Magnetic fields and conservation of energy

[esenor]

I'm a new teacher to E&M and confess I was a lowly Chemistry Major who needs some help.

A proton and a particle of equal mass without charge are somehow provided equal energy directed, at least at onset, in the form of translational energy. Will the uncharged particle move faster because some of the translational energy of the proton will be used to create a magnetic field around it as it moves?

 

[cesiumfrog]

You're basically (almost) asking, does charge have weight?

The answer is kind-of (look up "self force"). When you accumulate charges together, you increase the electric potential energy of your conglomerate. Since energy is equivalent to relativistic mass, the result is that the conglomerate requires more input of energy in order to increase its speed. So it should be harder to speed up a proton than to speed up the three separate quarks that a proton is made of. A pith-ball speeds up less when its surface acquires a significant deficit or excess of electrons.

If you had asked about a fundamental particle (say, electron vs "neutral fundamental particle with same rest mass as the electron") then classical E&M breaks down (gives infinity) and QM says that the charge contributes zero to the mass of the particle.

A problem with your original question is that one *definition* of mass *is* the amount of energy required to change something's speed. (It may also help avoid confusion if you look up "relativistic mass" vs "rest mass", and whether a hot potato is heavier than a cold one.)

 

[Meir Achuz]

I'm a new teacher to E&M and confess I was a lowly Chemistry Major who needs some help.

A proton and a particle of equal mass without charge are somehow provided equal energy directed, at least at onset, in the form of translational energy. Will the uncharged particle move faster because some of the translational energy of the proton will be used to create a magnetic field around it as it moves?

The self field of a charged particle does not effect the relation between its translational kinetic energy and its velocity. The charged and uncharged particles will have the same velocity.

 

[atyy]

First note that radiation is an electromagnetic wave, and so if a charge emits radiation, then it will lose energy.

A charge moving at constant velocity does not radiate. The way to see this is that the laws of electromagnetism are the same for any reference frames moving at a constant velocity (via a Lorentz transformation). So we can use a reference frame where the charge is stationary. And it's common sense that a stationary charge does not radiate. Since radiation is an electromagentic wave, and given that the speed of an electromagnetic wave is the same in any reference frame, if there is radiation in one reference frame, there is radiation in all reference frames. So there is no radiation, and the charge should not lose energy.

From here we see that only accelerating charges emit radiation. It's also clear that since only reference frames moving at constant velocity are allowed, an charge that is accelerating in one reference frame will accelerate in all reference frames. In fact, all accelerating (or decelerating) charges radiate.

 

[granpa]

The self field of a charged particle does not effect the relation between its translational kinetic energy and its velocity. The charged and uncharged particles will have the same velocity.

then where does the energy in the magnetic field come from?

of course, as cesiumfrog said, ' one *definition* of mass *is* the amount of energy required to change something's speed'. maybe that is what you meant.

 

[granpa]

First note that radiation is an electromagnetic wave, and so if a charge emits radiation, then it will lose energy.

A charge moving at constant velocity does not radiate. The way to see this is that the laws of electromagnetism are the same for any reference frames moving at a constant velocity (via a Lorentz transformation). So we can use a reference frame where the charge is stationary. And it's common sense that a stationary charge does not radiate. Since radiation is an electromagentic wave, and given that the speed of an electromagnetic wave is the same in any reference frame, if there is radiation in one reference frame, there is radiation in all reference frames. So there is no radiation, and the charge should not lose energy.

From here we see that only accelerating charges emit radiation. It's also clear that since only reference frames moving at constant velocity are allowed, an charge that is accelerating in one reference frame will accelerate in all reference frames. In fact, all accelerating (or decelerating) charges radiate.

the question wasnt about radiation.

 

[Meir Achuz]

then where does the energy in the magnetic field come from?

of course, as cesiumfrog said, ' one *definition* of mass *is* the amount of energy required to change something's speed'. maybe that is what you meant.

As I read the question, I took "provided equal energy directed, at least at onset, in the form of translational energy" to mean that the two particles had equal translational kinetic energy. Your question about the magnetic field energy relates to the fact that more energy input may be required to get the charged particle up to that kinetic energy.

' one *definition* of mass *is* the amount of energy required to change something's speed.'
does not make sense.