Does Conservation of Energy Still Apply in Time-Dependent Potentials?

[hokhani]

Consider a particle moving near the ground with the ground surface as the zero-point potential reference. If at time t we apply an electric field, say parallel to gravity force, where we should consider as a zero-potential reference point? Does the energy remain conserved (Is the energy equal to that before the time t)?

 

[Simon Bridge]

Consider a particle moving near the ground with the ground surface as the zero-point potential reference. If at time t we apply an electric field, say parallel to gravity force, where we should consider as a zero-potential reference point?

Wherever makes the maths easiest - depending on what we want to do.

Does the energy remain conserved (Is the energy equal to that before the time t)?

Yes - the law of conservation of (total) energy always applies.

Consider: the zero for potential energy is arbitrary - so set it anywhere and it makes no difference to the physics.
So: to help you think about it, at time t=0, set the gravitational potential and the electric potential at the particle position to be zero.

 

[hokhani]

So: to help you think about it, at time t=0, set the gravitational potential and the electric potential at the particle position to be zero.

Thanks. Let us discuss this question more exactly; Suppose that at $t=0$ the particle is at $h(0)=0$ (on the ground) moving with velocity $v(0)$ upwards. According to your proposal we choose the potential reference point at $h=0$. At $t=t_1$ the particle is at $h(t_1)=h_1$ and we turn on the external electric filed. Just before applying the electric field the total energy is $E=1/2mv^2(t_1) +h_1$ while right after it we have $E=1/2mv^2(t_1) +h_1+Qh_1$ (we consider a positive charge and a downward electric field). So, it seems that the energy is not conserved! I think we should consider the zero point of potential exactly at the point that particle is present there at the time $t_1$. This way the energy remains conserved.

 

[Simon Bridge]

What makes you think energy is not conserved?
You appear to have excluded the apparatus for turning on the electric field from the systrm.

 

[hokhani]

What makes you think energy is not conserved?

I think time dependency of potential, forces us to redefine a new potential reference point at $t=t_1$. Otherwise, energy would not be conserved.

You appear to have excluded the apparatus for turning on the electric field from the systrm.

Right. I have considered the particle as the system, gravitational and electric fields as the external fields. I am only dealing with the system itself so, sources of external fields are not important. It seems that the apparatus doesn't play any role in defining the potential reference.

 

[Simon Bridge]

I think time dependency of potential, forces us to redefine a new potential reference point at $t=t_1$. Otherwise, energy would not be conserved.

Beggin the question: what makes you think that this is required at all?

Right. I have considered the particle as the system, gravitational and electric fields as the external fields. I am only dealing with the system itself so, sources of external fields are not important. It seems that the apparatus doesn't play any role in defining the potential reference.

If you treat only the particle as the system then of course energy for the particle will not be conserved, it does not have to be since the system is not closed wrt to energy.

What is your question again?

 

[hokhani]

Beggin the question: what makes you think that this is required at all?

It should be noted that I don't know whether or not energy is conserved in this process. So, I meant that if energy conservation is required (because gravitational and electric fields are conservative forces) it is necessary to redefine the reference point of potential.

If you treat only the particle as the system then of course energy for the particle will not be conserved, it does not have to be since the system is not closed wrt to energy.

I think this is the main point I should discuss further about. According to your statement, a particle (as a system) with mass m in the gravitational force of the Earth doesn't have conserved energy because we have excluded the earth. Also, energy of an electric charge in an electric field is not conserved if we only consider the charge itself disregarding the external charges that induce the electric field. But the energies of these systems are conserved.

What is your question again?

Could you please help me with that?

 

[jbriggs444]

So, I meant that if energy conservation is required (because gravitational and electric fields are conservative forces)

A static gravitational or electric field is "conservative". One of the consequences (or even the definition) is that the work done by the gravitational or electrostatic force on a particle traversing a closed path is zero. If you allow the field to be turned on or off, this will fail to hold in general. That is to say that the work done on a particle as it traverses a closed path (in space) that extends over time need not be zero. Particle accelerators make use of this fact, as do power generation plants.

Energy is still conserved in such a case, but one must consider energy in the total system, including the particle, the field and the generating gear.

 

[Khashishi]

You can put the zero point wherever you want. You can choose the zero such that the energy of the particle doesn't change, if you want. The energy isn't something you can measure directly.

 

[hokhani]

Energy is still conserved in such a case, but one must consider energy in the total system, including the particle, the field and the generating gear.

Energy in the total system is always conserved even in the case of non-conservative forces such as friction. In this case, energy is just transferred from one thing to another and the total energy is conserved. I guess that conservation concept is used to provide the possibility of working with a system independent of the generating gears like Earth (in gravitational field) or external charges (which generate electric fields).