## Friday, November 17, 2023

### Momentum of Light: UNIZOR.COM - Physics 4 All - Waves - Electromagnetic ...

Notes to a video lecture on http://www.unizor.com

Momentum of Light

In this lecture we will deal with electromagnetic field in vacuum with a flat wave front consisting of sinusoidal monochromatic (same frequency) synchronous (same phase) oscillations, which we will simply refer to as light.

We know that light carries energy.
More precisely, as we have established earlier (lectures Electric Field Energy and Magnetic Field Energy of topic Energy of Waves of the current part Waves of the course), the total electric + magnetic energy density of electromagnetic field (that is, an amount of energy per unit of volume) is
PE+M = ½·[ε·E²+(1/μ)·B²]
where E(t,x,y,z) is an intensity of electric component and B(t,x,y,z) is a magnetic component's intensity of the field.

In our case of monochromatic sinusoidal oscillations in vacuum this energy density expression is simplified to (see lecture Electric Flux Density, where we used B=E/c relation)
PE+M = ε0·E²

When the light hits an absorbing surface of some object, this energy is transferred into this object causing moving its electrons, heating and other manifestations, even mechanical movement of an object.
Let's explain the nature of this transfer of energy. On the picture above the light propagates along X-axis, electric component E of the electromagnetic field oscillates along Y-axis and magnetic component B oscillates along Z-axis.

As vector E oscillates up and down, negatively charged electrons on an object's surface move down and up.
The Lorentz force on these electrons caused by their movement in the magnetic field B pushes them forward along X-axis.
When the direction of vector E changes to opposite, the direction of electrons' movement and vector B also change to opposite. As a result, the Lorentz force will still push the electrons forward along X-axis in the same direction.

So, electrons inside the surface layer of an object are always pushed in the same direction with pulsating force, exerting a radiation pressure on the entire object.
Let's analyze the quantitative characteristic of this pressure.

Assume the flat surface of some object is perpendicular to the direction of light propagation and its area is A.
Let the density of electric charge on this surface be σ, so the total charge on this surface is q=σ·A.

The sinusoidal electric component E(t) acts on this charge q, causing electrons to move up and down with velocity v(t).
The Lorentz force on this charge consists of electric component moving electrons along Y-axis and magnetic component pressing electrons perpendicularly to their velocity vector along X-axis.
The total force on the electrons in vector form is, therefore,
F = q·E + q·(vB)

Since only B component pushes electrons in the direction of light propagation and, as we mentioned above, B = E/c, the force that pushes object along the X-axis is
Fx(t) = q·v(t)·E(t)/c
Notice that v(t) is the speed of electrons along the Y-axis, the same axis the electric component of the field E(t) acts.

Let's deviate for a moment from the electromagnetic field and consider classical Newtonian Mechanics.
Recall the Newton's Second Law of Mechanics connecting the force F, mass m and acceleration a:
F(t) = m·a(t)

Since acceleration a(t) is a derivative of speed v(t) by time, the above can be transformed into a relation between an impulse of force and an increment of an object's momentum
F(t) = m·[dv(t)/dt] =
=
d
[m·v(t)]/dt
from which follows
F(t)·dt = d[m·v(t)] = dp(t)
where p(t) is a momentum of an object.

Using the above relation between an increment of impulse F(t)·dt and an increment of momentum dp(t), we can state that during an infinitesimal time from t to t+dt the force Fx(t) gives an object a push forward in the X-direction quantified as impulse Fx(t)·dt which is converted into an increase of momentum of an object px(t):
Fx(t)·dt = dpx(t)

Using the formula for Fx(t) above, the expression for an increment of a momentum of an object is
dpx(t) = (1/c)·q·v(t)·E(t)·dt

Let's analyze and interpret the right side of the equation above.

First of all, a product of a charge q and the intensity of an electric component E(t) is the electric force the field exerts upon a charge:
Fe(t) = q·E(t)

Secondly, expressing the speed v(t) as a derivative of the distance of electrons' movements along Y-axis, that is v(t)=dy(t)/dt, we can write Fe(t)·v(t) = Fe(t)·dy(t)/dt =
=
dW(t)/dt

where W(t) is work performed by the field to move electrons along Y-axis.

Substituting all the obtained equalities into the formula for an increment in momentum of an object above, we obtain
dpx(t) = (1/c)·[dW(t)/dt]·dt =
= (1/c)·
dW(t)

The equation above relates increment of momentum of an object in the direction of the light propagation (X-axis) and amount of work done by this light.

The Law of Conservation of Energy then dictates that the light by its electric intensity component E(t) has performed this work by transferring its own energy to move the electrons on an object's surface along Y-axis.
That, in turn, caused these electrons to push along X-axis because of action of light's magnetic intensity component B(t).
Finally, this caused a movement of an object along X-axis and increased its momentum in the direction of propagation of light.
The amount of this lost energy of light, divided by the speed of light, equals to an increment of a momentum of a movement of an object along X-axis.

But there is also the Law of Conservation of a Momentum. Therefore, that increment of a momentum of an object must be equal to a decrement in light's momentum.
That means:
(a) light carries an energy and a momentum
(b) when light is completely absorbed by a surface of some object, its energy and momentum are transferred to this object and the decrement of light's momentum equals to the decrement of light's energy divided by the speed of light.
dpx(t) = (1/c)·dW(t)
(c) an object absorbing light absorbs its energy and momentum in the same proportion

Incidentally, since
Fx(t)·dt = dpx(t)
we can express the force exerted by light on an object fully absorbing this light as
Fx(t) = dpx(t)/dt =
= (1/c)·
dW(t)/dt

The expression dW(t)/dt represents amount of energy carried by light in a unit of time.

Dividing both sides by the area A on which light falls, we will get a radiation pressure P(t)=Fx(t)/A on the left side and energy flux density divided by the speed of light on the right side:
Fx(t)/A = (1/c)·[dW(t)/dt]/A

As presented in the previous lecture Electromagnetic Energy Flux Density, the energy flux density can be expressed as Poynting vector S(t)=(1/μ)·E(t)B(t).
Therefore, in terms of Poynting vector, a radiation pressure on an object fully absorbing the light is
Pabs(t) = S(t)/c

Everything above was about the case of full absorption of the light by an object (the object fully absorbing the light is called blackbody).

Let's assume now, we have a fully reflective object.
That means that, if an incident light had momentum p, the reflected light will have momentum −p.

From the Law of Conservation of Momentum follows that, if the momentum of the light was p and became −p, the increment of the momentum of an object should be 2p.

That means, the force of radiation pressure in case of fully reflecting object should be twice as strong as in case of fully absorbing object, that is
Pref(t) = 2·S(t)/c
So, a light reflective object feels twice as strong radiation pressure than a light absorbing object.

The following experiment confirms this.

Four small squares with one side black and another being a reflective mirror are arranged like a propeller that can freely rotate on a needle inside a sphere with vacuum inside.

As soon as light falls on this propeller, it starts rotating because the mirror side has more radiation pressure than the black one. 