Energy Carried by Electromagnetic Waves- why is it wrong?

In summary, the author has a problem with the answer to a homework problem that calculates the average electric and magnetic field energy densities at a specific location. The author calculates the energy density using different equations and comes to the same answer, but the answer is wrong. The author concludes that the problem with the answer is in the equation given in the book, but is not able to provide an equation that works correctly.
  • #1
edlin
7
0
Energy Carried by Electromagnetic Waves- why is it wrong?

Hi! I have this homework problem that I just don't understand why it's wrong.

A monochromatic light source emits 110 W of electromagnetic power uniformly in all directions.

(a) Calculate the average electric-field energy density 3.00 m from the source.

(b) Calculate the average magnetic-field energy density at the same distance from the source.

(c) Find the wave intensity at this location.



First of all, I know that (a) and (b) will be the same. I have also calculate part (c), which is correct, and it is 0.9726 W/m^2.


I have used various equations that would work for this type of problems.

1. The Sav, or average Poynting vector, in terms of the magnetic and electric field. I always get the same answer, which is 6.4*10^-8. Then I use the energy density equation, which gives me 3*10^-9.

2. I use Sav=cu...since I have Sav, and I know c, then it seemed logical to use this. My answer is the same, 3*10^-9.

But its WRONG, and I don't understand WHY! PLEASE HELP ME.

Thankyou!
 
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  • #2
First of all what does it mean that "A monochromatic light source emits 110 W of electromagnetic power uniformly in all directions."? I think it means that you have a source that radiates spherical waves and the average flux of the Poynting vectors through a spherical surface centered on the source is of 110W. In particular, taking the sphere to be of radius 3.00 m, this gives

[tex]110=<\oint_{\mathbb{S}_3} \vec{S}(\vec{r}=3\hat{r})\cdot\hat{n}dA >= <\int_0^{2\pi}\int_{0}^{\pi} ||\vec{E}(\vec{r}=3\hat{r})\times\vec{H}(\vec{r}=3\hat{r})|| 3^2 sin\theta d\phi d\theta >=36\pi <E(\vec{r}=3\hat{r})H(\vec{r}=3\hat{r})>[/tex]

But in a monochromatic plane wave propagating in the void, [itex]H=E/\mu_0 c=[/itex] so we have the equation

[tex]\frac{110\mu_0 c}{36\pi } =<E(\vec{r}=3\hat{r})^2>[/tex]

Now you can substitute that into the formula for the average mean electric energy:

[tex]<u_E(\vec{r}=3\hat{r})>=\frac{\epsilon_0<E(\vec{r}=3\hat{r})^2>}{2} =\frac{110\epsilon_0\mu_0 c}{72\pi }=\frac{110}{72\pi c}\approx 1.62\times 10^{-9} [J/m^3][/tex]

What is the correct answer?

(I don't know what equations are given to you; if you write them I can help you sort out your problem using those equations)
 
Last edited:
  • #3
I took a bit more simplistic approach and just calculated the energy density from the rate of energy flowing through the surface (power) and the speed of light. Looks to me like the total energy density is

u = P/Ac = 3.24*10^-9 J/m³
 
  • #4
Actually...the equations that are given in the book are very general, and none of them involve integrals.

I have used the third equatin below Poynting vector.



ph207-6-eqn15.GIF


And I also used Saverage/c=uaverage...but it's still wrong.
 
  • #5
I just divided the 3.24 by 2, which gave me the answer.

The reason why I did not get it, I think, is because the book does not present it correctly.

The Smax equation in the picture above is actually given as Sav in my book, so that's why the answer was not correct.

Thank you so much!
 

1. How does energy travel through electromagnetic waves?

Energy travels through electromagnetic waves via a combination of electric and magnetic fields that oscillate at right angles to each other. The energy is transmitted from one point to another through this oscillation, without the need for a medium.

2. Why is it wrong to say that electromagnetic waves carry energy?

While it is commonly stated that electromagnetic waves carry energy, this is not entirely accurate. Electromagnetic waves do not physically transport energy from one place to another, but rather the energy is transferred through the oscillation of the electric and magnetic fields. This is similar to how energy is transferred through a vibrating guitar string, rather than the string itself moving from one point to another.

3. Can energy be created or destroyed in electromagnetic waves?

No, energy cannot be created or destroyed in electromagnetic waves. Energy can only be transferred from one form to another. In the case of electromagnetic waves, energy is converted from electrical energy to the energy carried by the electric and magnetic fields, and then back to electrical energy when the waves are absorbed by an object.

4. How does the energy of electromagnetic waves relate to their frequency and wavelength?

The energy of an electromagnetic wave is directly proportional to its frequency and inversely proportional to its wavelength. In other words, as the frequency increases, the energy also increases, while as the wavelength increases, the energy decreases. This is described by the equation E = hf, where E is energy, h is Planck's constant, and f is frequency.

5. Is the energy carried by electromagnetic waves the same for all types of waves?

No, the energy carried by electromagnetic waves can vary depending on the type of wave. For example, the energy carried by a radio wave is much lower than that of a gamma ray. This is because the energy is directly proportional to the frequency, and different types of electromagnetic waves have different frequencies. Additionally, the amplitude of the wave can also affect the amount of energy carried, with larger amplitudes resulting in more energy.

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