Core EM Wave Formulas

$\boxed{c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = 3 \times 10^8 \text{ m/s}} \quad [\text{L T}^{-1}]$

$\boxed{c = f\lambda} \quad [\text{L T}^{-1}] = [\text{T}^{-1}][\text{L}]$

$\boxed{\frac{E_0}{B_0} = c} \quad \frac{[\text{V m}^{-1}]}{[\text{T}]} = [\text{m s}^{-1}] \checkmark$

$\boxed{I = \frac{1}{2}\varepsilon_0 c E_0^2} \quad [\text{M T}^{-3}] = \text{W m}^{-2}$

Dimensional check: $[\varepsilon_0] = [\text{C}^2 \text{N}^{-1} \text{m}^{-2}]$; $[\varepsilon_0 c E_0^2] = [\text{C}^2 \text{N}^{-1} \text{m}^{-2}][\text{m s}^{-1}][\text{V}^2 \text{m}^{-2}]$. Since 1 V = 1 J/C = 1 N·m/C: $[\text{C}^2 \cdot \text{N}^{-1} \cdot \text{m}^{-2} \cdot \text{m s}^{-1} \cdot \text{N}^2 \text{m}^2 \text{C}^{-2} \cdot \text{m}^{-2}] = [\text{N m}^{-1} \text{s}^{-1}] = [\text{W m}^{-2}]$ ✓

Displacement Current

$\boxed{I_d = \varepsilon_0 \frac{d\Phi_E}{dt}} \quad [\text{A}] = [\text{C}^2 \text{N}^{-1} \text{m}^{-2}][\text{V m s}^{-1}]$

$\Phi_E = EA \Rightarrow \frac{d\Phi_E}{dt} = A\frac{dE}{dt} \quad [\text{V m s}^{-1}] = [\text{m}^2][\text{V m}^{-1} \text{s}^{-1}]$

Radiation Momentum and Pressure

$\boxed{p_{\text{absorbed}} = \frac{U}{c}} \quad [\text{M L T}^{-1}] = \frac{[\text{M L}^2 \text{T}^{-2}]}{[\text{L T}^{-1}]} \checkmark$

$\boxed{p_{\text{reflected}} = \frac{2U}{c}} \quad [\text{M L T}^{-1}]$

Ampere-Maxwell Law

$\boxed{\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0\left(I_c + \varepsilon_0 \frac{d\Phi_E}{dt}\right)}$

Dimensional check: $[\text{T m}] = [\text{H m}^{-1}][\text{A}]$ where $[H] = [\text{kg m}^2 \text{A}^{-2} \text{s}^{-2}]$: $[\text{kg m}^2 \text{A}^{-2} \text{s}^{-2}][\text{m}^{-1}][\text{A}] = [\text{kg m A}^{-1} \text{s}^{-2}] = [\text{T m}]$ ✓