# Electromagnetic Waves Question, regarding Magnetic Field and Electric Field

1. Dec 1, 2008

### TFM

1. The problem statement, all variables and given/known data

Write down the (real) electric and magnetic field for a monochromatic plane wave of amplitude E0, frequency ω, and phase angle zero that is

(a) Travelling in the negative x direction and polarized in the z direction;
(b) Travelling in the direction from the origin to the point (1, 1, 1), with polarization parallel to the x, z plane.

In each case, sketch the wave, and give the explicit Cartesian component of the vectors k and n.

2. Relevant equations

Not Sure I have:

$$E(z,t) = E_0e^{kz - \omega t}$$

and

$$B(z,t) = B_0e^{kz - \omega t}$$

3. The attempt at a solution

See, I have the formulas above, But I am not sure wether or not they are the right formulas for this equation? For instance, it is in terms of z, but the question has the first part moving in the x-axis???

TFM

2. Dec 1, 2008

### turin

You need to generalize
$$E,E_0,B,B_0,z,k$$
to vectors, i.e.
$$\vec{E},\vec{E}_0,\vec{B},\vec{B}_0,\vec{z},\vec{k}.$$
Also, then, you need to generalize the ordinary product,
$$kz$$
to the dot product
$$\vec{k}\cdot\vec{z}.$$

I'm sure this is what you meant, but there should probably be an $i\left(=\sqrt{-1}\right)$ in the exponent (or a "$j$" if you are an EE student).

Do you know vector calculus? Do you know Maxwell's equations?

Oh yeah, and don't forget that$v=c$, which imposes a relationship between $k$ and $\omega$ ...

Last edited: Dec 1, 2008
3. Dec 1, 2008

### TFM

I Thought that the I could be taken out, and the E has the squiggle above it (not a vetor, but a curly line) removed. My mistake.

So:

$$E(z,t) = E_0e^{i(kz - \omega t)}$$

$$B(z,t) = B_0e^{i(kz - \omega t)}$$

and turn into vector form. Should this be:

$$\vec{E}(\vec{z},\vec{t}) = \vec{E_0}e^{i(\vec{k} \cdot \vec{z} - \omega t)}$$

$$\vec{B}(\vec{z},\vec{t}) = \vec{B_0}e^{i(\vec{k} \cdot \vec{z} - \omega t)}$$

?

TFM

4. Dec 1, 2008

### turin

You are probably talking about the phasor, or whatever they call it in your class. They usually put a tilde ("squiggly line") above the "E" and "B" to indicate the complex generalizations of the electric and magnetic fields, and the complex phase is what represents the phase of the wave. You still need the exponent in the plane-wave part of the expression to be imaginary, otherwize the expression would represent an exponential decay or an exponentially increasing field, not an oscillating, propagating wave.

Yes, if you mean $$\vec{z}$$ is $$\hat{x}x+\hat{y}y+\hat{z}z.$$

5. Dec 1, 2008

### TFM

I'm not quite sure, where have you got this from?

TFM

6. Dec 1, 2008

### turin

I haven't decided if you're just having a problem with notation, or something deeper, but I don't know what else to say about the notation, so please don't get offended if the following is too basic for you.

Assume that space has three dimensions and that time is a single dimension. Then, in general, the wavevector is a three-dimensional object. A general plane wave is written
$$e^{i\left(\vec{k}\cdot\vec{r}-\omega{}t\right)},$$
where I have written $\vec{r}$ instead of $\vec{z}$ for the position vector, to save confusion. $\vec{r}$ has three components which are the three coordinates of the point in space. z is only one of those coordinates; x and y are the two others. Typically, one simply orients the coordinate system so that the wave vector points in the z-direction, which simply results in the expressions that you gave in your first post, because, in such a case, the y and z components of $\vec{k}$ would vanish. However, your problem is asking you to use the more general expressions.

Once you realize this, you must realize that a plane wave reduces Maxwell's equations to simple algebraic equations that relate the three vectors, $\vec{E},\vec{B},\vec{k}$. This is easy to show using vector calculus, but perhaps you are not at that level?

Tell me, are you a physics student, and engineering student, or what? And, what year? Actually, the best clue would be to tell me where you got those expressions for the plane wave in your first post.

7. Dec 1, 2008

### TFM

Okay. So that was a general case, so now

$$\vec{E}(\vec{\hat{x}x},\vec{t}) = \vec{E_0}e^{i(\vec{k} \cdot \hat{x}x - \omega t)}$$

$$\vec{E}(\vec{\hat{y}y},\vec{t}) = \vec{E_0}e^{i(\vec{k} \cdot \hat{y}y - \omega t)}$$

$$\vec{E}(\vec{\hat{z}z},\vec{t}) = \vec{E_0}e^{i(\vec{k} \cdot \hat{z}z - \omega t)}$$

and the same for B:

$$\vec{b}(\vec{\hat{x}x},\vec{t}) = \vec{B_0}e^{i(\vec{k} \cdot \hat{x}x - \omega t)}$$

$$\vec{B}(\vec{\hat{y}y},\vec{t}) = \vec{B_0}e^{i(\vec{k} \cdot \hat{y}y - \omega t)}$$

$$\vec{B}(\vec{\hat{z}z},\vec{t}) = \vec{B_0}e^{i(\vec{k} \cdot \hat{z}z - \omega t)}$$

And these will give you the x,y and z components?

TFM

Edit: I don't know how they are for you, but the latex isn't working for me. it should read: x(with hat)x, y(with hat)y and z(with hat)z

8. Dec 1, 2008

### turin

No, that wasn't the idea that I was trying to convey. What it looks like you're trying to write are three different specific cases: one when the wave propagates in the x direction, one when the wave propagates in the y direction, and one when the wave propagates in the z direction.

This is almost the idea that you need for the first part of the problem. For that, you also need to realize that $\vec{k}=-\hat{x}k_x$. Then, you need to realize the stuff about Maxwell's equations that I was talking about, but I don't want to say anymore about that until you humor my question regarding your area and level of coursework.

For the second part, you need to be more general that this. You need to realize that all three spatial components are relevant.

BTW, time does not usually get display as a vector. Just for yours and others' benefit, I will display what I think you meant.

$$\vec{E}\left(\hat{x}x,t\right)=\vec{E}_0e^{i\left(k_xx-\omega{}t\right)}$$
etc.
What we are actually going for is this:
$$\vec{E}\left(x,y,z,t\right)=\vec{E}_0e^{i\left(k_xx+k_yy+k_zz-\omega{}t\right)}$$
And, by choosing the components of k appropriately, you can choose the direction of the plane wave (i.e. $\hat{k}$ is the direction of the plane wave).

9. Dec 1, 2008

### TFM

Firstly, the vector over t is a Latex error, I didn't put it there (silly Latex )

I'm currently studying physics at 2nd Yr level.

Maxwell's Equations are:

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$

$$\nabla \cdot \vec{B} = 0$$

$$\nabla \times \vec{E} = \frac{\partial \vec{B}}{\partial t}$$

$$\nabla \times \vec{B} = \mu_0J + \mu_0\epsilon \frac{\partial \vec{E}}{\partial t}$$

For the last part, I thought about doing that, but for some unknown reason went against that.

$$\vec{E}\left(x,y,z,t\right)=\vec{E}_0e^{i\left(k_x x+k_yy+k_zz-\omega{}t\right)}$$

Okay?

TFM

10. Dec 1, 2008

### turin

Okay.

First, you need to decide what to set $\rho$ and $\vec{J}$ to. Then, you can put the general expressions for E and B into these four Maxwell's equations to obtain some "simple" algebraic equations that relate E, B, k, and omega. By simple, I mean that there will be one or two terms on each side of the equation that will be linear in (the components of) E, B, k, and omega. Or, you can leave them in abstract vector notation, but at any rate you need to get the algebraic equations as opposed to the differential equations.

Hint: the exponential factors should divide out after the derivatives act on E and B. Since they are always nonzero, this is OK to do.

Hint number 2: k and omega should be independent of space and time (that is part of the definition of the plane wave, although it usually is unstated).

11. Dec 2, 2008

### TFM

Well, $$\rho$$ ios the charge density on the plate, and J is the displacement current,

$$J = \epsilon_0 \frac{\partial E}{\partial t}$$

Will this help?

TFM

12. Dec 2, 2008

### turin

What plate?

No. In fact, you are talking here about Maxwell's crowning contribution, and the main reason why we name these equations after him. Notice in the equation with the curl of B that there are two terms on the other side. You are talking about the second term. This is one of the terms that will allow you to relate E, B, k, and omega (once you decide how to write the plane wave). The other term is the term that you have to "decide" outright.

Hint: what plate?
Hint 2: does the problem mention any charges, currents, or any material objects?
Hint 3: why did Maxwell introduce the displacement current?
Hint 4: the terms that you will calculate from the plane wave are the two divergences, the two curls, and the two time derivatives.

13. Dec 2, 2008

### TFM

I Think I got confused , the plate is for a capacitor - wrong question

J is the current density

$$\nabla \times \vec{B} = \mu_0J + \mu_0\epsilon \frac{\partial \vec{E}}{\partial t}$$

Maxwells Equation.

Maxwell added the final part because the div of the B curl = 0, but the other side didn't = 0.

TFM

14. Dec 3, 2008

### TFM

Is the above useful?

TFM

15. Dec 3, 2008

### TFM

I have also found some more general formulas that may or may not be useful:

$$\vec{E} (r,t) = \vec{E_0} e^{i(\vec{k} \cdot \vec{r} - \omega t)}\hat{n}$$

$$\vec{B} (r,t) = \frac{1}{C} \vec{E_0} e^{i(\vec{k} \cdot \vec{r} - \omega t)}(\hat{k} \times \hat{n} = \frac{1}{c}\hat{k} \times \vec{E}$$

Are these useful?

TFM

16. Dec 3, 2008

### turin

The mathematics of this problem are, what I would consider to be, quite trivial. I think the point of the exercise is to be able to decide what are the important equations to use. If you plug into Maxwell's equations, you will see immediately what these are. They may or may not be similar to the "equations" that you have written here. (I hope that those are merely LaTeX errors.)

I think that you are having a lot of trouble with the very basics of E&M, and you should concentrate your efforts on appreciating the meaning of Maxwell's equations, even if you don't solve this particular problem in time. I'll say it again: this problem becomes trivial once you understand what Maxwell's equations say (and what they don't say). I'll give you another hint, or way to think about it: the input to Maxwell's equations are the sources (rho and J); the output from Maxwell's equations are the fields (E and B). If you already assume that you have a plane wave, then you have already implicitly assumed what your sources are.

Hint number 2: the "plane wave solution" to Maxwell's equations is sometimes referred to as the "free-field solution".

17. Dec 4, 2008

### TFM

Okay, looked up the Maxwell Equations in the PF Glossary

$$\rho$$ is the Free charge density.

$$J$$ is the free current density

But I am not sure where these charge density and current density are? This is a wave made from Magnetic Fields and Electric Fields I know, but where would the current/charge come from?

TFM

18. Dec 4, 2008

### turin

OK, I'm just goint to tell you, partly because my hints seem to be worded in confusing terminology compared to the literature that you are using, and partly because this is painful to watch.

J=0, rho=0

These are assumptions that you make in order to get plane wave solutions to Maxwell's equations. (a better hint might have been "homogeneous solutions" - oh well)

Now, plug your plane waves into Maxwell's equations and show me your result. (Maybe try using plain text.)

19. Dec 4, 2008

### TFM

Okay so:

Curl E = dB/dt
Curl B = mu_0 0 + mu epsilon dE/dt = mu epsilon dE/dt

Div E = 0/epsilon_0 = 0
Div B = 0

So only the curls remain

Curl E = dB/dt
Curl B = = mu epsilon dE/dt

Also, For Future Reference, would you assume J = 0 and rho - 0 since we are dealing with a wave, and there isn't any real charge density/Current density?

TFM

20. Dec 4, 2008

### turin

A wave could travel through, for example a charged plasma. But basically, yes, when you consider the propagation of waves in E&M, you should assume J=0 and rho=0 unless you have a good reason not to. However, be careful, because there may be some charge or current somewhere else (e.g. a dipole source, or conducting surfaces) that affects the geometry of the waves (i.e. they may not always be plane waves).