Radiation pressure and electromagnetic waves

[jdismu1]

Homework Statement

A laser beam of power P and diameter D is directed upward at one circular face (of diameter d < D) of a perfectly reflecting cylinder. The cylinder is levitated because the upward radiation force matches the downward gravitational force. If the cylinder's density is ρ, what is its height H? State your answer in terms of the given variables, using c and g if needed.

http://edugen.wiley.com/edugen/courses/crs4957/art/qb/qu/c33/q26.jpg

Homework Equations

Intensity = P/A
F = 2IA/c <-- for totally reflecting surface
\rho = m/V
F(gravity) = mg

The Attempt at a Solution

i set the two forces equal to each other because the cylinder is floating.

F(due to beam)=F(gravity)
F(due to beam) = 2IA/c
F(gravity)=mg
F(gravity)=\pi*r^2*h*\rho)*g <-substituted density (constant rho) and volume (\pi*r^2*h) for mass

**i think this is where i may be going wrong**
F(due to beam)= 2IA/c = 2P/c <-- substituted in the equation of intensity where I = P/A, power divided by area.

after this i solved for H and got, i also substituted r = d/2 because r was not a variable to use in this problem

2P/(\pi*(d/2)^2*h*\rho*g*c) = H

 

[jdismu1]

Figured it out

first change the equations to eliminate all variables that are not used in the problem.

F_r = 2IA_disk/c <-- area in this equation is only concerned with the area of the disk that is reflecting the beam, not the total area of the beam

intensity was not given as a known value so we have to change it to appropriate variables

I = P/A_beam <-- this is where i was wrong, the area in this equation is with respect to the beam as a whole and is only concerned with the cross sectional area at the bottom of the cylinder

A_beam = (pi)*r²
because r was not given as one of the appropriate variable i had to change it to D
so now we have

A_beam = (pi)*D²/4 <-- (D/2)² = D²/4

plugging the new value of I into the equation you get

F_r = 8A_diskP/[c*(pi)*D²] <-- bring the 4 up to the top and combine it with 2 to get 8 XD

next for F_g=mg, m is not given as a know value so we have to change it to things that we know

m = V(rho) <-- rho is density here

volume is also not given so we have to change it yet again

V = (pi)*r²*h <-- for convenience ill use (pi)*r²*h = A_disk*h

now we have a value for m that we can use

m = A_disk*h*(rho)

finally F_g = mg
F_g = A_disk*h*(rho)*g

set the two equation equal to each other because forces are balance and the cylinder is floating in air

F_r=F_g

plug in know values for the forces

8A_diskP/[c*(pi)*D²]=A_disk*h*(rho)*g

Areas of the disk cancel and bring everything else except for H over to the other side and the answer is

h = 8P/[c*(pi)*D²*g*(rho)]