# What Does Seek a Vector Potential That Is Parallel Mean in Calculus?

• Niles
In summary, The conversation discusses finding a vector potential that is parallel to the current density and the curl of the magnetic field for static fields. A hint is given for finding this vector potential by setting it equal to k times the curl of the magnetic field. The person asking for help is unsure about how to apply this hint and is seeking assistance.
Niles

## Homework Statement

Hi all.

What do they mean by that hint?

## The Attempt at a Solution

First I want to find the divergence of the three vectors.

Second, for the vectors which have a divergence equal to zero, I want to find the vector A so the rotation of A = B. But this is the part I am in doubt about - can you help?

Niles.

Niles said:

## Homework Statement

Hi all.

What do they mean by that hint?

For static fields, the vector potential $\vec{A}$ is parallel to the current density $\vec{J}$. And, Ampere's law says that $\vec{\nabla}\times\vec{B}$ is parallel to the current density as well, under the same condition. Since the vector potential and the curl of the magnetic field are both parallel to the same vector, they are parallel to each other.

## The Attempt at a Solution

Second, for the vectors which have a divergence equal to zero, I want to find the vector A so the rotation of A = B. But this is the part I am in doubt about - can you help?

Here's a hint: If $\vec{A}$ is parallel to $\vec{\nabla}\times\vec{B}$, then $\vec{A}=k\vec{\nabla}\times\vec{B}$ for some $k$.

## 1. What is a vector potential in calculus?

A vector potential is a mathematical function used in vector calculus to represent the potential of a vector field. It is used to describe the behavior of vector fields in three-dimensional space.

## 2. What are the applications of vector potentials in calculus?

Vector potentials have various applications in physics and engineering, particularly in the study of electromagnetism. They are used to calculate the magnetic field, electric field, and other properties of electromagnetic fields.

## 3. How is a vector potential different from a scalar potential?

A scalar potential is a single-valued function that describes the potential of a scalar field, while a vector potential is a multi-valued function that describes the potential of a vector field. Additionally, scalar potentials are used to calculate electric fields, while vector potentials are used for magnetic fields.

## 4. How are vector potentials calculated in calculus?

Vector potentials are typically calculated using the equation ∇ × A = B, where A is the vector potential, B is the magnetic field, and ∇ is the gradient operator. This equation is known as the vector potential equation.

## 5. What are some common vector potential functions used in calculus?

Some common vector potential functions used in calculus include the Coulomb gauge potential, the Lorenz gauge potential, and the transverse gauge potential. These functions are chosen based on the specific application and desired properties of the vector field being studied.

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