Electromagnetic waves

∮ E·dl = -dΦB/dt

Step 1: Maxwell's four equations correspond to four fundamental laws of electromagnetism. Step 2: The first equation ∮ E·dA = Q/ε₀ is Gauss's law for electricity. Step 3: The second equation ∮ B·dA = 0 is Gauss's law for magnetism. Step 4: The third equation ∮ E·dl = -dΦB/dt is Faraday's law — it says a changing magnetic flux induces an electric field. Step 5: The fourth equation is the Ampere-Maxwell law. So the third Maxwell equation is Faraday's law.

17.7 A

Step 1: Write the formula for displacement current: id = ε₀ × (dΦE/dt). Step 2: Given values: ε₀ = 8.85 × 10⁻¹² C²/Nm², dΦE/dt = 2 × 10¹² V·m/s. Step 3: Substitute: id = 8.85 × 10⁻¹² × 2 × 10¹². Step 4: Calculate: id = 8.85 × 2 × 10⁻¹²⁺¹² = 17.7 × 10⁰ = 17.7 A. Step 5: The option 8.85 A would result if only ε₀ was used without multiplying by 2. The option 4.425 A would result from dividing instead of multiplying.

Gamma rays

Step 1: The electromagnetic spectrum is arranged in order of increasing frequency (or decreasing wavelength): Radio waves → Microwaves → Infrared → Visible → Ultraviolet → X-rays → Gamma rays. Step 2: Radio waves have frequency range of a few Hz to 10⁹ Hz. Step 3: Infrared rays have frequency range 10¹¹ Hz to 4 × 10¹⁴ Hz. Step 4: X-rays have frequency range 10¹⁷ Hz to 10¹⁹ Hz. Step 5: Gamma rays have frequencies of 10¹⁸ Hz and above — the highest in the EM spectrum. They are produced by radioactive nuclei.

Speed of the electromagnetic wave

Step 1: In an electromagnetic wave, the electric and magnetic fields oscillate simultaneously. Step 2: The relationship between the amplitudes of these fields in free space is: c = E₀/B₀. Step 3: This means the ratio E₀/B₀ = c = 3 × 10⁸ m/s, the speed of light in vacuum. Step 4: In any medium, v = E₀/B₀ where v is the speed of EM wave in that medium (which is less than c). Step 5: Refractive index n = c/v, which is derived from this ratio but is not directly the ratio itself.