(2π/λ)(vt - z)], By = B0 sin[(2π/λ)(vt - z)]. Find direction of propagation.
  1. The term vt - z means propagation along +z.
  2. Also E × B = x × y = z, confirming +z direction.
Final Answer: +z direction

Additional Solved Exam Drills

These are kept as extra practice from the broad EM waves notes. They avoid repeating the same Q1-Q21 style, but reinforce wavelength, frequency, field amplitude and vector direction.

Drill 1. An EM wave has frequency 6.0 × 1014 Hz. Find its wavelength in vacuum.
  1. Use λ = c/f.
  2. λ = (3.0 × 108) / (6.0 × 1014) m.
  3. λ = 5.0 × 10-7 m.
Final Answer: λ = 5.0 × 10-7 m
Drill 2. Magnetic field amplitude is 2.0 × 10-6 T. Find electric field amplitude.
  1. Use E0 = cB0.
  2. E0 = (3.0 × 108)(2.0 × 10-6) V/m.
  3. E0 = 600 V/m.
Final Answer: E0 = 600 V/m
Drill 3. If E is along +Y and B is along +Z, find direction of propagation.
  1. Direction = E × B.
  2. +Y × +Z = +X.
Final Answer: Propagation is along +X
Drill 4. A wave has λ = 0.6 m. Calculate wave number k.
  1. Use k = 2π/λ.
  2. k = 2π/0.6 = 10.47 rad m-1.
Final Answer: k = 10.47 rad m-1
Drill 5. A wave frequency is 5 × 1014 Hz. Find angular frequency.
  1. Use ω = 2πf.
  2. ω = 2π(5 × 1014) = 3.14 × 1015 rad s-1.
Final Answer: ω = 3.14 × 1015 rad s-1
Drill 6. If wave travels along +Z and E is along +X, find B direction.
  1. Need E × B = +Z.
  2. +X × +Y = +Z.
Final Answer: B is along +Y

PYQ-Style Practice and Revision

Common points from both pages are merged here: direction rule, amplitude relation and wave equation reading.

Most asked rule:
For a valid plane EM wave, E, B and propagation direction must be mutually perpendicular.
Most asked formula:
E0/B0 = c and c = νλ.
Most asked direction test:
Use E × B, not B × E.

CBSE Level

CBSE 1. State two characteristics of electromagnetic waves.

They are transverse waves and do not require a material medium. Their electric and magnetic fields are mutually perpendicular and perpendicular to propagation.

Answer: Transverse and can travel in vacuum.
CBSE 2. Write relation between E0 and B0.

For an EM wave in vacuum, c = E0/B0. Therefore E0 = cB0.

Answer: E0/B0 = c
CBSE 3. Why can EM waves travel in vacuum?

They consist of self-sustaining changing electric and magnetic fields, so they do not need particles of a medium.

Answer: No material medium is required.

NEET Level

NEET 1. If E is +X and B is -Z, direction is?

Direction = E × B = i × (-k). Since i × k = -j, i × (-k) = +j.

Answer: +Y
NEET 2. B0 = 10-8 T. Find E0.

E0 = cB0 = (3 × 108)(10-8) = 3 V/m.

Answer: 3 V/m
NEET 3. Are E and B in phase in a plane EM wave?

Yes. Maxima and minima of E and B occur at the same position and time.

Answer: Yes, E and B are in phase.

JEE Main Level

JEE Main 1. For λ = 500 nm, find frequency.

f = c/λ = (3 × 108)/(500 × 10-9) = 6 × 1014 Hz.

Answer: 6 × 1014 Hz
JEE Main 2. If k = 4π rad/m, find λ.

λ = 2π/k = 2π/(4π) = 0.5 m.

Answer: 0.5 m
JEE Main 3. If wave travels +Y and B is +X, find E.

Need E × +X = +Y. Since +Z × +X = +Y, E is along +Z.

Answer: E is along +Z

JEE Advanced Level

JEE Advanced 1. E = E0 sin(ωt + ky). If E is +Z, find B direction.

The term ωt + ky means propagation along -Y. Need +Z × B = -Y. Since k × (-i) = -j, B is -X.

Answer: B is along -X
JEE Advanced 2. If E = +Y and propagation is -X, find B.

Need +Y × B = -X. Since j × (-k) = -i, B is along -Z.

Answer: B is along -Z
JEE Advanced 3. Why can E and B not be parallel in a travelling plane EM wave?

The energy flow and propagation direction are along E × B. If E and B are parallel, E × B = 0, so no transverse propagation direction is formed.

Answer: E and B must be perpendicular for a valid plane EM wave.

Section 6: Practice Set

15 additional NEET/JEE level questions for quick practice.

1. If E is +x and B is +y, find propagation direction.
2. If propagation is +z and E is +y, find B direction.
3. y = A sin(5t - 2x). Find direction of propagation.
4. y = A sin(7t + 3z). Find direction of propagation.
5. E0 = 120 V/m. Find B0.
6. B0 = 4×10-8 T. Find E0.
7. λ = 400 nm. Find frequency.
8. f = 100 MHz. Find wavelength.
9. k = 10π rad/m. Find wavelength.
10. ω = 6.28×109 rad/s. Find frequency.
11. If wave travels -x and E is +z, find B.
12. If wave travels +y and B is +z, find E.
13. Check whether Ex with propagation +x is a valid plane EM wave.
14. Write E and B equations for +z propagation with E along +x.
15. If Erms = 30 V/m, find B0.

Common Mistakes to Avoid

1. Confusing wave direction from sign
For the form y = A sin(ωt - kx), the wave travels along +x. For y = A sin(ωt + kx), it travels along -x.
2. Using B × E instead of E × B
Propagation direction is E × B. Reversing the order gives the opposite direction.
3. Making E parallel to propagation
In a plane EM wave, E must be perpendicular to the direction of propagation. If E is along x, the wave cannot propagate along x.
4. Forgetting c = E0/B0
Use the amplitude relation carefully. If RMS electric field is given, first convert to peak using E0 = √2 Erms.
5. Mixing wavelength and wave number
Wave number k is not wavelength. Always use k = 2π/λ and λ = 2π/k.
6. Ignoring physical consistency
If a question gives E, B and propagation directions, first check that all three are mutually perpendicular and follow E × B.

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