Calculating Intensity of 20-W Laser Beam After 10,000 m Travel

[twalters84]

Homework Statement

If a 20-W laser beam, which has an initial diameter of 2.0 mm, spreads out to a diameter of 2.0 m after traveling 10,000 m, what is the final intensity of the beam?

A. 20 x 10-4 W/m2
B. 20 x 10-6 W/m2
C. 20 x 10-8 W/m2
D. 6.4 W/m2
E. 6.4 x 10-6 W/m2

Homework Equations

I found an equation that states if light hits a surface area A at normal incidence, the intensity I is equal to:

I = <P> / A

The Attempt at a Solution

If I draw this beam out, I have a frustrum of a right circular cone.

The lateral surface area of a frustrum = PI * S * (R + r) where R and r are the radia and s is the diagonal side.

After a little math, I found s to be about 10000. It is a little over but its neglible.

So the surface area would be = PI * 10000 * (.001 m + 1.0 m) = 31447.3 m^2

Wouldnt that mean the intensity I = 20 W / 31447.3 m^2 = .000636 W / m^2

Intensity = 6.4 * 10^-4 W / m^2

Obviously, that is not one of the answers. What am I doing wrong here?

Any help will be greatly appreciated. Thanks in advance.


Travis Walters

 

[dlgoff]

Think of it this way. How much energy (watts) do you have through a 2m diam. circle?

 

[twalters84]

Hey there,

For reference, the answer key says 6.4 W/m2 is the correct answer.

The answer I got was 6.4 * 10^-4 W / m^2, so I am just wondering the answer is a typo.

I am not sure where you are getting at dlgoff? Please explain :)

Thanks in advance.


Travis Walters

 

[dlgoff]

Watts/area. 20W/(3.14xr^2) where r is the radius (1meter).