Undergrad Total Energy of Particle in Potential: SR Explanation

[greypilgrim]

Hi.

What is the total energy of a particle in a potential? Is it
$$E=\gamma m_0 c^2+E_pot$$
or is it still
$$E=\gamma m_0 c^2$$
where $m_0$ is a bigger mass than the particle would have in absence of the potential?

 

[stevendaryl]

Specifically for the electromagnetic field, the conserved energy is given by:

E = \gamma mc^2 + q \Phi

where \Phi is the electric potential.

 

[Jonathan Scott]

Note that if you have two or more charged particles, this simple model would say that each particle in a pair gets the full potential energy due to the other, which double counts the potential energy. For a collection of particles, one can simply allocate half of the potential energy to each one, or one can use an alternative model which takes into account energy density in the field.

(For gravity, where energy acts as a source so everything is non-linear, this gets much more complicated and as far as I know there isn't any satisfactory answer to where the equivalent of potential energy resides, not even in GR).

 

[greypilgrim]

Specifically for the electromagnetic field, the conserved energy is given by:

E = \gamma mc^2 + q \Phi

where \Phi is the electric potential.

But how does this work with
$$E^2=c^2\cdot \mathbf{p}^2+m^2\cdot c^4 \enspace ?$$
If we look at two identical particles with the same velocity where one is in an electric potential and the other is not, the right sides of this equation are the same, but not the energy squared on the left?

 

[Jonathan Scott]

Electric potential energy is not part of the energy of the particle and does not contribute to its inertia. It is part of the energy of the system which includes the particle and the field, and the standard explanation is that it resides in the field, with an energy density proportional to the square of the field. Within the squared field expression, there are terms made up of the scalar product of the field components due to each pair of charged particles, and when each scalar product term is integrated over all space the result is equal to the potential energy between that pair of particles.

In contrast, gravitational potential energy (which is negative relative to the local rest mass) is part of the energy of the particle and is assumed to contribute its inertia, but to get the usual conservation laws to work (at least for a weak field approximation) there also has to be positive energy in the field which compensates for the double effect of each particle having the whole potential energy.