Static electric and magnetic fields and energy.

In summary, systems tend towards minimal potential energy, including static electric fields. Maxwell's equations can be obtained from the principle of stationary action, which involves minimizing energy for given boundary conditions. This is true for electrostatic fields in a region with no free charge, where the possible configurations have the same potential on the boundary surface, but do not necessarily follow the Gauss law. For the vacuum, the minimum theorem also holds for linear dielectric, but the possible fields must have the same electric displacement on the boundary. A proof for this can be found through a link provided by someone.
  • #1
chingel
307
23
It is well known that all sorts of systems tend towards minimal potential energy. I was wondering if this applies to static electric fields also, i.e. is an electric field such that it's energy integrated over all space is minimal? For example if we have a bounded source free region and if the electric field on the boundary is defined, does the electric field inside have minimal energy of all possible configurations? With possible configurations I mean all such which follow the Gauss law, but not necessarily the other Maxwell's laws.

The same question for magnetic fields.
 
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  • #2
Yes. Maxwell's equations can be obtained from the principle of stationary action. In the time-independent case, this reduces a minimisation of the energy, for the given boundary conditions.
 
  • #3
This is true for the electrostatic field in a region where there is no free charge. The possible configurations are any electrostatic fields that have the same potential on the boundary surface. The electric intensity on the surface is not fixed, and the fields do not have to obey the Gauss law. If the field obeys the Gauss law and the boundary conditions, it is already the solution with minimal energy.

EDIT: This is true for vacuum. If we have linear dielectric and define energy by

$$
W = \int \frac{1}{2}\mathbf E\cdot\mathbf D dV
$$
the minimum theorem holds too, but the possible fields have to have the same electric displacement ##\mathbf D## on the boundary.
 
Last edited:
  • #4
Thanks for the answers. Does anyone have a link to a proof? I would be interested in reading it.
 
  • #5


I can confirm that the concept of minimal potential energy applies to both static electric and magnetic fields. In fact, the principle of minimum energy is a fundamental principle in physics and is known as the principle of least action.

In the case of electric fields, the electric potential energy is defined as the work required to bring a unit positive charge from infinity to a specific point in the electric field. This potential energy is directly related to the electric field strength, and the electric field configuration that results in the lowest potential energy is the one that satisfies the Gauss law. Therefore, in a bounded source-free region with a defined electric field on the boundary, the electric field inside will have the minimal energy among all possible configurations that follow the Gauss law.

Similarly, for magnetic fields, the magnetic potential energy is defined as the work required to bring a unit magnetic pole from infinity to a specific point in the magnetic field. The magnetic field configuration that results in the lowest potential energy is the one that satisfies the Maxwell's equations, including the Gauss law for magnetism. So, in a system with a defined magnetic field on the boundary, the magnetic field inside will have the minimal energy among all possible configurations that follow the Maxwell's laws.

Overall, the concept of minimal potential energy is a powerful tool in understanding the behavior of electric and magnetic fields. It allows us to predict the most stable configurations and provides insight into the underlying laws governing these fields. I hope this explanation answers your question.
 

1. What are static electric and magnetic fields?

Static electric and magnetic fields are areas of energy that do not change over time. They are created by the presence of electric charges and currents, and can exert forces on other charges and currents.

2. How are static electric and magnetic fields different?

Static electric fields are created by stationary electric charges, while magnetic fields are created by moving electric charges. Additionally, electric fields exert forces on both electric charges and currents, while magnetic fields only exert forces on moving electric charges.

3. Are static electric and magnetic fields harmful to humans?

There is currently no conclusive evidence that exposure to static electric and magnetic fields at typical levels found in everyday environments is harmful to human health. However, some studies suggest that prolonged exposure to high levels of these fields may have potential health effects.

4. How can static electric and magnetic fields be measured?

Static electric and magnetic fields can be measured using specialized instruments such as gaussmeters and electrometers. These instruments measure the strength and direction of the fields and can provide information about the sources of the fields.

5. Can static electric and magnetic fields be shielded?

Yes, static electric and magnetic fields can be shielded using materials that are good conductors of electricity, such as copper or aluminum. These materials can redirect the fields and reduce their strength in a specific area.

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