Question

A plane electromagnetic wave traveling in the positive direction of anxaxis in vacuum has components${\mathit{E}}_{\mathbf{x}}\mathbf{=}{\mathit{E}}_{\mathbf{y}}\mathbf{=}\mathbf{0}$and${\mathit{E}}_{\mathbf{z}}\mathbf{=}\left(2.0\text{V/m}\right)\mathbf{cos}\left[\left(\pi \times {10}^{15}{\text{s}}^{-1}\right)\left(t-x/c\right)\right]$. (a) What is the amplitude of the magnetic field component? (b) Parallel to which axis does the magnetic field oscillate? (c) When the electric field component is in the positive direction of thezaxis at a certain pointP, what is the direction of the magnetic field component there?

Short answer

a) Theamplitude of the magnetic field componentis$6.7\mathrm{nT}$ .

b)Magnetic fieldcomponent will be oscillating in the y direction.

c)The direction of the magnetic field component is –y when the electric field is in the +z direction.

Step-by-step solution

Listing the given quantities

The electric field component $E_z=\left(2.0\text{V/m}\right)\cos \left[\left(\pi \times {10}^{15}{\text{s}}^{-1}\right)\left(t-x/c\right)\right]$.

Understanding the concepts of electric and magnetic fields

The electric and magnetic fieldsandare always perpendicular to the direction in which the wave is traveling.

The electric and magnetic fields are related as, $c=\frac{E_z}{B_y}$

(a) Calculations of the amplitude of the magnetic field component

The electric and magnetic fields E and B are always perpendicular to the direction in which the wave travels.

The plane electromagnetic wave is traveling in x direction hence $E_x$and $E_y$are zero.

From the given, the electric field is in the z direction; hence the magnetic field is in the y direction.

The electric and magnetic fields are related as,

$c=\frac{E_z}{B_y}$

Hence, the amplitude of the magnetic field will be,

$B_y=\frac{E_z}{c}$

Where the speed of the light is, $c=3\times {10}^8\text{m/s}$.

Substitute the values in the above expression, and we get,

role="math" localid="1662971654502" $\begin{array}{rcl}{B}_y& =& \frac{2.0\text{V/m}}{3\times {10}^8\text{m/s}}\\ B_y& =& 6.67\times {10}^{-9}\mathrm{T}\\ & =& 6.7\mathrm{nT}\end{array}$

Thus, theamplitude of the magnetic field component is $6.7\mathrm{nT}$.

(b) Explanation of the magnetic field oscillates

The EM wave is traveling in the x direction. The electric field component is in the z-direction.

Hence, the magnetic field component will oscillate in the y direction.

(c) Direction of magnetic field

From Faraday’s law of induction, the direction of the magnetic field is –y when the electric field is in the +z direction.

Thus, the direction of the magnetic field is –y.