Find the intensity of this wave and calculate the energy flowing?

Question

A traveling sinusoidal electromagnetic wave in vacuum has an electric field amplitude of 52.7 V/m.

Find the intensity of this wave and calculate the energy flowing during 16.9 s through an area of 0.0275 m2 that is perpendicular to the wave's direction of propagation.

Intensity= ? W/m^2

Energy= ? J

I keep getting this problem wrong and I think I might be using the wrong formulas.

Any help would be appreciated & writing out the steps help a lot!

Answer 1

For the calculation of I = E_rms² / (cμ₀), I get

I = (52.7 / √2)² / [(2.99 x 10⁸) (1.26 x 10^-6)] = 3.69 W/m²

 

t = 16.9 s

 

A = 0.0275 m²

 

U = energy = IAt = (3.69) (0.0275) (16.9) J = 1.71 J

Answer 2

Hi Payton!

 

The intensity of the wave, the time rate of energy flow per unit area (there's a mouthful! :) ), is related to a time-averaged version of a quantity called the Poynting vector, and takes the formula:

 

I = (1/cμ_o)E²

 

This E is often taken to be what is called the "root-mean-square" electric field (a kind of average).  The way the problem is worded above, it sounds like you are given the amplitude (or the maximum amplitude) of the electric field.  In this case, you would have to calculate the "root-mean-square" or "rms" value through the relationship:

E_rms = E_max/√2

 

Also:

 

c = speed of light

μ_o = permeability of free space (a universal constant)

 

Both of these values, as you probably know, can be looked up.

 

With these assumptions, I get:

 

I = (1/[(2.99x10⁸ m/s)(1.26x10^-6)])[(52.7 V/m)/√2]² = 3.69 W/m²

 

[Is that correct for this part?  If not, they could mean you to put the E value they give directly into the intensity expression, in which case you would not divide by √2 at the end, and you would get I = 0.14 W/m²]

 

The intensity should be pretty small here, as this is not a very large magnitude for an electric field.

 

I = 3.69 W/m² = 3.69 J/(m²·s)

 

So this is the energy flowing per second per meter squared for the wave.  To get the total flowing through some area in some time, you just have to multiply the intensity by the total area and the total time you have.

 

Energy in 16 s through 0.0275 m²:

 

Energy = 3.69 J/(m²·s) * (0.0275 m²)*(16.9 s) = 1.71 J

 

Let me know if that is not correct, and we can look at it more closely together.  But these should be the quantities and formulas to use, I suspect.  I hope this helps some!