Power of EM Wave Absorbed by Circular Disk of Radius 2m

[NA19]

Homework Statement

A plane polarized electromagnetic wave propagates
with Erms = 30 V/m. What is the power transmitted to
a circular disk of radius r = 2m, if all of the light is
absorbed by the disk and S is perpendicular to the
disk?

Homework Equations

There is a passage that goes along with this that states that I =Erms^2/(cμ0), and that I = P/A. u0 = 4 × 10–7 N•s^2/C^2.

The Attempt at a Solution

Basically, I plugged in the values of Erms and μ0 into the Intensity equation and then multiplied by A (pi*4) to get 94.2.

But the answer key says the answer is 30 and that instead of multiplying by pi*r^2, you multiply by r^2. Please help!

 

[Simon Bridge]

Looks like you tried to apply the formulas without understanding them.
(You also left out your units.)

See also:
http://farside.ph.utexas.edu/teaching/302l/lectures/node119.html

You could lose a pi if it cancels out someplace, or it could be a mistake.

 

[NA19]

Where is the other pi?

 

[Simon Bridge]

You'll just have to try to understand the problem to see.

 

[Nickie]

As a scientist, it is important to approach problems like this with a critical and analytical mindset. While your initial attempt at solving the problem may seem reasonable, it is important to carefully consider all factors and equations involved.

Firstly, it is important to note that the equation I = Erms^2/(cμ0) is the correct equation for calculating the intensity of an electromagnetic wave. This equation takes into account the amplitude of the electric field (Erms), the speed of light (c), and the permeability of free space (μ0). The equation I = P/A is a simplified version that only applies to a plane wave propagating through a uniform medium.

Next, let's consider the units involved in the problem. The given electric field amplitude (Erms) has units of V/m, while the permeability of free space (μ0) has units of N•s^2/C^2. In order to get the intensity in units of W/m^2, we need to convert the electric field amplitude to its equivalent in units of W/m^2. This can be done by squaring the amplitude and dividing by the impedance of free space, which is the product of the speed of light and the permeability of free space (Z0 = cμ0). So, the correct equation for calculating intensity in this problem would be I = (Erms^2/Z0).

Now, let's consider the area of the circular disk. The given radius is 2m, so the area of the disk would be A = πr^2 = 4π m^2. Note that the area of the disk is not simply r^2, as stated in the answer key.

Putting all of this together, the correct solution would be:

I = (Erms^2/Z0) * A
= (30 V/m)^2 / (cμ0) * 4π m^2
= (900 V^2/m^2) / (3*10^8 m/s * 4*10^-7 N•s^2/C^2) * 4π m^2
= (900 * 10^-6 W/m^2) * (4π m^2)
= 3.6π mW/m^2

So the power absorbed by the circular disk would be 3.6π mW/m^2. It is important to carefully consider all equations and units