Which Direction Is the Electromagnetic Wave Traveling?

[roam]

Homework Statement

At a particular point, the electric field associated with an electromagnetic wave is oriented in the +y direction, and the magnetic field in the +x direction. In which direction is the wave travelling?

Also, what if they ask:

At a particular point, the electric field associated with an electromagnetic wave is oriented in the -y direction, and the magnetic field in the -x direction. In which direction is the wave travelling?

The Attempt at a Solution

I know that if an electromagnetic wave travels in the positive x direction then the electric field E is in the y direction and the magnetic field is in the z direction. But here in the first question for example, what does the it mean by saying "electromagnetic wave is oriented in the +y direction"? How do I work out the direction of the wave from that?
I have a test tomorrow so I appreciate if anyone could show me how to answer problems of this sort.

 

[kuruman]

At a particular point, the electric field associated with an electromagnetic wave is oriented in the +y direction, and the magnetic field in the +x direction. In which direction is the wave travelling?

There is a rule involving a cross product that relates the direction of the electric and magnetic fields to the direction of propagation. What is this rule?

Also, what if they ask:
At a particular point, the electric field associated with an electromagnetic wave is oriented in the -y direction, and the magnetic field in the -x direction. In which direction is the wave travelling?

Follow the rule the same way.

But here in the first question for example, what does the it mean by saying "electromagnetic wave is oriented in the +y direction"? How do I work out the direction of the wave from that?

You skipped part of the question. It says

At a particular point, the electric field associated with an electromagnetic wave is oriented in the +y direction, and the magnetic field in the +x direction. In which direction is the wave travelling?

It is the electric field that has the direction, not the wave.

 

[ehild]

"The electric field is oriented in the y direction" means that the electric field vector points in the +y direction.
If the wave travels in the positive x direction the E vector can point in any direction in the (y,z) plane, and the magnetic field vector is also in the (x,z) plane but perpendicular to E.
The electromagnetic wave is a transverse wave so both E and H are perpendicular to the direction of propagation in free space.
There is a vector associated with the direction of propagation of the electromagnetic wave. What is the name and how is it defined?

ehild

 

[roam]

Are you referring to the Poynting vector?

$$\vec{S}=\frac{1}{\mu_0}\vec{E}\times \vec{B}$$

The poyniting vector is along the direction of wave propagation. So do I just need to substitute the signs for E (electric field) and B (magnetic field) into it?

For question 1: $$\vec{S}=\frac{1}{\mu_0}\vec{+y}\times \vec{+x}$$

where $$\mu_0 = 4 \pi \times 10^{-7}$$

How can I tell the direction from this? Besides I don't know what the cross product equals to... I mean if we had ExB=EB we'd get $$S= \frac{EB}{\mu_0}$$. How do I need to evaluate the cross product?

 

[kuruman]

Are you referring to the Poynting vector?

Yes, I am referring to the Poynting vector. It (being a vector) has a direction, which is the direction of the wave propagation. How do you find the direction of a cross product? You only need the direction for this, not the magnitude.

 

[ehild]

http://mathworld.wolfram.com/CrossProduct.htmlehild

 

[roam]

I think the direction must be perpendicular to the other two (it shouldn't align with electric field and magnetic field). Since E=+y and B=+x, the poynting vector must be perpendicular to them, thus it must have direction +z. But this is not the correct answer. Why?

 

[kuruman]

I think the direction must be perpendicular to the other two (it shouldn't align with electric field and magnetic field). Since E=+y and B=+x, the poynting vector must be perpendicular to them, thus it must have direction +z. But this is not the correct answer. Why?

I agree that it must be perpendicular to the other two. But "perpendicular" is either along +z or along -z. How can you tell (without knowing the correct answer) which of the two is the case? Back to my original question that you have not answered so far: How do you find the direction of the cross product ExB?

 

[roam]

How do you find the direction of the cross product ExB?

I don't know what you mean by finding the direction of cross product ExB? Could you show me the process? The direction of ExB relative to E and B can be found by a "right-hand rule" since it's perpendicular to the other two. But you are right, even using this rule we can get either along +z or along -z. But how could we possibly know which one is the right answer? I really need to know this right now!

 

[ehild]

Apply the right-hand rule. What does it state?

ehild

 

[roam]

Apply the right-hand rule. What does it state?

ehild

It says if the fingers of right hand are cupped so that they curl to form E toward B in the direction of rotation that takes E to B in less than 180 degs, the the thumb will point in the direction of ExB. So if I hold my hand like that... then my thumb points up in +z direction. But this not the correct answer!

 

[kuruman]

Let's make sure you got the direction of the axes correct. Suppose you draw x and y axes on a sheet of paper in the conventional manner, +y is vertically up (bottom of page to top) and +x is horizontally to the right. The +z axis must be perpendicular to the page but in which direction do you think. Out of the page or into the page?

 

[ehild]

Did Roam give up? We used the right-screw rule to get the direction of the vector product. Turn the first vector into the second one with the angle less than 180°. If you would do the same motion with a screw driver, the motion of a right screw is in the direction of the product vector.

If the unit vectors of the Cartesian system of coordinates are denoted by i for the x direction, j for y and k for z,
ixj=k. If x points to the right and y up, turning the x-axis into y will unscrew that screw...

ehild

 

[roam]

Yeah... I gave up! I got this question in the test and got it wrong. The correct answer to the first one was -z not +z! :(