How much energy does light transfer to an object?

[alanf]

On my E&M final, a question asked 1) how much energy a light beam transferred to a non-reflective object in x seconds (3 joules), 2) what pressure the light exerted on the object during that time, and 3) what energy would be transferred to a perfectly reflective mirror by the same beam of light. The answer to 3 was zero, which seems intuitive - 100% of the energy has to be returned into space - yet the pressure on the mirror should be twice that on the absorptive object. So it should gain momentum. So how can it gain no energy?

 

[jbriggs444]

The answer to 3 was zero, which seems intuitive - 100% of the energy has to be returned into space - yet the pressure on the mirror should be twice that on the absorptive object. So it should gain momentum. So how can it gain no energy?

It does gain an exceedingly tiny bit of energy. As you say, the mirror gains a tiny bit of momentum. In the frame of reference where the mirror starts at rest, the light that it reflects is red-shifted by a tiny amount and thereby loses a tiny bit of energy.

On the other hand, if you adopt the frame of reference where the mirror ends at rest, the mirror loses energy and the light is blue-shifted.

 

[A.T.]

So it should gain momentum. So how can it gain no energy?

In the limiting case, where the mass of the mirror goes to infinity, the gain in kinetic energy goes to zero while the change in momentum stays finite. That's because one depends on velocity squared, while the other just on velocity.

 

[anorlunda]

The reference frame is missing from the problem statement. Kinetic energy is frame dependent, therefore the energy question can not be answered without choosing a reference. jbriggs444 said it in post #3

It does gain an exceedingly tiny bit of energy. As you say, the mirror gains a tiny bit of momentum. In the frame of reference where the mirror starts at rest, the light that it reflects is red-shifted by a tiny amount and thereby loses a tiny bit of energy.

On the other hand, if you adopt the frame of reference where the mirror ends at rest, the mirror loses energy and the light is blue-shifted.

 

[alanf]

Thank everyone. I knew that the mirror had to pick up some energy from the light. It was kind of an odd question to put on the final, since the prof had made an effort in lectures to point out that reflective objects are under twice the radiation pressure of absorptive objects.

 

[russ_watters]

Just to be clear, though: pressure isn't energy and doesn't necessarily cause an energy transfer.

In the limiting case, where the mass of the mirror goes to infinity, the gain in kinetic energy goes to zero while the change in momentum stays finite.

Or if the light source and mirror are both fixed to the same object, the energy transfer is exactly zero.

 

[A.T.]

Or if the light source and mirror are both fixed to the same object, the energy transfer is exactly zero.

Why?

 

[russ_watters]

Why?

Just to clarify, that was kinetic energy due to the motion of the object that is hit. If they are stationary wrt each other, they can't get kinetic energy wrt each other.

Now that I think about it though, that's still an open system and the object they are sitting on could still gain kinetic energy wrt an outside frame.

 

[jbriggs444]

I do not view it as having anything to do with an outside object. It is a simple transaction between the light and the mirror. The energy that is transferred between them will net to zero -- whatever is gained by the light will be lost by the mirror and vice versa. However, which one gains and which loses is determined by the choice of reference frame, not by anything physical.

 

[A.T.]

Now that I think about it though, that's still an open system and the object they are sitting on could still gain kinetic energy wrt an outside frame.

There are no "outside" and "inside" frames, just inertial and non-inertial ones. The rest frame of the connected mirror and source is non-inertial, so energy isn't even conserved in that frame.

 

[russ_watters]

There are no "outside" and "inside" frames, just inertial and non-inertial ones. The rest frame of the connected mirror and source is non-inertial, so energy isn't even conserved in that frame.

I know I'm changing the scenario even more, but if we add a 3rd object, also connected to the mirror and source, located behind the source and colored black, doesn't this device now become non-accelerating by virtue of the fact that it is now closed, so the photons don't escape?

 

[jbriggs444]

Is the light considered to be part of the device or separate from the device? You don't close a system by putting physical boundaries around it. You close a system by putting notional boundaries around it.

 

[A.T.]

I know I'm changing the scenario even more, but if we add a 3rd object, also connected to the mirror and source, located behind the source and colored black, doesn't this device now become non-accelerating by virtue of the fact that it is now closed, so the photons don't escape?

Yes, except for the period when you switch on the the lights source, but the photons haven’t reached the black screen yet.