Determination of electric field E

In summary, the theorem in electromagnetism states that if the divergence and curl of a field are known, the field is uniquely determined. However, in the case of electric fields, the divergence equation provides 1 constraint and the curl equation provides 3 constraints, totaling to 4 constraints. This suggests that the system of equations may be overdetermined and may have dependency. This means that there is no solution or one or more equations are superfluous. To fully reconstruct the field, both divergence and curl of the field are necessary. A helpful resource on this topic is the paper "Maxwell's Uniqueness Theorem" by C. Baird.
  • #1
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Theorem in electromagnetism says if the divergence [itex]\nabla\cdot E[/itex] and the curl [itex]\nabla\times E[/itex] are known, then E is uniquely determined. So my question is, the divergence equation gives 1 constraint and the curl equation gives 3 constraints, which add up to 4 constraints. On the other hand, the electric filed E has only 3 unknown components. Does it mean 4 equations are overdetermined and thus has dependency?
 
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  • #2
If the system of equations is overdetermined, it means there is no solution.

If the equations are dependent, it means one or more equations are superfluous (do not give new constraint).

Divergence and curl of a field are both necessary to reconstruct the field. See

http://mathworld.wolfram.com/HelmholtzsTheorem.html
 
  • #3
zhangwfjh said:
Theorem in electromagnetism says if the divergence [itex]\nabla\cdot E[/itex] and the curl [itex]\nabla\times E[/itex] are known, then E is uniquely determined. So my question is, the divergence equation gives 1 constraint and the curl equation gives 3 constraints, which add up to 4 constraints. On the other hand, the electric filed E has only 3 unknown components. Does it mean 4 equations are overdetermined and thus has dependency?
You might like this paper

http://faculty.uml.edu/cbaird/95.657(2013)/Maxwell_Uniqueness.pdf
 

What is electric field E?

Electric field E is a physical quantity that describes the strength and direction of the electric force on a charged particle at any given point in space.

How is the electric field E calculated?

The electric field E is calculated by dividing the electric force on a test charge by the magnitude of the test charge. It can also be calculated by using Coulomb's law, which states that the electric field E is equal to the electric force divided by the magnitude of the test charge squared.

What are the units of electric field E?

The units of electric field E are Newtons per Coulomb (N/C) in the SI system of units. In other systems, such as the CGS system, the units are dynes per statcoulomb (dyn/cm * sc).

What factors affect the electric field E?

The electric field E is affected by the magnitude and sign of the source charges, the distance between the source charges and the test charge, and the medium in which the charges are located. It is also affected by the presence of other charges in the vicinity.

What are some real-world applications of determining electric field E?

Determining electric field E is important in understanding the behavior of charged particles in various systems, such as in electronics, plasma physics, and electromagnetism. It is also crucial in the design and operation of devices such as capacitors, generators, and motors.

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