class 12 notes

Class 12 Physics Notes Chapter 8 (Electromagnetic Waves) – Examplar Problems (English) Book

Philoid

Philoid

— 7 min read

Examplar Problems (English)
Alright class, let's delve into Chapter 8: Electromagnetic Waves. This is a relatively short but conceptually important chapter, bridging electricity, magnetism, and optics. It's frequently tested in competitive exams, so pay close attention to the details.

Electromagnetic Waves: Detailed Notes

1. Introduction & Maxwell's Displacement Current

  • Inconsistency in Ampere's Circuital Law: Ampere's Circuital Law ((\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed})) was found to be logically inconsistent when applied to situations involving time-varying electric fields, such as the charging or discharging of a capacitor.
    • Consider the space between the capacitor plates during charging. There's no conduction current ((I_c = 0)), yet a magnetic field exists around it, which contradicts the original Ampere's law if we consider a loop enclosing only the space between the plates.
  • Maxwell's Correction: James Clerk Maxwell proposed that a changing electric field (or electric flux) in space generates a magnetic field, just like a conduction current does. He introduced the concept of Displacement Current ((I_d)).
  • Displacement Current ((I_d)): It is the current that comes into play in a region wherever the electric field and hence the electric flux is changing with time.
    • Formula: (I_d = \epsilon_0 \frac{d\Phi_E}{dt})
      • Where (\epsilon_0) is the permittivity of free space and (\frac{d\Phi_E}{dt}) is the rate of change of electric flux.
    • For a parallel plate capacitor charging, (E = \frac{Q}{\epsilon_0 A}), so (\Phi_E = E A = \frac{Q}{\epsilon_0}).
    • Therefore, (I_d = \epsilon_0 \frac{d}{dt} \left( \frac{Q}{\epsilon_0} \right) = \frac{dQ}{dt} = I_c). This shows that the displacement current between the plates is exactly equal to the conduction current in the wires during charging/discharging, restoring continuity.
  • Ampere-Maxwell Law (Modified Ampere's Circuital Law):
    • (\oint \vec{B} \cdot d\vec{l} = \mu_0 (I_c + I_d) = \mu_0 \left( I_c + \epsilon_0 \frac{d\Phi_E}{dt} \right))
    • This law states that the line integral of the magnetic field around any closed loop is equal to (\mu_0) times the total current (conduction current + displacement current) passing through the surface enclosed by the loop.
    • Significance: A changing electric field is a source of magnetic field.

2. Maxwell's Equations (Conceptual Foundation of Electromagnetism)

Maxwell unified the laws of electricity and magnetism into four fundamental equations:

  1. Gauss's Law for Electricity: (\oint \vec{E} \cdot d\vec{S} = \frac{Q_{enclosed}}{\epsilon_0}) (Relates electric field to charge distribution; implies isolated charges exist).
  2. Gauss's Law for Magnetism: (\oint \vec{B} \cdot d\vec{S} = 0) (Relates magnetic field lines; implies magnetic monopoles do not exist, magnetic field lines are closed loops).
  3. Faraday's Law of Induction: (\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}) (A changing magnetic flux induces an electric field - EMF).
  4. Ampere-Maxwell Law: (\oint \vec{B} \cdot d\vec{l} = \mu_0 I_c + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}) (Conduction current and/or a changing electric flux induce a magnetic field).

3. Electromagnetic Waves: Prediction and Source

  • Prediction: Maxwell showed that these four equations collectively predict the existence of waves of oscillating electric and magnetic fields that propagate through space, even vacuum. These are Electromagnetic Waves (EM Waves).
  • Mechanism: A time-varying magnetic field produces a time-varying electric field (Faraday's Law). A time-varying electric field produces a time-varying magnetic field (Ampere-Maxwell Law). This self-sustaining process results in the propagation of EM waves.
  • Source: The fundamental cause of EM waves is an accelerated electric charge.
    • A stationary charge produces only an electric field.
    • A charge moving with uniform velocity produces both electric and magnetic fields (steady current), but not EM waves.
    • An oscillating or accelerated charge produces oscillating electric and magnetic fields, which radiate outwards as EM waves. Example: An oscillating charge in an LC circuit antenna.

4. Nature and Properties of EM Waves

  • Transverse Nature: The oscillating electric field ((\vec{E})) and magnetic field ((\vec{B})) vectors are perpendicular to each other and also perpendicular to the direction of propagation of the wave ((\vec{k}) or direction (\vec{v})).
    • If the wave propagates along the Z-axis, (\vec{E}) could be along the X-axis and (\vec{B}) along the Y-axis (or vice versa).
    • (\vec{E} \times \vec{B}) gives the direction of wave propagation.
  • Phase: The electric and magnetic fields oscillate in phase. They reach their maximum and minimum values at the same time and location.
  • Speed in Vacuum (c): Maxwell's equations predicted the speed of EM waves in vacuum to be:
    • (c = \frac{1}{\sqrt{\mu_0 \epsilon_0}})
    • Where (\mu_0 = 4\pi \times 10^{-7}) T m/A (permeability of free space) and (\epsilon_0 = 8.854 \times 10^{-12}) C²/N m² (permittivity of free space).
    • Calculating this value gives (c \approx 3 \times 10^8) m/s, which is the speed of light in vacuum. This confirmed that light is an electromagnetic wave.
  • Speed in a Material Medium (v):
    • (v = \frac{1}{\sqrt{\mu \epsilon}})
    • Where (\mu = \mu_0 \mu_r) (permeability of the medium) and (\epsilon = \epsilon_0 \epsilon_r) (permittivity of the medium). (\mu_r) is the relative permeability and (\epsilon_r) is the relative permittivity (dielectric constant, K).
    • (v = \frac{1}{\sqrt{\mu_0 \epsilon_0 \mu_r \epsilon_r}} = \frac{c}{\sqrt{\mu_r \epsilon_r}})
    • The refractive index (n) of the medium is given by (n = \sqrt{\mu_r \epsilon_r}).
    • So, (v = \frac{c}{n}).
  • Relation between E and B: The magnitudes of the electric and magnetic field vectors in an EM wave are related:
    • In vacuum: (E_0 = c B_0) or (E_{rms} = c B_{rms}) or simply (E = cB) at any instant.
    • In a medium: (E_0 = v B_0) or (E = vB).
  • Energy: EM waves carry energy which is shared equally between the electric and magnetic fields.
    • Energy density of electric field: (u_E = \frac{1}{2} \epsilon_0 E^2) (Instantaneous)
    • Energy density of magnetic field: (u_B = \frac{1}{2\mu_0} B^2) (Instantaneous)
    • In EM waves, (u_E = u_B).
    • Total instantaneous energy density: (u = u_E + u_B = \epsilon_0 E^2 = \frac{B^2}{\mu_0})
    • Average energy density: (u_{avg} = \frac{1}{2} \epsilon_0 E_0^2 = \frac{B_0^2}{2\mu_0} = \epsilon_0 E_{rms}^2 = \frac{B_{rms}^2}{\mu_0})
  • Intensity (I): Energy crossing per unit area per unit time perpendicular to the direction of propagation. It is the power per unit area.
    • (I = u_{avg} \times \text{speed} = u_{avg} \times c) (in vacuum)
    • (I = \frac{1}{2} \epsilon_0 E_0^2 c = \frac{B_0^2}{2\mu_0} c = \epsilon_0 E_{rms}^2 c = \frac{B_{rms}^2}{\mu_0} c)
  • Momentum (p): EM waves carry linear momentum. If a total energy U is absorbed by a surface, the momentum transferred is:
    • (p = \frac{U}{c}) (Complete absorption)
    • If the wave is totally reflected, the momentum transferred is (p = \frac{2U}{c}).
  • Radiation Pressure (P): The pressure exerted by EM waves on a surface.
    • (P = \frac{\text{Force}}{\text{Area}} = \frac{\Delta p / \Delta t}{A})
    • For complete absorption: (P = \frac{I}{c}) (I = Intensity)
    • For complete reflection: (P = \frac{2I}{c})
  • Medium: EM waves do not require a material medium for propagation; they can travel through vacuum.
  • Wave Phenomena: They exhibit reflection, refraction, interference, diffraction, and polarization. Polarization demonstrates the transverse nature of EM waves.

5. Electromagnetic Spectrum

The classification of EM waves according to their frequency (or wavelength) in a specific order is called the electromagnetic spectrum. There are no sharp boundaries between different types; they overlap.

Type Wavelength Range (approx.) Frequency Range (Hz) (approx.) Production Detection Applications
Radio Waves > 0.1 m < (3 \times 10^9) Accelerated motion of charges in wires (LC oscillators, antennas) Receiver aerials Radio & TV broadcasting, Cellular phones, RADAR (Radio Detection and Ranging)
Microwaves 0.1 m to 1 mm (3 \times 10^9) to (3 \times 10^{11}) Klystron, Magnetron tubes Point contact diodes RADAR, Microwave ovens, Telecommunication (satellite, long distance)
Infrared (IR) 1 mm to 700 nm (3 \times 10^{11}) to (4 \times 10^{14}) Vibration of atoms and molecules (hot bodies) Thermopiles, Bolometer, IR photographic film Remote controls, Thermal imaging (night vision), Physiotherapy, Greenhouse effect
Visible Light 700 nm (Red) to 400 nm (Violet) (4 \times 10^{14}) to (7.5 \times 10^{14}) Electrons in atoms emitting light (Incandescent lamps, LEDs, flames) Eye, Photographic film, Photodiodes, Photocells Optics, Vision, Photography, Photosynthesis
Ultraviolet (UV) 400 nm to 1 nm (7.5 \times 10^{14}) to (3 \times 10^{16}) Inner shell electron transitions in atoms, Very hot bodies (Sun), UV lamps Photocells, Photographic film Sterilization (killing germs), Detecting forged documents, Vitamin D production, LASIK eye surgery
X-rays 1 nm to (10^{-3}) nm (3 \times 10^{16}) to (3 \times 10^{19}) Bombardment of high-energy electrons on a metal target (X-ray tube), Inner shell electron transitions Photographic film, Geiger tubes, Scintillation counters Medical imaging (detecting fractures), Cancer treatment (radiotherapy), Study of crystal structure
Gamma Rays < (10^{-3}) nm > (3 \times 10^{19}) Radioactive decay of atomic nuclei, Nuclear reactions Photographic film, Geiger tubes, Scintillation counters Cancer treatment (radiotherapy), Sterilizing medical equipment, Detecting flaws in castings

Key Order: (Remember in increasing frequency / decreasing wavelength)
Radio Makes It Very Unusual Xmas Gift (Radio, Micro, IR, Visible, UV, X-ray, Gamma)

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed:

  1. The concept of displacement current was introduced by:
    (a) Ampere
    (b) Faraday
    (c) Maxwell
    (d) Hertz

  2. Which of the following is the primary source of electromagnetic waves?
    (a) A stationary charge
    (b) A charge moving with constant velocity
    (c) An accelerated charge
    (d) A charge in equilibrium

  3. In an electromagnetic wave propagating in vacuum, the ratio of the magnitude of the electric field to the magnitude of the magnetic field (E/B) is equal to:
    (a) (c) (speed of light)
    (b) (1/c)
    (c) (c^2)
    (d) (\sqrt{c})

  4. The direction of propagation of an electromagnetic wave is given by:
    (a) (\vec{E} \cdot \vec{B})
    (b) (\vec{E} \times \vec{B})
    (c) (\vec{B} \times \vec{E})
    (d) (\vec{E})

  5. Which part of the electromagnetic spectrum is used in RADAR systems?
    (a) Infrared waves
    (b) Ultraviolet rays
    (c) Microwaves
    (d) X-rays

  6. The speed of electromagnetic waves in a material medium with relative permittivity (\epsilon_r) and relative permeability (\mu_r) is given by:
    (a) (c \sqrt{\mu_r \epsilon_r})
    (b) (c / \sqrt{\mu_r \epsilon_r})
    (c) (\sqrt{\mu_r \epsilon_r} / c)
    (d) (c)

  7. If (u_E) and (u_B) are the average energy densities due to electric and magnetic fields respectively in an EM wave, then:
    (a) (u_E > u_B)
    (b) (u_E < u_B)
    (c) (u_E = u_B)
    (d) Relation depends on the medium

  8. Which of the following electromagnetic waves has the highest frequency?
    (a) Radio waves
    (b) Microwaves
    (c) X-rays
    (d) Gamma rays

  9. An electromagnetic wave transfers momentum to a surface on which it falls. If the wave is totally reflected, the momentum transferred ((p)) for energy (U) is:
    (a) (p = U/c)
    (b) (p = 2U/c)
    (c) (p = U/2c)
    (d) (p = Uc)

  10. The existence of electromagnetic waves was experimentally confirmed by:
    (a) Maxwell
    (b) Hertz
    (c) Marconi
    (d) Faraday

Answers to MCQs:

  1. (c) Maxwell
  2. (c) An accelerated charge
  3. (a) (c) (speed of light)
  4. (b) (\vec{E} \times \vec{B})
  5. (c) Microwaves
  6. (b) (c / \sqrt{\mu_r \epsilon_r})
  7. (c) (u_E = u_B)
  8. (d) Gamma rays
  9. (b) (p = 2U/c)
  10. (b) Hertz

Make sure you understand the reasoning behind each answer. This chapter relies heavily on understanding the fundamental concepts and formulas. Good luck with your preparation!

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