Vector potential

Summary

In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field.

Formally, given a vector field v, a vector potential is a vector field A such that

Consequence edit

If a vector field v admits a vector potential A, then from the equality

 
(divergence of the curl is zero) one obtains
 
which implies that v must be a solenoidal vector field.

Theorem edit

Let

 
be a solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases at least as fast as   for  . Define
 

Then, A is a vector potential for v, that is, 

 
Here,   is curl for variable y. Substituting curl[v] for the current density j of the retarded potential, you will get this formula. In other words, v corresponds to the H-field.

You can restrict the integral domain to any single-connected region Ω. That is, A' below is also a vector potential of v;

 

A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.

By analogy with Biot-Savart's law, the following   is also qualify as a vector potential for v.

 

Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law.

Let   and let the Ω be a star domain centered on the p then, translating Poincaré's lemma for differential forms into vector fields world, the following   is also a vector potential for the  

 

Nonuniqueness edit

The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is

 
where   is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.

This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.


See also edit

References edit

  • Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.