Chapter 8: Electromagnetic Waves
Q1. Arrange the following in order of increasing wavelength: X-rays, infrared rays, microwaves, UV rays, visible light, radio waves, gamma rays.
Solution
Increasing wavelength:
Gamma rays < X-rays < UV rays < Visible light < Infrared rays < Microwaves < Radio waves
Q2. Name the electromagnetic waves used (i) in radar (ii) for cooking (iii) in radiotherapy (iv) for sterilizing surgical instruments.
Solution
(i) Radar: Microwaves
(ii) Cooking: Microwaves (2.45 GHz)
(iii) Radiotherapy: Gamma rays
(iv) Sterilizing: UV rays
Q3. Prove that electromagnetic waves are transverse in nature.
Solution
Maxwell's equations show that E and B fields are always perpendicular to the direction of propagation. Since the fields oscillate perpendicular to the direction of travel (and to each other), EM waves are transverse in nature.
Q4. The magnetic field in a plane electromagnetic wave is given by By = 2 × 10−7 sin(0.5 × 103x + 1.5 × 1011t) T. Find (i) wavelength (ii) frequency.
Solution
Comparing with B = B₀ sin(kx + ωt):
k = 0.5 × 103 rad/m, ω = 1.5 × 1011 rad/s
(i) λ = 2π/k = 2π/(500) = 1.26 × 10−2 m = 1.26 cm
(ii) f = ω/(2π) = (1.5 × 1011)/(2π) = 2.39 × 1010 Hz = 23.9 GHz
Q5. What is the frequency of electromagnetic waves produced by an oscillating charge of frequency 10 MHz?
Solution
The frequency of the EM waves is equal to the frequency of the oscillating charge.
f = 10 MHz
Q6. Write down Maxwell's equations in integral form and explain the physical significance of each.
Solution
1. Gauss's law for E: ∮ E · dA = q/epsilon;₀ → Electric charges produce electric fields.
2. Gauss's law for B: ∮ B · dA = 0 → No magnetic monopoles exist.
3. Faraday's law: ∮ E · dl = −dΦ₂/dt → Changing magnetic flux induces EMF.
4. Ampere-Maxwell law: ∮ B · dl = μ₀(I + ε₀dΦ₁/dt) → Current and changing electric flux produce magnetic fields.
Q7. What is displacement current? How did Maxwell modify Ampere's circuital law?
Solution
Displacement current: Id = ε₀(dΦ₁/dt)
Maxwell added the displacement current term to Ampere's law to make it consistent for charging capacitors.
Modified Ampere's law:
∮ B · dl = μ₀(Iconduction + Idisplacement)
= μ₀(I + ε₀dΦ₁/dt)
Q8. Name the part of the EM spectrum used for (i) treatment of muscle strain (ii) satellite communication (iii) purification of water (iv) studying crystal structure.
Solution
(i) Treatment of muscle strain: Infrared rays (heat therapy)
(ii) Satellite communication: Microwaves
(iii) Purification of water: UV rays (kills bacteria)
(iv) Studying crystal structure: X-rays
Q9. An EM wave has a wavelength of 0.5 mm. What type of EM wave is it? Find its frequency and the time period of the wave.
Solution
Wavelength 0.5 mm = 5 × 10−4 m → This is in the microwave region.
f = c/λ = (3 × 108)/(5 × 10−4) = 6 × 1011 Hz = 600 GHz
T = 1/f = 1.67 × 10−12 s = 1.67 ps
Q10. The electric field in an EM wave is given by E = 100 sin(6π × 108 t) V/m. Find (i) the frequency (ii) the wavelength (iii) the magnetic field amplitude.
Solution
ω = 6π × 108 rad/s
(i) f = ω/(2π) = 3 × 108 Hz = 300 MHz
(ii) λ = c/f = (3 × 108)/(3 × 108) = 1 m
(iii) B₀ = E₀/c = 100/(3 × 108) = 3.33 × 10−7 T
Q11. How are electromagnetic waves produced? Explain why an accelerated charged particle radiates EM waves.
Solution
EM waves are produced by accelerated charged particles.
When a charged particle accelerates, its electric field changes. This changing electric field produces a changing magnetic field, which in turn produces a changing electric field. This self-sustaining oscillation propagates as an EM wave.
A stationary charge produces only a static E-field. A charge moving at constant velocity produces static E and B fields. Only acceleration causes radiation.
Q12. Radio waves of wavelength 300 m are used for communication. Find the frequency. What type of EM waves are these?
Solution
f = c/λ = (3 × 108)/300 = 106 Hz = 1 MHz
These are radio waves in the medium wave (MW) band.
Q13. Show that the energy density of an EM wave is equally divided between the electric and magnetic field energy densities.
Solution
Electric energy density: uE = ½ε₀E²
Magnetic energy density: uB = B²/(2μ₀)
For EM waves: E = cB, c = 1/√(μ₀ε₀)
uB = E²/(2μ₀c²) = E²/(2μ₀ × 1/(μ₀ε₀))
= ½ε₀E² = uE
Therefore, uE = uB: energy is equally divided.