How Is the Formula for Radiation Pressure on a Reflecting Surface Derived?

[premed]

for a reflecting surface, the radiation pressure is equal to 2I/c where I is the intensity of light and c is the speed of light. I saw this in my physics homework manual but it was never mentioned in the book. thanks.

 

[olgranpappy]

First, recall that the intensity I is the average of the poynting (sp?) vector, so it is the energy flux. I.e., it's the number of incoming photons times their energy divided by time and divided by area (that's what "flux" means "whatever" flux is "whatever" per unit time per unit area)... that is
$$I=\frac{\hbar\omega N}{\rm time*Area}$$
where N is the number of photons and \hbar\omega is the energy of a single photon.

So

$${\rm Pressure}=\frac{\rm Force}{\rm Area}$$
and the Force is the total momentum transferred divided by the time. For reflection the photon bounces off so that it transfers an amount of momentum $\Delta p=2p$,
so the total force due to N reflecting photons is (2pN)/(time)
$${\rm Pressure}=\frac{2pN}{\rm time*Area}=\frac{2\hbar\omega N}{c{\rm * time*Area}} =\frac{2I}{c}$$

wheras, for absorption the momentum imparted by a single photon is $\Delta p=p$, which gives
$$P=\frac{I}{c}$$

 

[sir-pinski]

The formula for radiation pressure on a reflecting surface is derived from the principles of electromagnetism and the behavior of light. When light strikes a surface, it exerts a force known as radiation pressure. This force is caused by the transfer of momentum from the photons of light to the surface.

The intensity of light, represented by the variable I, is a measure of the amount of energy carried by the light per unit area. This means that the higher the intensity of light, the more photons are present and the greater the force of radiation pressure.

The speed of light, represented by the variable c, is a fundamental constant in physics that describes the speed at which light travels in a vacuum. This constant is used to convert the energy of light (measured in watts) to a force (measured in newtons).

In order to derive the formula for radiation pressure, we use the equation for momentum, which states that the momentum of an object is equal to its mass multiplied by its velocity. In this case, the object is a photon of light and its velocity is the speed of light.

By equating the momentum of a photon to the force of radiation pressure, we can derive the formula 2I/c. This means that for every unit of intensity, there is a corresponding force of 2/c acting on the reflecting surface.

It is important to note that this formula is only applicable to perfectly reflecting surfaces, as some energy will be absorbed or transmitted by other surfaces. Additionally, this formula assumes that the light is hitting the surface at a perpendicular angle.

In summary, the formula for radiation pressure on a reflecting surface is derived from the principles of electromagnetism and the behavior of light. It is a fundamental equation in physics that helps us understand the effects of light on surfaces.