Master Formula Sheet

# Result Formula
1 Displacement current id=ε0dΦEdt=ε0AdEdt=CdVdti_d = \varepsilon_0\dfrac{d\Phi_E}{dt} = \varepsilon_0 A\dfrac{dE}{dt} = C\dfrac{dV}{dt}
2 Ampere-Maxwell law ∮B⃗⋅dl⃗=μ0ic+μ0ε0dΦEdt\oint\vec B\cdot d\vec l = \mu_0 i_c + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}
3 B inside capacitor gap B=μ0idr2πR2B = \dfrac{\mu_0 i_d r}{2\pi R^2} (r < R)
4 Plane wave Ex=E0sin⁡(kz−ωt)E_x = E_0\sin(kz-\omega t), By=B0sin⁡(kz−ωt)B_y = B_0\sin(kz-\omega t)
5 Wave numbers k=2π/λk = 2\pi/\lambda; ω=2πν\omega = 2\pi\nu; ω=ck\omega = ck; νλ=c\nu\lambda = c
6 Amplitude lock B0=E0/cB_0 = E_0/c
7 Vacuum speed c=1/μ0ε0=3×108c = 1/\sqrt{\mu_0\varepsilon_0} = 3\times10^8 m/s
8 Speed in a medium v=1/με=c/μrεrv = 1/\sqrt{\mu\varepsilon} = c/\sqrt{\mu_r\varepsilon_r}
9 Direction rule propagation along E^×B^\hat E\times\hat B

Maxwell's equations: Gauss (E) ∮E⃗⋅dA⃗=Q/ε0\oint\vec E\cdot d\vec A = Q/\varepsilon_0; Gauss (B) ∮B⃗⋅dA⃗=0\oint\vec B\cdot d\vec A = 0; Faraday ∮E⃗⋅dl⃗=−dΦB/dt\oint\vec E\cdot d\vec l = -d\Phi_B/dt; Ampere-Maxwell as row 2.

Wave Facts to Quote Verbatim

  • EM waves are transverse: E⃗⊥B⃗⊥k⃗\vec E\perp\vec B\perp\vec k, with E and B in phase.
  • Sourced only by accelerated charges; the wave inherits the source's frequency; radiated energy comes from the source.
  • Self-sustaining (Faraday + Ampere-Maxwell regeneration): no medium needed.
  • c in vacuum is identical for all wavelengths — so precisely known it defines the metre.
  • Maxwell predicted; Hertz demonstrated (1887); Bose made millimetre waves; Marconi sent them kilometres.
  • The displacement current restores symmetry: changing B makes E, changing E makes B — though no magnetic monopoles exist, so the symmetry is imperfect.
  • For the charging capacitor: wire has ici_c, gap has idi_d, and they are equal.

The Spectrum Table (Increasing Frequency)

Band λ\lambda Produced by Detected by Star uses
Radio > 0.1 m electrons in aerials receiver aerials broadcasting (AM 530-1710 kHz, FM 88-108 MHz, TV 54-890 MHz)
Microwave 0.1 m-1 mm klystron/magnetron/Gunn diode point contact diodes radar, speed guns, ovens (water resonance)
Infrared 1 mm-700 nm molecular vibrations, hot bodies thermopile, bolometer greenhouse effect, remotes, physiotherapy, satellites
Visible 700-400 nm atomic electron transitions eye, photocells, film sight
UV 400 nm-0.6 nm special lamps, very hot bodies, sun photocells, film LASIK, germicidal lamps; ozone absorbs; glass blocks
X-rays 10 nm-10−410^{-4} nm electrons striking metal targets film, Geiger tube, ionisation chamber medical imaging, cancer therapy
Gamma < 10−1010^{-10} m nuclei (reactions, radioactivity) film, Geiger tube, ionisation chamber destroying cancer cells

Boundaries overlap; production decides borderline cases. Memory hook for increasing frequency: Rascals Make Ill Visitors Uneasy X-mas Gifts.

One-Glance Revision Flow

The chapter in five steps:

  1. Ampere's law fails at a charging capacitor → displacement current ε0 dΦE/dt\varepsilon_0\,d\Phi_E/dt repairs it → Ampere-Maxwell law.
  2. Now changing E and changing B source each other → self-sustaining EM waves, radiated by accelerated charges.
  3. Wave anatomy: transverse, in-phase fields, B0=E0/cB_0 = E_0/c, direction E^×B^\hat E\times\hat B, speed 1/μ0ε01/\sqrt{\mu_0\varepsilon_0}.
  4. That speed equals light's → light is an EM wave (Maxwell predicted, Hertz confirmed).
  5. The spectrum spans 18 orders of magnitude — classified by production/detection, from aerials to nuclei.

Morning-of-exam checklist: idi_d needs changing E … gap current = wire current … E and B IN PHASE (not 90 degrees) … direction by cross product, not memory … frequency never changes between media … ozone absorbs UV, glass blocks UV … X-rays from electron-target, gamma from nuclei … oven = water resonance … all bands travel at c in vacuum. Done — go score.