Chapter 9.4, Problem 1aT

Shown below are mathematical and pictorial representation of an electromagnetic plane wave propagating through empty space. The electric field is parallel to the z-axis; the magnetic field is parallel to the y-axis ( $\hat{x},\hat{y}\text{and}\hat{z}$ are unit vectors along the $+x,+y,\text{and}+z-$ directions.)
$\boxed{\stackrel{\to }{E}\left(x,y,z,t\right)=E_o\sin \left(kx+\omega t\right)\hat{z}\stackrel{\to }{B}\left(x,y,z,t\right)=B_o\sin \left(kx+\omega t\right)\hat{y}}$

1. In which direction is the wave propagating? Explain how you can tell from the expression for the electric field and magnetic field.

Is the wave transverse or longitudinal? Explain in terms of the quantities that are oscillating.

Is the wave transverse or longitudinal? Explain in terms of the quantities that are oscillating.

2. The points 1—4 in the diagram above lie in the x—z plane.

For the instant shown, rank these points according to the magnitude of the electric field. If the electric field is zero at any point, state that explicitly.

Is your ranking Consistent with the mathematical expression for the electric field shown above? 1f not, resolve any inconsistencies. (For example, how, if at all, does changing the value of z affect the value of $\stackrel{\to }{E}\left(x,y,z,t\right)$ ?)

For the instant shown, rank points 1—4 according to the magnitude of the magnetic field. Check that your ranking is consistent with the expression for the magnetic field, $\stackrel{\to }{B}\left(x,y,z,t\right)$ , above.

In the diagram at right, the four points labeled $"\times "$ are all located in a plane parallel to the y-z plane. One of the labeled points is located on the x-axis.

On the diagram, sketch vectors to show the direction and relative magnitude of the electric field at the labeled points.

Justify the use of the term plane wave for this electromagnetic wave.

To determine

To Explain:

The direction of wave propagation and the reason behind the nature of the wave by using the expressions for the electric and magnetic field.

Answer to Problem 1aT

The wave propagates in negative x- direction. The wave is a transverse wave.

Explanation of Solution

Introduction

A disturbance that travels through a medium from one place to another place is termed as a wave.It transfers energy, and also, comprises a periodic, repetitive movement.

By addition of sine waves, all waves can be pictured. The phase, amplitude, wavelength as well as frequency are determined by the shape of sine wave.

Poynting vector of an electromagnetic wave is the cross product of the electric field and magnetic field of the wave and the direction of the cross product tells the direction of the propagation of the wave. Basically, pointing vector is the energy per unit area per unit time carried by the wave and

Hence individual can identify the vector:

  $\overrightarrow{S}=\frac{1}{{\mu }_0}\text{}\overrightarrow{E}\times \overrightarrow{B}$

Electromagnetic waves are formed of an electric field ( $\overrightarrow{E}$ ) and also, magnetic field ( $\overrightarrow{B}$ ) oscillating perpendicular to another furthermore mutually perpendicular to direction of the propagation of wave.

So, if we consider the direction only and if the direction of the pointing vector is, $\text{<mover accent="true"><mo>S</mo><mo>→</mo></mover>}$ then

  

It means the wave is propagating in a negative $\text{x}$ -direction.

As the electric field $\left(\text{<mover accent="true"><mo>E</mo><mo>→</mo></mover>}\right)$ and the magnetic field $\left(\text{<mover accent="true"><mo>B</mo><mo>→</mo></mover>}\right)$ are oscillating along $\left(\text{<mover accent="true"><mo>z</mo><mo>^</mo></mover>}\right)$ and $\left(\text{<mover accent="true"><mo>y</mo><mo>^</mo></mover>}\right)$ directions respectively, the wave is said to be a transverse wave but the wave is propagating perpendicular to those oscillating fields or along the $\left(\text{ -<mover accent="true"><mo>x</mo><mo>^</mo></mover>}\right)$ direction.

Conclusion:

Therefore, the wave is propagating along negative $\text{x}$ direction.

To determine

To Explain:Whether ranking is consistent with the mathematical expression for the given electric field.

Answer to Problem 1aT

The rank points are consistent. Also, the rank points of diagram are $\left\{1=2>3=4\right\}$ .

Explanation of Solution

• The electric field is parallel to $\text{z}$-axis.

  $\text{<mover accent="true"><mo>E</mo><mo>→</mo></mover>}\left(\text{x,y,z,t}\right){\text{=E}}_{\text{0}}\text{sin}\left(\text{kx+wt}\right)\text{<mover accent="true"><mo>z</mo><mo>^</mo></mover>}$

• The magnetic field is parallel to $\text{y}$-axis

  $\text{<mover accent="true"><mo>B</mo><mo>→</mo></mover>}\left(\text{x,y,z,t}\right){\text{=E}}_{\text{0}}\text{sin}\left(\text{kx+wt}\right)\text{<mover accent="true"><mo>y</mo><mo>^</mo></mover>}$

Both, electric and magnetic field areoscillating in space and time. As this is a plane wave, the field will be constant on a z-y plane. Hence, the plane consisting the points 1 and 2 will have the same field whereas the plane consisting the points 3 and 4 will have the same field. But at the point 1 field is greater compared to point 4, as shown by the length of the arrows.

Therefore, the electric field gives rank as $\left\{1=2>3=4\right\}$

The above ranking is consistent.

The ranking of the magnetic field is given by $\left\{1=2>3=4\right\}$

By changing the value $\left(\text{z}\right)$ , the magnitude of an electric or the magnetic field at any point will not change.

Conclusion:

Therefore, the rank points are consistent. Also, the rank points of diagram are $\left\{1=2>3=4\right\}$ .

To determine

To Explain:The direction and magnitude of the electric field of the given electromagnetic wave on the given plane.

Explanation of Solution

Introduction

A surface on which locus of constant phase of a wave forms a plane then waveis known as a plane wave.

An electromagnetic wave is travelling in x direction with magnetic field along y and electric field along z direction wave shown in Figure 1a. In Figure 1b, a z-y plane is considered on which the direction and magnitude of electric field is sketched.

  

Figure 1:(a) An electromagnetic wave travelling in x direction with magnetic field along y and electric field along z direction wave (b) The magnitude and the direction of electric field on a plane at a given location on x axis

The direction of theelectric field is shown with green arrows and the length of the arrow shows the magnitude of the field which is same at all points on the plane.

Conclusion:

The wave is a plane wave as the constant field is present on the given plane.