Derivation of [itex] p = \frac{E}{c} [/itex] using Maxwell's light

[Adesh]

Imagine that we have an electromagnetic wave or light propagating in $x$ direction, and $\mathbf{E}$ is oscillating in $z$ direction and $\mathbf{B}$ in $y$ direction. The picture looks something like this

Now, if there exists a charged particle $q$ on the xx axis at rest, then our B field can't do anything, but the electric field E will pull it upwards and as soon as it starts moving the magnetic field B comes into action and it goes like this $$ \mathbf{F_{mag}} = q (\mathbf{v} \times \mathbf{B}) $$ Since, E is going to cause a velocity in $z$ direction i.e. perpendicular to magnetic field B , therefore $$ F_{mag} = q~v~B$$
$$ F_{mag} = q~v~\frac{E}{c}$$
$$F_{mag} = v~\frac{(qE)}{c}$$
$$ F_{mag} = \frac{F_{electric}~v}{c}$$
$$ F_{mag} = \frac{dW_{electric}/dt}{c}$$ and after doing this he writes
That light carries energy we already know. We now understand that it also carries momentum, and further, that the momentum carried is always 1/c times the energy

After considering this line, I thought the last equation could be written as $$ \frac{dp}{dt} = \frac{1}{c} ~ \frac{dW}{dt}$$
$$ dp = \frac{dW}{c} $$
$$ p= \frac{E}{c}$$
But I find this thing sloppy because when I wrote $F_{mag} = \frac{dp}{dt}$ it was for the particle, that is force applied on the particle was equated with the rate of change of particle's momentum and not the light's momentum and similarly $dW$ is the energy imparted to the particle and not the energy of light itself.

So, these are my problems. I earnestly request you to please help me over here.

 

[mfb]

Use conservation of momentum. The momentum the particle gets must come from the radiation. Same for the energy.
It is not completely free of hand-waving, because you can also ask if the motion of the particle will emit new radiation on its own, but that is a smaller effect.

 

[Adesh]

Use conservation of momentum. The momentum the particle gets must come from the radiation. Same for the energy.
It is not completely free of hand-waving, because you can also ask if the motion of the particle will emit new radiation on its own, but that is a smaller effect.

But that should involve complete absorption of light then only we can call the charged particle’s momentum and energy to be equal to that of light, isn’t ?

 

[vanhees71]

The non-handwaving derivation of the conservation laws leads to Noether's theorem. However you need some more advanced mathematics than you find in the Feynman Lectures, i.e., Hamilton's principle for fields. Then the electromagnetic energy-momentum tensor follows from invariance under space-time translations (+ a gauge-symmetry argument to derive a manifestly gauge invariant symmetric energy-momentum tensor).

 

[mfb]

But that should involve complete absorption of light then only we can call the charged particle’s momentum and energy to be equal to that of light, isn’t ?

It doesn't matter how much energy transferred, as long as energy and momentum are transferred at a rate of pc=E. The remaining light will have the same energy-momentum relation as before, that is only possible if pc=E applies to the overall light as well.

 

[Adesh]

The non-handwaving derivation of the conservation laws leads to Noether's theorem. However you need some more advanced mathematics than you find in the Feynman Lectures, i.e., Hamilton's principle for fields. Then the electromagnetic energy-momentum tensor follows from invariance under space-time translations (+ a gauge-symmetry argument to derive a manifestly gauge invariant symmetric energy-momentum tensor).

What is handwaving?

 

[Ibix]

What is handwaving?

Handwaving is being non-rigorous. I think it comes from the habit many people have of waving their hand in the air when dismissing a question about details of an argument. So non-handwaving is rigorous.

 

[Adesh]

Handwaving is being non-rigorous. I think it comes from the habit many people have of waving their hand in the air when dismissing a question about details of an argument. So non-handwaving is rigorous.

Thank you. But why that derivation of Feynman was called hand-waving?

 

[Ibix]

Thank you. But why that derivation of Feynman was called hand-waving?

Oh I see! Sorry - early in the morning. I'll have to let vanhees71 answer that.

 

[vanhees71]

I haven't called it handwaving in the first place. I think, however, that most E&M textbooks just state the expressions for energy-momentum density and then derive the conservation laws by just using Maxwell's equations in a lengthy calculation using standard 3D vector analysis. That's fine but doesn't lead to a deeper understanding where the conservation laws come from, which is the great work by Emmy Noether of 1918. It's just relating the 10 general conservation laws to the symmetries of Galilei-Newton or Einstein-Minkowski spacetime.