Electromagnetic waves - Formula Sheet

Fundamental Formulas

Displacement Current: \(i_d = \epsilon_0 \frac{d\Phi_E}{dt}\)
Defines the current produced by a changing electric flux \(\Phi_E\) with permittivity of free space \(\epsilon_0\).

Ampere-Maxwell Law: \(\oint B \cdot dl = \mu_0 (i_c + i_d)\)
Relates the line integral of magnetic field \(B\) to both conduction current \(i_c\) and displacement current \(i_d\).

Maxwell’s Equations (Integral Form):
1. \(\oint E \cdot dA = \frac{q}{\epsilon_0}\) (Gauss Law for Electricity)
2. \(\oint B \cdot dA = 0\) (Gauss Law for Magnetism)
3. \(\oint E \cdot dl = -\frac{d\Phi_B}{dt}\) (Faraday’s Law)
4. \(\oint B \cdot dl = \mu_0 i_c + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}\) (Ampere-Maxwell Law)

Wave Propagation

Speed of Light in Vacuum: \(c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8\) m/s
Determined by permeability \(\mu_0\) and permittivity \(\epsilon_0\) of free space.

Speed in a Medium: \(v = \frac{c}{n} = \frac{1}{\sqrt{\mu \epsilon}}\)
Where \(n\) is the refractive index, \(\mu\) is permeability, and \(\epsilon\) is permittivity of the medium.

Refractive Index: \(n = \sqrt{\epsilon_r \mu_r}\)
Relates to relative permittivity \(\epsilon_r\) and relative permeability \(\mu_r\).

Field Amplitude Ratio: \(c = \frac{E_0}{B_0}\) (or \(v = \frac{E_0}{B_0}\) in a medium)
Relates peak electric field \(E_0\) to peak magnetic field \(B_0\).

Energy and Momentum

Average Energy Density: \(u = \epsilon_0 E_{rms}^2 = \frac{B_{rms}^2}{\mu_0}\)
Total energy per unit volume in the electromagnetic field.

Linear Momentum: \(p = \frac{U}{c}\)
Momentum carried by a wave of energy \(U\) in a vacuum.

Poynting Vector: \(S = \frac{1}{\mu_0}(E \times B)\)
Describes the direction and magnitude of power per unit area.