Magnitude and average value of energy intensity of light beam

[dl447342]

Homework Statement

Explain why the magnitude of $\vec{S}$ , denoted $S$, fluctuates at twice the frequency of the wave. Explain why the average value of $S$ is half the instantaneous peak value.

Relevant Equations

I know $\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B},$ where $\mu_0$ is the magnetic permeability of the vacuum, and the other two variables represent the electric field vector and magnetic field vector. Also, $c = \lambda f,$ where $c$ is the speed of light, $\lambda$ is the wavelength, and $f$ is the frequency, and the momentum of a photon is $\frac{h}{\lambda}$, where $h$ is Planck's constant.

I tried using the equations above, but I wasn't really able to come up with an intuitive explanation. From my understanding, the electric field vector only varies in the x-y plane while the magnetic field vector only varies in the z-y plane. Also, both vary sinusoidally and both reach extrema (peaks or troughs) at the same points in time.

 

[hutchphd]

Herer is the short answer $$2\cos^2 \theta =1+ \cos 2\theta$$

 

[dl447342]

Herer is the short answer $$2\cos^2 \theta =1+ \cos 2\theta$$

Why is that the answer? I know that's true from basic trignometric identities, but what is $\theta$?

Could you provide some web links so I can learn more about this concept?

 

[hutchphd]

This is because E and B both contain $\cos$ and so the cross-product contains a $\cos^2$.
$\theta=2\pi\frac {(x-ct)}\lambda$
Does that help?

 

[Steve4Physics]

Homework Statement:: Explain why the magnitude of $\vec{S}$ , denoted $S$, fluctuates at twice the frequency of the wave. Explain why the average value of $S$ is half the instantaneous peak value.

The statements you are asked to explain apply to sinusoidal waves. They may apply to some other waveforms too, but I guess whoever set the question intended you to assume that the field strengths vary as, say, sin(ωt ) (or as sin(ωt + φ) if you want to be a bit more general).

Ideally, there should have been some mention of 'sinusoidal' in the question,

 

[dl447342]

The statements you are asked to explain apply to sinusoidal waves. They may apply to some other waveforms too, but I guess whoever set the question intended you to assume that the field strengths vary as, say, sin(ωt ) (or as sin(ωt + φ) if you want to be a bit more general).

Ideally, there should have been some mention of 'sinusoidal' in the question,

Okay. I think the waves have particular formulas that would make the computation of the magnitude of $\vec{S}$ easier.

 

[hutchphd]

So can you write an explicit equation for the magnitude of S ??

 

[dl447342]

So can you write an explicit equation for the magnitude of S ??

The formula for $\vec{E}$ is $\vec{E} = \hat{j} E_{max} \cos(kx - \omega t)$ and the formula for $\vec{B}$ is $\vec{B} = \hat{k} B_{max} \cos(kx - \omega t)$, where $\omega$ is the angular frequency and $k$ is the wave number, equal to $2\pi$ divided by the wavelength, $t$ represents time and $x$ represents the positive with respect to the origin along the x-axis.
Using the right-hand rule, where $\vec{B}$ is assumed to be along the z-axis while $\vec{E}$ is assumed to be on the y-axis), I get that $\vec{S}$ should point in the direction of the positive x-axis. So for an explicit equation, I was thinking of something like $\frac{E_{max}B_{max}}{2 \mu_0} (1+\cos 2(kx - \omega t))\hat{i}$ where $\hat{i}$ is the unit vector in the direction of the x-axis.

$\omega = 2\pi f$ where f is the frequency of "each electromagnetic wave" and so the magnitude of $S$ fluctuates at twice this value with respect to time as it has a factor of $2$ in $\cos 2(kx - \omega t)$.

 

[Steve4Physics]

Haven't checked the details but looks good. The working you hand-in should show the use of the trig' identity you used: cos²x = (1+cos(2x))/2.

Really there's no need to consider the vector aspects for the question as posed. The directions don't affect the answers.

To answer the part about the average being half the peak, you might want to consider S expressed in terms of cos²(...). Do you know what the graph y = cos²x looks like?

 

[hutchphd]

This looks good except you have the $\vec B$ in the $\hat x$ direction. It should be in $\hat z$ then $\hat y \times \hat z=\hat x$ which is what you want for $\hat S$. Also what is the "ax" the expressions?
In general I prefer $\hat x, \hat y, \hat z$ to i,j,k mostly because k is the wavevector also

 

[dl447342]

This looks good except you have the $\vec B$ in the $\hat x$ direction. It should be in $\hat z$ then $\hat y \times \hat z=\hat x$ which is what you want for $\hat S$. Also what is the "ax" the expressions?
In general I prefer $\hat x, \hat y, \hat z$ to i,j,k mostly because k is the wavevector also

Thanks. I fixed it.