Chapter 5 — Exercise Solutions
Electromagnetic Waves is a compact scoring chapter. Focus on spectrum order, uses, and basic properties.
How to Use This Page
- Read the Key Concepts table first for a quick recap of definitions you must know.
- Attempt MCQs, Fill in the Blanks, and True or False on your own before checking answers.
- Use the Match the Following section to reinforce spectrum order and wave-uses pairings.
- Read Worked Examples carefully and re-calculate each step yourself.
- Before the exam, run through the Exam Checklist to confirm you have not missed anything.
Key Concepts (Quick Recap)
| Term | Definition |
|---|
| Electromagnetic waves | Waves consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation |
| Displacement current | Current produced by a time-varying electric field, introduced by Maxwell to complete Ampere's law |
| Conduction current | Current due to flow of free electrons in a conductor |
| Maxwell's equations | Set of four fundamental equations describing electric and magnetic fields and their sources |
| EM spectrum | Ordered arrangement of electromagnetic waves based on increasing frequency or decreasing wavelength |
| Transverse nature | Electric and magnetic field vectors are perpendicular to the direction of wave propagation |
| Speed of EM waves | c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s in vacuum |
| Electromagnetic induction | Production of EMF in a coil when magnetic flux linked with it changes |
| Poynting vector | Vector S = (1/μ₀)(E × B) representing the direction and magnitude of energy flow in an EM wave |
| Radio waves | EM waves with longest wavelengths (> 0.1 m), used in broadcasting and communication |
| Microwaves | EM waves with wavelengths from 0.1 mm to 0.1 m, used in radar and cooking |
| Infrared | EM waves with wavelengths from 700 nm to 0.1 mm, used in remote controls and thermal imaging |
| Ultraviolet | EM waves with wavelengths from 400 nm down to 1 nm, used in sterilisation |
| X-rays | EM waves with wavelengths from 0.01 nm to 10 nm, used in medical imaging |
| Gamma rays | EM waves with shortest wavelengths (< 0.01 nm), produced in nuclear reactions, used in cancer treatment |
Key Points
- Electromagnetic waves do not need a material medium.
- Electric field and magnetic field are perpendicular to each other.
- Both fields are perpendicular to direction of propagation.
- EM waves travel in vacuum with speed
c = 3 x 10^8 m/s.
Spectrum Order
Radio waves -> Microwaves -> Infrared -> Visible -> Ultraviolet -> X-rays -> Gamma rays
Uses Table
| Wave | Use |
|---|
| Radio waves | Broadcasting |
| Microwaves | Radar, microwave oven |
| Infrared | Remote control, thermal imaging |
| Visible light | Vision |
| Ultraviolet | Sterilisation |
| X-rays | Medical imaging |
| Gamma rays | Cancer treatment |
I. Choose the Best Answer (MCQs)
1. The current introduced by Maxwell to make Ampere's law consistent is called:
Answer
(a) Conduction current   (b) Displacement current   (c) Eddy current   (d) Drift current
Maxwell added displacement current to account for the current between capacitor plates where no conduction current flows.
2. In an electromagnetic wave, the electric and magnetic fields are:
Answer
(a) Parallel to each other   (b) Perpendicular to each other and to the direction of propagation   (c) Perpendicular to each other but parallel to direction of propagation   (d) Randomly oriented
EM waves are transverse; both field vectors are mutually perpendicular and also perpendicular to the direction of wave travel.
3. Which of the following EM waves has the shortest wavelength?
Answer
(a) X-rays   (b) Ultraviolet   (c) Infrared   (d) Gamma rays
Gamma rays have the highest frequency and shortest wavelength in the EM spectrum.
4. The speed of electromagnetic waves in vacuum depends on:
Answer
(a) Amplitude of the wave   (b) Frequency of the wave   (c) Permeability and permittivity of free space   (d) Intensity of the wave
The speed is c = 1/√(μ₀ε₀) and depends only on these two constants of free space.
5. Microwaves are used in:
Answer
(a) Sterilisation   (b) Radar and cooking   (c) Medical imaging   (d) Cancer treatment
Microwaves have wavelengths suitable for radar detection and for exciting water molecules in food.
6. Displacement current flows in the gap between capacitor plates when:
Answer
(a) A steady DC voltage is applied   (b) A time-varying voltage is applied   (c) No voltage is applied   (d) A constant magnetic field is present
Displacement current arises from a changing electric field, so it requires a time-varying voltage.
7. The correct order of EM spectrum in increasing frequency is:
Answer
(a) Gamma rays, X-rays, UV, Visible, IR, Microwaves, Radio waves   (b) Radio waves, Microwaves, IR, Visible, UV, X-rays, Gamma rays   (c) Radio waves, IR, Microwaves, Visible, UV, X-rays, Gamma rays   (d) Microwaves, Radio waves, IR, Visible, UV, X-rays, Gamma rays
Frequency increases from radio waves (lowest) to gamma rays (highest).
8. Which EM wave is used in remote controls?
Answer
(a) Ultraviolet   (b) Infrared   (c) Microwave   (d) X-ray
Infrared radiation is emitted by TV remote control LEDs to send signals.
9. The Poynting vector gives:
Answer
(a) Energy density of EM wave   (b) Direction and magnitude of energy flow   (c) Electric field direction only   (d) Magnetic field direction only
The Poynting vector S = (1/μ₀)(E × B) gives the directional energy flux of an EM wave.
10. If the frequency of an EM wave is doubled, its wavelength in vacuum:
Answer
(a) Doubles   (b) Remains the same   (c) Becomes half   (d) Becomes four times
Since c = fλ and c is constant, if f doubles, λ must halve.
II. Fill in the Blanks
1. Electromagnetic waves are produced by ______ charged particles. — accelerated 2. The speed of electromagnetic waves in vacuum is approximately ______ m/s. — 3 × 10⁸ 3. The current that exists in the gap of a capacitor during charging is called ______ current. — displacement 4. In the EM spectrum, ______ rays have the highest frequency. — gamma 5. Electromagnetic waves are ______ in nature (transverse/longitudinal). — transverse 6. The relation between speed, frequency, and wavelength of an EM wave is given by ______. — c = fλ 7. ______ waves are used for sterilisation of surgical instruments. — Ultraviolet 8. The Poynting vector is given by the expression ______. — S = (1/μ₀)(E × B)
III. True or False (with Corrections)
1. Electromagnetic waves require a material medium for propagation. — False. Electromagnetic waves do not require a material medium; they can propagate through vacuum.
2. The electric and magnetic fields in an EM wave are parallel to each other. — False. The electric and magnetic fields are perpendicular to each other and to the direction of propagation.
3. Displacement current has the same physical effects as conduction current. — True. Displacement current produces a magnetic field just like conduction current.
4. Gamma rays have a longer wavelength than X-rays. — False. Gamma rays have a shorter wavelength and higher frequency than X-rays.
5. All electromagnetic waves travel at the same speed in vacuum. — True. All EM waves travel at c ≈ 3 × 10⁸ m/s in vacuum regardless of frequency.
6. Infrared radiation is used in sterilisation. — False. Ultraviolet radiation is used in sterilisation; infrared is used in remote controls and thermal imaging.
7. The Poynting vector points in the direction of wave propagation. — True. S = (1/μ₀)(E × B) gives the direction of energy flow, which is the direction of propagation.
IV. Match the Following
| Column A | Column B |
|---|
| 1. Radio waves | (a) Cancer treatment |
| 2. Microwaves | (b) Broadcasting |
| 3. Infrared | (c) Radar |
| 4. Ultraviolet | (d) Remote control |
| 5. X-rays | (e) Sterilisation |
| 6. Gamma rays | (f) Medical imaging |
Answer1-b, 2-c, 3-d, 4-e, 5-f, 6-a
V. Answer Briefly (2-mark)
1. What is displacement current?
Answer
Displacement current is the current that flows in the gap between the plates of a capacitor when a time-varying voltage is applied. It was introduced by Maxwell to make Ampere's circuital law consistent. It is given by I_d = ε₀(dΦ_E/dt) and produces a magnetic field just like conduction current.
2. State two properties of electromagnetic waves.
Answer
Electromagnetic waves do not require a material medium for propagation. They are transverse in nature — the electric and magnetic field vectors are perpendicular to each other and to the direction of wave propagation. They travel in vacuum at a speed c = 3 × 10⁸ m/s.
3. Why are electromagnetic waves transverse in nature?
Answer
In an EM wave, the electric field oscillates in one plane and the magnetic field oscillates in a perpendicular plane. Both field vectors are perpendicular to the direction of propagation, which is the defining characteristic of transverse waves. This can be demonstrated experimentally using polaroid films.
4. Write the expression for the speed of EM waves in vacuum.
Answer
The speed of EM waves in vacuum is given by c = 1/√(μ₀ε₀), where μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A) and ε₀ is the permittivity of free space (8.85 × 10⁻¹² C²/N·m²). Substituting these values yields c ≈ 3 × 10⁸ m/s.
5. What is the Poynting vector? Give its expression.
Answer
The Poynting vector represents the directional energy flux (power per unit area) of an electromagnetic wave. It gives both the magnitude and direction of energy flow. Its expression is S = (1/μ₀)(E × B), where E is the electric field, B is the magnetic field, and μ₀ is the permeability of free space. Its SI unit is W/m².
VI. Answer in Detail (5-mark)
1. Describe the production and propagation of electromagnetic waves.
Answer
Electromagnetic waves are produced by accelerated charged particles. When a charged particle accelerates, it produces a time-varying electric field. This changing electric field generates a time-varying magnetic field, and the changing magnetic field in turn regenerates the electric field. This mutual regeneration allows the disturbance to propagate through space as an electromagnetic wave. No material medium is required for this propagation.
In the wave, the electric field E and magnetic field B oscillate perpendicular to each other and both are perpendicular to the direction of propagation. The relationship between the magnitudes is E₀/B₀ = c, where c is the speed of the wave. The energy carried by the wave is described by the Poynting vector S = (1/μ₀)(E × B). EM waves carry energy and momentum and can exert radiation pressure on surfaces they strike.
2. Explain displacement current and show how it modifies Ampere's circuital law.
Answer
When a capacitor is being charged, conduction current flows through the connecting wires but not through the gap between the plates. Maxwell observed that this creates an inconsistency in Ampere's circuital law. To resolve this, he introduced the concept of displacement current, which exists in the gap between the capacitor plates where the electric flux is changing with time.
Displacement current is defined as I_d = ε₀(dΦ_E/dt), where Φ_E is the electric flux. It has the same dimensions as conduction current and produces the same magnetic effects. The modified Ampere's law becomes ∮ B · dl = μ₀(I_c + I_d) = μ₀(I_c + ε₀(dΦ_E/dt)), where I_c is conduction current and I_d is displacement current. This modification ensures continuity of current and makes the set of Maxwell's equations self-consistent.
3. Describe the electromagnetic spectrum. List the different types of EM waves, their sources, and key properties.
Answer
The electromagnetic spectrum is the ordered arrangement of all electromagnetic waves based on increasing frequency (or decreasing wavelength). Starting from the lowest frequency: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. All these waves travel at the same speed c = 3 × 10⁸ m/s in vacuum but differ in frequency and wavelength.
Radio waves are produced by accelerated charged particles in antennas and have wavelengths greater than 0.1 m. Microwaves (0.1 mm to 0.1 m) are produced by vacuum tubes. Infrared (700 nm to 0.1 mm) is produced by hot bodies. Visible light (400 nm to 700 nm) is produced by excited atoms. Ultraviolet (1 nm to 400 nm) is produced by very hot objects. X-rays (0.01 nm to 10 nm) are produced by inner electron transitions. Gamma rays (< 0.01 nm) are produced in nuclear reactions. As frequency increases, the penetrating power and energy of the radiation increase.
4. Discuss the uses and hazards of different electromagnetic waves.
Answer
Radio waves are used in radio and television broadcasting, mobile communication, and satellite transmission. They have no known harmful effects at normal exposure levels. Microwaves are used in radar systems, microwave ovens, and satellite communication. Prolonged microwave exposure can cause internal heating of body tissues.
Infrared is used in remote controls, thermal imaging cameras, and physiotherapy treatments. Excessive infrared can cause skin burns. Visible light enables vision and is used in fibre optic communication. Intense visible light can damage the retina. Ultraviolet is used in sterilisation of water and surgical instruments, and in security marking. UV exposure causes skin tanning, and prolonged exposure leads to skin cancer and cataracts.
X-rays are used in medical imaging (fractures, dental), security scanning, and cancer radiotherapy. X-ray overexposure damages living cells and can cause mutations. Gamma rays are used in cancer treatment (radiotherapy), sterilisation of medical equipment, and food preservation. Gamma radiation is highly penetrating and causes severe damage to living tissue at high doses. Proper shielding is essential when working with X-rays and gamma rays.
Short Answer Pattern
For spectrum questions:
1. Name the wave. 2. Write wavelength/frequency nature. 3. Write one or two uses. 4. Mention risk if asked.
Worked Examples
Example 1 — Wavelength from frequency
Find the wavelength of an FM radio wave of frequency 100 MHz.
λ = c / f = (3×10⁸) / (100×10⁶) = 3×10⁸ / 10⁸ = 3 m
Example 2 — Frequency from wavelength
Find the frequency of green light of wavelength 500 nm.
f = c / λ = (3×10⁸) / (500×10⁻⁹) = 3×10⁸ / 5×10⁻⁷ = 6×10¹⁴ Hz
Example 3 — Speed of EM waves
Verify c using μ₀ = 4π×10⁻⁷ and ε₀ = 8.85×10⁻¹².
c = 1 / √(μ₀ε₀) = 1 / √(4π×10⁻⁷ × 8.85×10⁻¹²)
≈ 1 / √(1.11×10⁻¹⁷) ≈ 3×10⁸ m/s
Common Mistakes to Avoid
- ❌ Writing spectrum order as "gamma rays first" → ✅ Spectrum order in increasing frequency: Radio → Microwave → IR → Visible → UV → X-ray → Gamma
- ❌ Saying EM waves need a material medium → ✅ EM waves can propagate through vacuum; they do not need a medium
- ❌ Confusing displacement current with conduction current → ✅ Displacement current arises from a changing electric field; conduction current is due to flow of free charges
- ❌ Confusing X-rays and gamma rays → ✅ X-rays are produced by inner electron transitions; gamma rays come from nuclear processes
- ❌ Writing
c = fλ and then using inconsistent units → ✅ Always convert frequency to Hz and wavelength to metres before applying c = fλ - ❌ Stating that infrared is used for sterilisation → ✅ Ultraviolet is used for sterilisation; infrared is used in remote controls and thermal imaging
Exam Checklist
- Spectrum order (increasing frequency): Radio → Microwave → IR → Visible → UV → X-ray → Gamma
- Each EM wave type has at least two specific uses memorised
- Speed of EM waves in vacuum:
c = 3 × 10⁸ m/s - Speed formula
c = 1/√(μ₀ε₀) is remembered and can be verified numerically - Displacement current definition and formula
I_d = ε₀(dΦ_E/dt) - Modified Ampere's law includes both conduction and displacement current
- Transverse nature: E ⊥ B ⊥ direction of propagation
- Poynting vector:
S = (1/μ₀)(E × B) — direction and meaning - Unit conversions for MHz, GHz, nm, and Ångströms are practiced
- Hazards of UV, X-rays, and gamma rays can be stated for exam questions
- Displacement current formula
I_d = epsilon0 (dPhi_E / dt) is memorised for derivations - Can you write all four of Maxwell's equations and explain the physical significance of each?
- Can you distinguish between conduction current and displacement current with examples?
Additional Numerical Practice
Example 4 — Energy of a photon
Find the energy of an X-ray photon of wavelength 0.5 Ångströms.
lambda = 0.5 Å = 0.5 × 10⁻¹⁰ m = 5 × 10⁻¹¹ m
E = hc / lambda = (6.63 × 10⁻³⁴ × 3 × 10⁸) / (5 × 10⁻¹¹)
= 1.989 × 10⁻²⁵ / 5 × 10⁻¹¹
= 3.978 × 10⁻¹⁵ J
In eV: E = 3.978 × 10⁻¹⁵ / 1.6 × 10⁻¹⁹ ≈ 24,860 eV ≈ 24.9 keV
Example 5 — Displacement current
A parallel plate capacitor has plates of area 0.01 m² separated by 1 mm. If the electric field between the plates changes at a rate of 10¹¹ V/m/s, find the displacement current.
I_d = epsilon0 × A × (dE/dt)
= (8.85 × 10⁻¹²) × 0.01 × 10¹¹
= 8.85 × 10⁻¹² × 10⁹
= 8.85 × 10⁻³ A ≈ 8.85 mA
