How much energy is required to accelerate a charge?

[calinvass]

Lets suppose we have a charge and we want to accelerate it from a point A to B and back. Does the energy requirement increases if we move the charge quicker?
I can make an analogy with mass. To accelerate an object over a distance d requires energy. The objects resist to change in velocity. The faster the acceleration the greater the force hence more energy is required over the same distance.
When accelerating a charge, a magnetic field is also created. I suppose the effect is that the charge will resist to the change in velocity.

 

[berkeman]

Lets suppose we have a charge and we want to accelerate it from a point A to B and back. Does the energy requirement increases if we move the charge quicker?
I can make an analogy with mass. To accelerate an object over a distance d requires energy. The objects resist to change in velocity. The faster the acceleration the greater the force hence more energy is required over the same distance.
When accelerating a charge, a magnetic field is also created. I suppose the effect is that the charge will resist to the change in velocity.

Well, you've posted this question in the technical Physics forums, with an "I"="intermediate"/undergraduate prefix, so it's appropriate for me to ask you to post the Relevant Equations and your Attempt at the Solution.

If use more power over a shorter time, how does that come into play? Can you post the equations for a couple scenarios that apply to your question? And also include the deceleration work that applies to a round-trip scenario like you mention...

 

[haruspex]

Lets suppose we have a charge and we want to accelerate it from a point A to B and back. Does the energy requirement increases if we move the charge quicker?
I can make an analogy with mass. To accelerate an object over a distance d requires energy. The objects resist to change in velocity. The faster the acceleration the greater the force hence more energy is required over the same distance.
When accelerating a charge, a magnetic field is also created. I suppose the effect is that the charge will resist to the change in velocity.

Does this help: https://en.m.wikipedia.org/wiki/Larmor_formula?

 

[calinvass]

For a mass, the total energy over an x distance is E=max. The greater the acceleration the greater the energy. But for a charge I would need some time to write the equations and to compare them to an harmonic oscillator.

 

[haruspex]

For a mass, the total energy over an x distance is E=max

That assumes constant acceleration. If the objective is to get the mass from rest at A to the point B in the shortest time for the given energy (and you have no way to recoup invested KE), is constant acceleration the best way? I think not.

But for a charge

Presumably the charge also has mass. Even for an electron, I suspect that the energy that would go into the KE would dwarf that lost to radiation unless the acceleration is very great. (I note the a²/c³ factors in the Larmor formula.)

 

[calinvass]

That assumes constant acceleration. If the objective is to get the mass from rest at A to the point B in the shortest time for the given energy (and you have no way to recoup invested KE), is constant acceleration the best way? I think not.

my objective is to see if the energy of the wave generated depends on frequency. So constant acceleration should be a reasonable approximation.

Presumably the charge also has mass. Even for an electron, I suspect that the energy that would go into the KE would dwarf that lost to radiation unless the acceleration is very great. (I note the a²/c³ factors in the Larmor formula.)

The charge should have mass but I think the energy required to move the mass can be initially calculated separately. The massive particle resists to a change in velocity and the energy spent goes into kinetic energy. The charge doesn't hold any extra energy if the object travels faster, so I suppose all the energy gets into the field. But accelerating a mass also creates a gravitational wave. That would mean the oscillator looses energy into the wave as well.

 

[Drakkith]

But accelerating a mass also creates a gravitational wave. That would mean the oscillator looses energy into the wave as well.

True, but the amount of energy lost to a gravitational wave is absolutely miniscule and can be entirely ignored here.

 

[calinvass]

The charge should have mass but I think the energy required to move the mass can be initially calculated separately. The massive particle resists to a change in velocity and the energy spent goes into kinetic energy. The charge doesn't hold any extra energy if the object travels faster, so I suppose all the energy gets into the field. But accelerating a mass also creates a gravitational wave. That would mean the oscillator looses energy into the wave as well.

The energy spent to aceletate the charge should be radiated as EM waves.

 

[Drakkith]

The energy spent to aceletate the charge should be radiated as EM waves.

Only part of the energy will be lost as EM radiation.

 

[haruspex]

if the energy of the wave generated depends on frequency

What frequency? The way you described the problem it is doing one round trip.
If you are going to have it bouncing back and forth, you could have a magnetic field at each end to turn it around. With that arrangement you would have no wasted KE, so all of the lost energy would be as EM.

 

[calinvass]

If we complete an oscillation in a certain period of time and the waves travel at c, we can have the frequency of the classical EM wave.

 

[haruspex]

If we complete an oscillation in a certain period of time and the waves travel at c, we can have the frequency of the classical EM wave.

I had always accepted the general view that even a charged particle undergoing uniform acceleration radiates. What I had not thought about was what the frequency spectrum would be. In the absence of oscillation, there is no obvious reason for any particular frequency. Presumably this should be resolved by quantum mechanics, much as it resolved the violet catastrophe.
However, I just read (skimmed) http://www.mathpages.com/home/kmath528/kmath528.htm, which says Feynman rejected the notion that there would be any radiation.

 

[Vanadium 50]

The OP is confused, and I don't think it helps him to discuss different situations than the one he posted.

I read the situation as the particle starts at A, at rest, travels to B, and then returns to A, again at rest. Strictly speaking, it was not stated that it begins or ends at A at rest - it could have started whizzing by - but it doesn't say otherwise.

The OP states constant acceleration. Fine - the acceleration will be a square wave: for the first and fourth quarters of the cycle it will be directed B-ward, and for the second and third, it will be directed A-ward.

While the particle travels, you have a changing dipole, and a changing dipole will radiate. You will in fact get a square wave. If you feel that you need to see this expressed in terms of sine waves, you can take a Fourier transform. The energy for this radiation has to come from somewhere, and it comes from the force that accelerates the particle.

The more acceleration, the more it radiates, and the more power you need to apply to achieve this acceleration.

 

[A.T.]

I had always accepted the general view that even a charged particle undergoing uniform acceleration radiates. What I had not thought about was what the frequency spectrum would be. In the absence of oscillation, there is no obvious reason for any particular frequency. Presumably this should be resolved by quantum mechanics, much as it resolved the violet catastrophe.
However, I just read (skimmed) http://www.mathpages.com/home/kmath528/kmath528.htm, which says Feynman rejected the notion that there would be any radiation.

Note that a charge resting on your table undergoes constant 1g proper acceleration. It can rest there forever without energy expenditure.

 

[calinvass]

Note that a charge resting on your table undergoes constant 1g proper acceleration. It can rest there forever without energy expenditure.

That would mean there is no proper acceleration but the charge is in equilibrium, but in GR if we think of gravity as curved spacetime the charge accelerates at 1g. Now, for the forces that repel the charge you can perhaps add new dimensions and say there are not forces either and we restore the equilibrium.

 

[calinvass]

The OP is confused, and I don't think it helps him to discuss different situations than the one he posted.

I read the situation as the particle starts at A, at rest, travels to B, and then returns to A, again at rest. Strictly speaking, it was not stated that it begins or ends at A at rest - it could have started whizzing by - but it doesn't say otherwise.

The OP states constant acceleration. Fine - the acceleration will be a square wave: for the first and fourth quarters of the cycle it will be directed B-ward, and for the second and third, it will be directed A-ward.

While the particle travels, you have a changing dipole, and a changing dipole will radiate. You will in fact get a square wave. If you feel that you need to see this expressed in terms of sine waves, you can take a Fourier transform. The energy for this radiation has to come from somewhere, and it comes from the force that accelerates the particle.

The more acceleration, the more it radiates, and the more power you need to apply to achieve this acceleration.

Yes the more acceleration the more power, but this doesn't show why increasing the frequency increases the energy of a limited duration wave.

 

[Vanadium 50]

this doesn't show why increasing the frequency increases the energy

I didn't say energy. I said power. The power increases because the radiation is happening in a shorter time. (There is also an energy effect, but this is the clearest way to see why the power goes up)

 

[A.T.]

That would mean there is no proper acceleration but the charge is in equilibrium, but in GR if we think of gravity as curved spacetime the charge accelerates at 1g.

The proper acceleration is 1g.

 

[calinvass]

The transformation of fields between inertial frames, should explain how higher frequency wave packets have more energy. I suppose just as increasing the frequency of a wave packet increases its total energy, increasing the oscillation frequency of a charge increases the emitted wave packet energy.