Electromagnetic Waves — Key Concepts

Overview

• Electromagnetic (EM) waves are formed by mutually generating changing electric and magnetic fields.
• In EM waves, electric field, magnetic field, and direction of propagation are all mutually perpendicular.
• Light is an electromagnetic wave that travels at the speed of light in vacuum.
• EM waves carry energy, have intensity, and obey an inverse square law with distance.
• EM spectrum covers radio waves to gamma rays, with frequency and wavelength inversely related.

Formation and Nature of Electromagnetic Waves

• A changing magnetic field produces an electric field; a changing electric field produces a magnetic field.

• This continuous mutual production leads to waves of electric and magnetic fields that propagate through space.

• These waves are called electromagnetic waves.

• Example source: an AC current-carrying wire

  • Current alternates: positive half-cycle and negative half-cycle.
  • Electric field direction oscillates as charge changes sign (positive/negative).
  • Magnetic field circles around the wire; its direction reverses when current reverses.

• Orientation of fields in a plane wave:

  • Electric field (E) along y-direction.
  • Magnetic field (B) along z-direction.
  • Wave propagates along x-direction.
  • E ⟂ B and both ⟂ direction of propagation.

• Near the oscillating wire:

  • Magnetic field lines form closed loops that expand outward.
  • As loops move far from the source, they become large and nearly parallel.
  • Far from the source, the EM wave can be treated as a plane wave.

• Conclusion:

  • Combined changing electric and magnetic fields give EM waves.
  • E and B are perpendicular to each other and to the direction of propagation.

Speed of Light and E–B Relationship

• Ratio of electric field magnitude to magnetic field magnitude in an EM wave is constant:

  • ( \dfrac{E}{B} = c )
  • ( c ) is the speed of light in vacuum.

• Maxwell used known constants:

  • Permittivity of free space: ( \varepsilon_0 )
  • Permeability of free space: ( \mu_0 )
  • Relationship:
    • ( c = \dfrac{1}{\sqrt{\varepsilon_0 \mu_0}} )

• Numerical value:

  • ( c \approx 3 \times 10^8 ,\text{m/s} ) (in vacuum).

Historical Measurements of Speed of Light

• Early astronomers estimated large value of c using observations of Jupiter’s moons.

• Michelson experiment:

  • Used an eight-sided rotating mirror and a stationary mirror.
  • Light reflected from rotating mirror to stationary mirror and back.
  • From rotation speed of mirror and distance to stationary mirror, speed of light was calculated.

• Another method with two mirrors:

  • Two mirrors separated by distance d.
  • Light travels from one mirror to the other and back: distance = 2d.
  • Using speed formula: ( v = \dfrac{\text{distance}}{\text{time}} = \dfrac{2d}{t} ) to find c.

Electromagnetic Spectrum and Wave Relations

• Light as EM wave:

  • Visible wavelength range: approximately 400 nm to 750 nm.

• Relation between wavelength and frequency:

  • ( c = \lambda f )
    • ( c ): speed of light.
    • ( \lambda ): wavelength.
    • ( f ): frequency.

• EM waves travel at speed c in vacuum, regardless of frequency or wavelength.

Electromagnetic Spectrum Order

• From lowest frequency (longest wavelength) to highest frequency (shortest wavelength):
  • Radio waves
  • Microwaves
  • Infrared
  • Visible light (small band)
  • Ultraviolet
  • X-rays
  • Gamma rays

General Rules for EM Waves

• Higher-frequency EM waves:

  • Are more energetic.
  • Penetrate materials more deeply than low-frequency waves.

• Information capacity:

  • High-frequency waves can carry more information per unit time than low-frequency waves.

• Resolution rule:

  • Shorter-wavelength EM waves probing a material can resolve smaller details.

Frequency–Wavelength Inverse Relation

• From ( c = f \lambda ) and constant c:

  • ( f \propto \dfrac{1}{\lambda} )
  • Higher frequency → shorter wavelength.
  • Lower frequency → longer wavelength.

• Extreme examples:

  • Gamma rays: highest frequency, shortest wavelength.
  • Radio waves: lowest frequency, longest wavelength.

• Conceptual view with waves:

  • Frequency: number of cycles per second.
  • Wave with more cycles in same time has higher frequency and shorter wavelength.
  • Wave with fewer cycles in same time has lower frequency and longer wavelength.

Use of Different EM Bands

• Gamma rays:

  • Very short wavelength, high frequency.
  • Harder to resolve signal information because of very small wavelength.

• Radio waves:

  • Long wavelength, low frequency.
  • Easier to resolve signals, suitable for transmitting radio and TV signals.

Energy in Electromagnetic Waves

• EM waves carry energy due to their electric and magnetic fields.
• Some EM radiation produces noticeable warmth (e.g., microwaves).
• Others, like gamma rays, can penetrate the body and destroy living cells without a felt warmth.

Resonance and Energy Transfer

• If EM wave frequency matches the natural frequency of a system:
  • Resonance occurs.
  • Resulting oscillation has double amplitude compared to a single wave.
  • Energy transfer to the system is more efficient; more energy is deposited.

Dependence on Amplitude

• Energy carried by an EM wave is proportional to square of amplitude:
  • Proportional to ( E^2 ) and ( B^2 ).
• Larger amplitude of electric and magnetic fields:
  • Produces larger forces on charges.
  • Larger forces do more work, so more energy is transferred.

Intensity of Electromagnetic Waves

• Intensity definition:

  • Energy transported through unit area per unit time.

• Main intensity formulas:

• Average intensity:

  • For oscillating waves, average intensity is half of each peak expression:
    • ( I_{\text{avg}} = \dfrac{1}{2} \varepsilon_0 c E_0^{2} )
    • ( I_{\text{avg}} = \dfrac{1}{2} \dfrac{c}{\mu_0} B_0^{2} )
    • ( I_{\text{avg}} = \dfrac{1}{2} \dfrac{E_0 B_0}{\mu_0} )

• Peak vs average:

  • Peak intensity is twice average intensity:
    • ( I_{\text{peak}} = 2 I_{\text{avg}} )_

Example: Microwave Oven

• Given:

  • Output power: ( P = 1,\text{kW} ).
  • Area: ( 30,\text{cm} \times 40,\text{cm} ) (convert to m²).

• Steps:

  • Find average intensity:
    • ( I_{\text{avg}} = \dfrac{P}{A} ).
  • Peak intensity:
    • ( I_{\text{peak}} = 2 I_{\text{avg}} ).

• If problem statement does not say “peak” or “maximum,” assume given intensity or power corresponds to average value._

Finding Field Amplitudes from Intensity

• From average intensity:
  • Use ( I_{\text{avg}} = \dfrac{1}{2} \varepsilon_0 c E_0^{2} ) to get peak electric field:
    • ( E_0 = \sqrt{\dfrac{2 I_{\text{avg}}}{\varepsilon_0 c}} )
  • Use relation ( \dfrac{E_0}{B_0} = c ) to find peak magnetic field:
    • ( B_0 = \dfrac{E_0}{c} )

Inverse Square Law for Intensity

• EM waves spreading out from a point source:

  • Assume they radiate uniformly in all directions as a sphere.

• Spherical spreading:

  • For radius r, area of sphere: ( A = 4 \pi r^2 )
  • Intensity:
    • ( I = \dfrac{P}{4 \pi r^2} )
  • Therefore:
    • ( I \propto \dfrac{1}{r^2} ) (inverse square law).

• As r increases (r → 2r → 3r), intensity decreases rapidly:

  • Example ratios:
    • At r: ( I )
    • At 2r: ( \dfrac{I}{4} )
    • At 3r: ( \dfrac{I}{9} )

Demonstration with Light Bulbs and Photometer

• Setup:

  • One side: four light bulbs.
  • Other side: one light bulb.
  • Separation between sets: 2 meters.
  • Photometer: two wax blocks separated by tin foil; each side glows with nearby light.

• Observations:

  • At midpoint, side facing four bulbs is brighter than side facing one bulb.
  • Moving photometer closer to four bulbs makes that side even brighter.
  • Equal brightness occurs when photometer is twice as close to the single bulb as to the four bulbs.

• Reason:

  • Intensity ( \propto \dfrac{1}{r^2} ).
  • Four bulbs on one side vs one bulb on the other.
  • To balance intensities:
    • Single bulb must be at half the distance compared to four bulbs.

Key Terms and Formulas

Action Items / Next Steps

• Practice using ( c = \lambda f ) to convert between wavelength and frequency for different EM spectrum parts.
• Solve problems calculating:
  • Intensity from power and area.
  • Electric and magnetic field amplitudes from intensity.
• Apply inverse square law to compare intensities at different distances from a source.
• Review resonance concept and examples where EM waves match system natural frequencies.