How much energy is required to accelerate a charge?

• calinvass
In summary, when accelerating a charge from point A to B and back, the energy requirement increases if the charge is moved quicker due to the greater force required for faster acceleration. This force is necessary to overcome the resistance to change in velocity of the charge. Additionally, a magnetic field is also created when accelerating the charge, which further resists the change in velocity. The energy spent to accelerate the charge is radiated as electromagnetic (EM) waves, with the frequency of the waves depending on the period of oscillation and the speed of light. However, with a setup where the charge bounces back and forth between points A and B, the wasted kinetic energy can be minimized and the lost energy will solely be in the form of EM radiation.
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.

Primordial_Coridal
calinvass said:
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...

calinvass
calinvass said:
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 and Primordial_Coridal
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.

calinvass said:
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.
calinvass said:
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.)

haruspex said:
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.

calinvass said:
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
calinvass said:
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.

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

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

calinvass said:
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.

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.

calinvass said:
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.

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.

haruspex said:
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.

A.T. said:
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.

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.

calinvass said:
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)

calinvass said:
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.

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.

What is the definition of energy?

Energy is the ability to do work or cause change. It exists in many forms, including kinetic energy, potential energy, thermal energy, and electromagnetic energy.

How is energy related to the acceleration of a charge?

The amount of energy required to accelerate a charge depends on the magnitude and direction of the charge's acceleration. The greater the acceleration, the more energy is required. This is described by the equation E = q•V, where E is energy, q is the charge, and V is the acceleration voltage.

What units are used to measure energy?

The SI unit for energy is the joule (J), but other common units include calories, kilowatt-hours, and electron volts.

Can energy be converted from one form to another?

Yes, energy can be converted from one form to another. This is known as the law of conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another. For example, electrical energy can be converted into kinetic energy when a charge is accelerated.

How does the mass of a charge affect the amount of energy required to accelerate it?

The mass of a charge does not directly affect the amount of energy required to accelerate it. However, a larger mass may require a greater force to accelerate it, which in turn would require more energy. This is described by Newton's second law, F=ma, where F is the force, m is the mass, and a is the acceleration.

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