Question

Using B = µ₀ H, find the ratio E₀/H₀ for a plane electromagnetic wave propagating through vacuum. Show that it has the dimensions of electric resistance. This ratio is a universal constant called the impedance of free space.

Answer 1

Given, B = µ₀H

For vacuum we can rewrite this equation as,
B₀ = µ₀H₀ ...(i)

Relation between magnetic field and electric field for vacuum is given as,
B₀ = µ_{0${\in }_0$}cE₀ ...(ii)

From equation (i) by (ii),
$$\begin{gathered} {\mathrm{\mu }}_0{H}_0={\mathrm{\mu }}_0{\in }_{0}c{E}_0 \\ \Rightarrow \frac{E_0}{H_0}=\frac{1}{{\in }_{0}c} \\ \Rightarrow \frac{E_0}{H_0}=\frac{1}{8.85\times {10}^{-12}\times 3\times {10}^8} \\ \Rightarrow \frac{E_0}{H_0}\approx 377\mathrm{\Omega } \end{gathered}$$
$$\begin{gathered} \mathrm{Dimension}\text{of}\frac{1}{{\in }_{0}c}=\frac{1}{\left[{\mathrm{LT}}^{-1}\right]\left[{\mathrm{M}}^{-1}{\mathrm{L}}^{-3}{\mathrm{T}}^4{\mathrm{A}}^2\right]} \\ =\frac{1}{{\mathrm{M}}^{-1}{\mathrm{L}}^{-2}{\mathrm{T}}^3{\mathrm{A}}^2} \\ ={\mathrm{M}}^4{\mathrm{L}}^2{\mathrm{T}}^{-3}{\mathrm{A}}^{-2}=\left[R\right] \end{gathered}$$