# What is the Magnetic Vector Potential for a Given Magnetic Field?

• beth92
In summary, the magnetic vector potential A for a B field with only a z component, in cylindrical coordinates, is given by A∅ = -k/s2 + C/s, where C is an arbitrary constant. This is based on the equations Curl of A = B and Divergence of A = 0.
beth92

## Homework Statement

For the magnetic field B=k/s3 z determine the magnetic vector potential A. For simplicity, assume that A does not have a component in the s direction.

(I don't know if this is relevant but this was a follow up question to one in which I was required to find the induced current for a bar moving along a semicircular loop of wire - like a slide wire generator bent into a semicircular shape - and then the torque on the bar due to the magnetic force and then the position ∅ at which the bar comes to rest.)

## Homework Equations

Curl of A = B

Divergence of A = 0

## The Attempt at a Solution

The z component of the curl of A in cylindrical coordinates is:
1/s[d(sA)/ds - d(As)/d∅]

The B field we are considering has only a z component so the s and components of the curl of A can be disregarded. Also, we are told in the problem that As = 0 so the only surviving term, equal to B, is:

1/s d(A)/ds = k/s3

Separating the variables:

∫ d(sA) = ∫ (k/s2) ds

After integration, we get:

A = -k/s2 + C/s

Where C is the integration constant.

This is as far as I got...I'm not sure how how to find out what C is. Any tips would be appreciated!

hi beth92!

vector potential is like scalar potential …

only the potential difference matters, so you can add any constant vector you like!

## 1. What is magnetic vector potential?

Magnetic vector potential is a mathematical concept used to describe the magnetic field in a given space. It is a vector quantity that represents the direction and strength of the magnetic field at any point in space.

## 2. How is magnetic vector potential different from magnetic field?

The magnetic vector potential is related to the magnetic field, but it is not the same thing. The magnetic field is a physical quantity that can be measured, while the magnetic vector potential is a mathematical construct used to describe the magnetic field.

## 3. What is the significance of magnetic vector potential in electromagnetism?

Magnetic vector potential plays a crucial role in electromagnetism as it allows us to calculate and understand the behavior of magnetic fields. It is used in various equations and formulas to determine the strength and direction of the magnetic field in a given space.

## 4. How is magnetic vector potential calculated?

Magnetic vector potential can be calculated using Maxwell's equations, which are a set of equations that describe the behavior of electromagnetic fields. The specific formula for calculating magnetic vector potential may vary depending on the situation and the type of magnetic field being studied.

## 5. What are the practical applications of magnetic vector potential?

Magnetic vector potential has various practical applications in fields such as engineering, physics, and geology. It is used to analyze and design electromagnetic devices, study the behavior of magnetic fields in different materials, and understand the Earth's magnetic field.

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