Radiation Pressure: Solving for Magnetic Field

[jarfungus]

Hello! First time on this site, so I hope I do this right. I have a homework question that I could use some help on:

(From Physics for Scientists and Engineers, Third Ed. Fishbane, Gasiorowicz, Thornton)

Chapter 34 #39.
Suppose that you want to use the radiation pressure from a beam of light to suspend a piece of paper in a horizontal position; the paper has an area of 50 cm^2 and a mass of 0.20 g. Assume that their is no problem with the balance, that the paper is dark and absorbs the beam fully, and that the entire beam can be used to hold the paper against the pull of gravity. How many watts must the light produce? Given your answer, what do you think will happen to the paper?

Since power=c*u*A, I can easily calculate the amount, right? The problem comes when solving for u. I know that u=1/u0*B^2. But how can I find the magnitude of the magnetic field with the information given? Would it be right to manipulate the formulas so that B=E^2/(sqrt(1-c^2))? (Then what is E?) ...

 

[jarfungus]

I got it:)

Nevermind! I think I figured it out. I forgot the u=F/A, and the force is m*g...so I am all set:)
Thanks for all the viewing.

 

[quicknote]

Hello! I would approach this problem by first understanding the concept of radiation pressure. Radiation pressure is the force exerted by electromagnetic radiation on an object. In this case, we are using a beam of light to exert a force on the paper.

To solve for the magnetic field, we need to use the formula P=rho*c, where P is the radiation pressure, rho is the energy density of the radiation, and c is the speed of light. We can rearrange this formula to solve for the energy density, rho=P/c.

Next, we can use the formula for the energy density of electromagnetic radiation, rho=(1/2)*epsilon_0*E^2, where epsilon_0 is the permittivity of free space and E is the electric field strength. We can rearrange this formula to solve for the electric field strength, E=sqrt(2*rho/epsilon_0).

Now, we can use the formula u=1/u_0*B^2 to solve for the magnetic field strength, where u_0 is the permeability of free space. Substituting our value for E into this formula, we get u=2*rho/(u_0*epsilon_0). Finally, we can substitute our value for rho into this equation and solve for B.

Once we have the value for the magnetic field, we can use the formula P=rho*c to solve for the power of the light beam. This power is the amount of energy per unit time that the light beam is delivering to the paper.

Based on the calculations, it seems that the power of the light beam needed to suspend the paper is very high. This suggests that the paper will likely be pushed away from the light beam due to the force of radiation pressure. However, this also depends on the distance between the light source and the paper, as well as the orientation of the paper with respect to the light beam. Further experiments and calculations would be needed to determine the exact behavior of the paper. I hope this helps!