# Magnetic potential derivation from B

• likephysics
In summary, the conversation discusses the derivation of the vector potential A for an infinite straight wire carrying a static electric current I in free space. The vector potential can be expressed as B = ∇ x A and can be evaluated using Stokes' theorem by choosing a loop parallel to the wire and setting one side of the loop at a certain distance from the wire and the other side at infinity. The vector potential is parallel to the wire and constant along its axis. However, the evaluation of the flux may be difficult due to potential infinities, but it can be solved by considering a finite point instead of infinity.
likephysics
π²³ ∞ ° → ~ µ ρ σ τ ω ∑ … √ ∫ ≤ ≥ ± ∃ · θ φ ψ Ω α β γ δ ∂ ∆ ∇ ε λ Λ Γ ô

## Homework Statement

Derive the vector potential A produced by an infinite straight wire of negligible thickness, located in free space and carrying a static electric current I.

## The Attempt at a Solution

I tried to start with the expression for Mag flux density B= µH
∇xH = jωD+J (ampere's law).
I substituted H=B/µ and couldn't proceed any further.

$$\vec{A}=\frac{\mu_0I}{4\pi}\int^\infty_{-\infty}\frac{\vec{dl}}{r}$$

and note the vector A is parallel to the wire since the vectors dl are all along the wire.

Last edited:
I don't think that formula works if the current doesn't go to 0 in infinity. Try using Stokes instead.

$$\int_{\partial S} \vec{A} \cdot d\vec{s}= \iint_S (\nabla \times \vec{A}) \cdot d\vec{S}$$

The second integral should be very familiar.

I have to derive an expression for A. How can I start with A= something?

I have to derive an expression for A. How can I start with A= something?

In the same way when you're asked to find the electric field of a charge distribution you start with E=something, something being the general expression for E. You then usually integrate and find E for that specific problem. Besides the hint I gave you in post 3 doesn't start with an expression A=something at all.

The vector potential can be expressed as follows:

B = $$\nabla$$$$\times$$A

Magnetic field curls around an infinitely long wire, so it's clear that the vector potential must be parallel to the wire current. Also, from symmetry the vector potential must be constant along the wire axis.

Using stokes' theorem, might be able to evaluate A as a function of the radius (or "cheat" mathematically to do so) by choosing a loop to be a square loop of a certain length parallel to the wire. Set one side of the rectangle at the distance where you want to evaluate the vector potential magnitude, and the other side (of length l) at an infinite distance from the wire. Then, the dotted integral around this loop will just be A times L, the length of the side of the rectangle.

It may be difficult to evaluate the flux because you could encounter infinities, but because the vector potential can differ by the gradient of an arbitrary function, you might be able to evaluate the loop integral with respect to a certain finite point rather than infinity.

## 1. How is magnetic potential derived from magnetic field (B)?

Magnetic potential (V) is derived from magnetic field (B) through the use of the equation V = -∫B·dl, where B represents the magnetic field and dl represents an infinitesimal path element. This equation is based on the relationship between magnetic potential and magnetic field strength, known as the magnetic field line integral.

## 2. What is the significance of magnetic potential in physics?

Magnetic potential plays a crucial role in understanding the behavior of magnetic fields. It is a scalar quantity that describes the energy associated with a magnetic field and is used to calculate the work done by the field on a charged particle. Magnetic potential is also used to determine the forces and movements of charged particles in magnetic fields.

## 3. Can magnetic potential be measured directly?

No, magnetic potential cannot be measured directly. It is a theoretical concept used to describe the behavior of magnetic fields. However, it can be indirectly observed and calculated through its relationship with magnetic field strength and the measurement of magnetic field lines.

## 4. How does magnetic potential differ from electric potential?

Magnetic potential and electric potential are two different concepts, although they both describe the energy associated with fields. Magnetic potential is a scalar quantity that describes the energy of a magnetic field, while electric potential is a scalar quantity that describes the energy of an electric field. Additionally, the equations used to calculate magnetic and electric potentials differ.

## 5. What are some real-life applications of magnetic potential?

One of the most common applications of magnetic potential is in electric motors and generators. The interaction between magnetic fields and charged particles, described by magnetic potential, is essential for these devices to function. Magnetic potential is also used in particle accelerators, magnetic resonance imaging (MRI) machines, and other technologies that utilize magnetic fields.

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