Correct Answer - Option 2 : The magnetic field of the wave will be
\(H = \frac{{{E_0}}}{{{\eta _0}}}\cos \left( {\omega t + \frac{\omega }{c}x} \right){a_y}\)
Analysis:
The general equation of an electric field intensity of a plane wave propagating in free space in -ax direction having amplitude E0 and frequency ω is given as:
E = E0 cos (ωt + βx) an
Where,
β = phase constant of the wave
an = unit vector in the direction of polarization of the wave.
The polarization of a plane wave is the “figure traced by the tip of the electric field vector as a function of time, at a fixed point in space.”
Since EM Wave is polarized in +az direction, we have:
an = az
\(E = {E_0}\cos \left( {\omega t + \beta x} \right){a_z}\)
\(\beta = \frac{{2\pi }}{\lambda } = \frac{{2\pi }}{c} \cdot f\)
\(\therefore \beta = \frac{\omega }{c}\)
\(E = {E_0}\cos \left( {\omega t + \frac{\omega }{c} x} \right){a_z}\)
The electric field and magnetic field of a plane wave satisfies the Poynting theorem i.e.
\(\vec P = \vec E \times \vec H\)
Also, the electric field magnitude and magnetic field magnitude of a plane wave are related as:
\(\frac{{{E_0}}}{{{H_0}}} = {η _0}\)
η0 = intrinsic impedance of free space.
∴ The magnetic field intensity direction must satisfy the Poynting theorem i.e.
\(\mathop {\mathop {\vec P}\limits_{ - {{\hat a}_x}} }\limits_{} = \mathop {\mathop {\vec E}\limits_{{{\hat a}_z}} }\limits_{} \times \mathop {\mathop {\vec H}\limits_{\begin{array}{*{20}{c}} {The~direction}\\ {must~be~{a_y}~to}\\ {satisfy~the}\\ {Poynting}\\ {theorem.} \end{array}} }\limits_{} \)
So, with \({H_0} = \frac{{{E_0}}}{{{η _0}}}\), and direction vector ây, the magnetic field will be:
\(H = \frac{{{E_0}}}{{{η _0}}}\cos \left( {\omega t + \frac{\omega }{c}x} \right){\hat a_y}\)
