Problem 2

 

2.) (15 points) Suppose you want to use the radiation presure from a beam of light to suspend a piece of paper horizontally. The paper has an area of 50 cm² and a mass of .2 g. Assume there is no problem with balancing the paper and that it is dark and absorbs all the radiation incident upon it.  
a)   How many watts of power would there have to be in the beam, assuming that area of the beam was the same as the area of the paper?
 
b)   Assuming you could find a laser as powerful as needed, why is this experiment still not physically possible?

Answer:
The forces must balance for the paper to be suspended.
m g = P_r A
Where P_r is the radiation pressure and A is the area.
The radiation pressure is given by
$P_{r} = \frac{S}{c}$
Where the value S is the magnitude of the Poynting vector. The power is given by
P = S A
So, combining these equations, we have for the power,
$S = \frac{m g c}{A}$
and $P = \frac{m g c}{A} A$
P = m g c
Putting in the numbers, we have
$P = (.0002 kg)(9.8 m/s^{2})(3 \times 10^{8} m/s)$
$P = 5.88 \times 10^{5} W$
P = 588 kW
 
That much power incident on a piece of paper weighing .2 g, would incinerate it completely in a very small fraction of a second.