Lernzettel: Electromagnetic Waves

📋 Course Outline

1. Maxwell’s Electromagnetic Unification
2. Displacement Current
3. Ampere–Maxwell Law
4. Sources of Electromagnetic Waves
5. Structure of Electromagnetic Waves
6. Wave Speed and Energy
7. Electromagnetic Spectrum
8. Applications of Spectrum Regions

📖 1. Maxwell’s Electromagnetic Unification

★ Must-know

• James Clerk Maxwell (1831–1879) showed that a time-varying electric field generates a magnetic field, completing the symmetry with Faraday’s result that a time-varying magnetic field generates an electric field.

• Maxwell’s equations, together with the Lorentz force formula, express the basic laws of electromagnetism and predict electromagnetic waves as coupled time-varying electric and magnetic fields propagating through space. — James Clerk Maxwell

• The predicted speed of electromagnetic waves was close to $3 \times 10^8\ \mathrm{m\,s^{-1}}$, the measured speed of light, leading to the conclusion that light is an electromagnetic wave. — James Clerk Maxwell

Further detail

• Hertz experimentally demonstrated electromagnetic waves in 1885, and their technological use by Marconi and others led to modern communication.

💡 Memory Hook

Changing electric fields → magnetic fields → electromagnetic waves

📖 2. Displacement Current

🔑 Key Concepts & Definitions

• Displacement current : Maxwell — the current associated with a time-varying electric field and is given by $i_d=\varepsilon_0\frac{d\Phi_E}{dt}$

★ Must-know

• During capacitor charging, the electric flux between the plates changes because the plate charge changes, producing a displacement current equal to the conduction current in the connecting wire.

📌 In the charging-capacitor example, outside the plates $i_c=i$ and $i_d=0$, while between the plates $i_c=0$ and $i_d=i$.

Further detail

• Displacement current has the same physical magnetic effects as conduction current and can exist in a region containing no conduction current.

💡 Memory Hook

Conduction current outside the capacitor; displacement current between its plates

📖 3. Ampere–Maxwell Law

★ Must-know

📐 Formula — The total current is the sum of conduction and displacement currents: $i=i_c+i_d=i_c+\varepsilon_0\frac{d\Phi_E}{dt}$.

📐 Formula — The Ampere–Maxwell law states that the circulation of the magnetic field equals $\oint \mathbf{B}\cdot d\mathbf{l}=\mu_0\left(i_c+\varepsilon_0\frac{d\Phi_E}{dt}\right)$.

Further detail

• Maxwell’s equations in vacuum are:

  • Gauss’s law for electricity
  • Gauss’s law for magnetism
  • Faraday’s law
  • The Ampere–Maxwell law

• The symmetry between Faraday’s law and the Ampere–Maxwell law is incomplete because no magnetic monopoles analogous to electric charges are known.

📖 4. Sources of Electromagnetic Waves

★ Must-know

• Stationary charges produce electrostatic fields, and charges in uniform motion produce magnetic fields that do not vary with time; neither is a source of electromagnetic waves.

• An oscillating charge produces an oscillating electric field, which produces an oscillating magnetic field; the two fields regenerate each other as the wave propagates.

• The frequency of an electromagnetic wave produced by an oscillating charge equals the oscillation frequency of the charge.

Further detail

• Hertz produced and detected electromagnetic waves in the radio-wave region in 1887 because electronic circuits could not oscillate at the approximately $6\times10^{14}\ \mathrm{Hz}$ frequency of yellow light.

💡 Memory Hook

Stationary or uniformly moving charges do not radiate; accelerated charges do

📖 5. Structure of Electromagnetic Waves

★ Must-know

• In a plane electromagnetic wave, the electric field, magnetic field, and direction of propagation are mutually perpendicular.

📐 Formula — For a wave propagating in the z-direction, the fields may be written as $E_x=E_0\sin(kz-\omega t)$ and $B_y=B_0\sin(kz-\omega t)$.

📐 Formula — The amplitudes of the electric and magnetic fields satisfy $B_0=\frac{E_0}{c}$.

Further detail

📐 Formula — The wave number and angular frequency satisfy $k=\frac{2\pi}{\lambda}$ and the fields oscillate with the common phase $kz-\omega t$.

💡 Memory Hook

Three mutually perpendicular arrows: E, B, and the propagation direction

📖 6. Wave Speed and Energy

★ Must-know

📐 Formula — The speed of electromagnetic waves in vacuum is $c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}$ and is approximately $3\times10^8\ \mathrm{m\,s^{-1}}$.

📐 Formula — The frequency, wavelength, and speed of an electromagnetic wave satisfy $\nu\lambda=c$.

📐 Formula — In a material medium with permittivity $\varepsilon$ and permeability $\mu$, the speed of electromagnetic waves is $v=\frac{1}{\sqrt{\mu\varepsilon}}$.

Further detail

• Electromagnetic waves can propagate through vacuum without a material medium and carry energy from one place to another.

📖 7. Electromagnetic Spectrum

★ Must-know

• The electromagnetic spectrum includes gamma rays, X-rays, ultraviolet rays, visible rays, infrared rays, microwaves, and radio waves, with no sharp boundaries between adjacent regions.

• Electromagnetic waves in vacuum all have the same speed, while their wavelengths and frequencies differ.

Further detail

• The wavelength range of the electromagnetic spectrum extends approximately from $10^{-12}\ \mathrm{m}$ for gamma rays to $10^6\ \mathrm{m}$ for long radio waves.

💡 Memory Hook

Gamma rays → X-rays → ultraviolet → visible → infrared → microwaves → radio waves

📖 8. Applications of Spectrum Regions

★ Must-know

• Radio waves are produced by accelerated electrons in conducting wires and are used in radio and television communication systems.

• Microwaves have frequencies in the gigahertz range, are produced by klystrons, magnetrons, or Gunn diodes, and are used in radar and microwave ovens.

• Visible rays occupy approximately $4\times10^{14}\ \mathrm{Hz}$ to $7\times10^{14}\ \mathrm{Hz}$, corresponding to wavelengths from about 700 nm to 400 nm, and are detected by the human eye.

• Gamma rays have wavelengths from about $10^{-10}\ \mathrm{m}$ to less than $10^{-14}\ \mathrm{m}$ and are produced by nuclear reactions and radioactive nuclei.

Further detail

• Infrared waves are produced by hot bodies and molecular vibrations, are absorbed by molecules such as water, and are used in physical therapy, Earth observation, and remote controls.

• Ultraviolet radiation spans approximately 400 nm to 0.6 nm, is largely absorbed by atmospheric ozone, and can cause tanning and biological damage.

• X-rays have wavelengths from about 10 nm to $10^{-4}\ \mathrm{nm}$ and are produced by bombarding metal targets with high-energy electrons or by inner-shell electronic transitions.