A plane electromagnetic wave traveling in a vacuum is characterized by the electric and magnetic fields as:

\(\vec E = \hat i\left( {30\pi \;v/m} \right)\;e^{\;i\left( {\omega t + kz} \right)}\)

\(\vec H = \hat j\left( {{H_0}\;A/m} \right)e^{\;i\left( {\omega t + kz} \right)}\)

It ω and k are positive, then the value of H₀ must be:

Concept:

In electromagnetic waves, the ratio of amplitudes of the electric field and the magnetic field is equal to the velocity of the electromagnetic waves in free space.

\(\frac{{{E_0}}}{{{B_0}}} = c\)

E = B × c

Where: 

E0 = Electric field

B0 = Magnetic field

c = speed of light = 3 × 108 m/sec

An electromagnetic wave traveling in the given direction must satisfy the Poynting theorem, as it describes the magnitude and direction of the flow of energy in electromagnetic waves.

Mathematically, the Poynting vector is the cross-product of the Electric field vector and the magnetic field vector, i.e.

\(P = \vec E × \vec H\;Watt/{m^2}\)

Calculation:

For an electromagnetic wave:

\(\vec E = \hat i\left( {30\pi \;v/m} \right)\;e^{\;i\left( {\omega t + kz} \right)}\)

\(\vec H = \hat j\left( {{H_0}\;A/m} \right)e^{\;i\left( {\omega t + kz} \right)}\)

Here E and H are having the same phase and moving perpendicular to each other.

\(\frac{{\left| E \right|}}{{\left| B \right|}} = c\)

B = μ₀H

\(\frac{{{E_0}}}{{{\mu _0}{H_0}}} = c\)

\({H_0} = \frac{{{E_0}}}{{c{\mu _0}}}\)

\({H_0} = \frac{{30\;\pi }}{{4\;\pi \; \times \;{{10}^{ - 7}}\; \times \;3\; \times \;{{10}^8}}}\)

= \(\frac{{10}}{{4\; \times \;10}} = 0.25\;A/m\)