In free space, \({\bf{E}}\left( {{\rm{z}},{\rm{t}}} \right) = {10^3}{\rm{sin}}\left( {{\rm{\omega t}}-{\rm{\beta z}}} \right)\hat y\). Obtain H(z,t).

Concept:

Any vector quantity can be represented as the product of magnitude and vector quantity, i.e.

\(\vec A = \left| A \right|.\;\hat u\)

|A| = Magnitude of the vector quantity.

û = Unit direction vector.

Poynting vector:

It states that the cross product of electric field vector (E) and magnetic field vector (H) at any point is a measure of the rate of flow of electromagnetic energy per unit area at that point that is 

\( \vec P = \vec E \times \vec H\)

Where P = Poynting vector, E = Electric field and H = Magnetic field

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves.

The unit of the Poynting vector is watt/m2

The ratio \(\frac{{\left| E \right|}}{{\left| H \right|}}\) for the wave is equal to the intrinsic impedance of the medium, η.

\(\frac{E}{H} = {\eta _0} = 377\;{\rm{\Omega }} = 120\;\pi \)

Analysis:

The direction of propagation = +z direction

Electric field vector = +y direction

The direction of the magnetic field will be such that the Poynting vector theorem is satisfied

i.e. P = E × H

∴ The magnetic field must be along (-a_x) direction.

The magnitude will be:

\(\frac{{\left| E \right|}}{{\left| H \right|}} = 377\)

\(H = \frac{{\left| E \right|}}{{377}} = \frac{{{{10}^3}}}{{377}} = 2.65\)

∴ The magnetic field vector will be:

\(\vec H\) (z, t) = |H| sin (ωt – βz) (-a_x)

= -2.65 sin (ωt - βz) a_x