Sufficient Condition for Existence of Vector Potential

In summary, the sufficient condition for the existence of the vector potential is that the field must be divergence free. This is both necessary and sufficient in Euclidean space, as shown in the example of magnetostatics. The 2 form representing the field must be closed, and by Poincare's Lemma, it is the exterior derivative of a 1 form which has a curl equal to the field.
  • #1
ManuelF
8
0
Hello!
Can you tell me the sufficient condition for the existence of the vector potential?
Thank you very much!
 
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  • #2
ManuelF said:
Hello!
Can you tell me the sufficient condition for the existence of the vector potential?
Thank you very much!

when a field is divergence free it has a vector potential.
 
  • #3
That is a necessary condition, but I do not think is enough!
Are you sure?
Thank you.
 
  • #4
ManuelF said:
That is a necessary condition, but I do not think is enough!
Are you sure?
Thank you.

It is necessary and sufficient in Euclidean space.

For instance, in magnetostatics a magnetic field always has a vector potential.

If the field is B = (u,v,w) consider the 2 form udy^dz -vdx^dz + wdx^dy

Since B has zero divergence, the 2 form is closed. By Poincare's Lemma it is therefore the exterior derivative of a 1 form adx + bdy + cdx. The curl of (a,b,c) equals B.
 
  • #5
Thank you very much!
 

What is a sufficient condition for the existence of a vector potential?

A sufficient condition for the existence of a vector potential is that the vector field must be conservative, meaning that the line integral of the vector field over any closed loop is equal to zero. This condition is also known as the Helmholtz condition.

Why is the existence of a vector potential important?

The existence of a vector potential is important because it allows us to simplify the calculation of the magnetic field in a region. By using the vector potential, we can express the magnetic field as the curl of the vector potential, which can make solving problems in electromagnetism much easier.

What are some examples of vector fields that satisfy the sufficient condition for existence of a vector potential?

Some examples of vector fields that satisfy the sufficient condition for existence of a vector potential are gravitational fields, electric fields, and magnetic fields. These fields are conservative, and therefore have a corresponding vector potential.

Can a vector field have a vector potential if it is not conservative?

No, a vector field cannot have a vector potential if it is not conservative. This is because the Helmholtz condition is a necessary and sufficient condition for the existence of a vector potential. If the vector field is not conservative, the line integral over a closed loop will not be equal to zero, and therefore a vector potential cannot exist.

Is the existence of a vector potential always guaranteed?

No, the existence of a vector potential is not always guaranteed. While the Helmholtz condition is a sufficient condition, it is not a necessary condition. This means that there may be cases where a vector field is conservative, but a vector potential does not exist. Additionally, in certain regions where the vector field is not defined, a vector potential may not exist.

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