What is Magnetic vector potential: Definition and 48 Discussions

Magnetic vector potential, A, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field:

×

A

=

B

{\textstyle \nabla \times \mathbf {A} =\mathbf {B} \,}
. Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Therefore, many equations of electromagnetism can be written either in terms of the fields E and B, or equivalently in terms of the potentials φ and A. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.
Historically, Lord Kelvin first introduced vector potential in 1851, along with the formula relating it to the magnetic field.

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1. I Why Does the Negative Sign Appear in the Vector Potential Equation?

We have motivated the derivation of the vector potential in the following way: However, I cannot understand where the ##-## sign in the second equality came from. I thought that it was because the gradient was with respect to the ##y##-variable, and then using the product rule one could...
2. A Relationship between magnetic potential and current density in Maxwell

I am currently studying to solve Maxwell's equations using FEM. I have a question about Maxwell's equations while studying. I understood that the magnetic potential becomes ▽^2 Az = -mu_0 Jz when the current flows only in the z-axis. I also understood the effect of the current flowing in a...
3. Helix Angle or Pitch Angle for this Solenoid carring Current?

I honestly have such a dumb question- it says “a pitch angle” but cannot find that in relation to a helix. It is not defined in the textbook. Looking on google I found a helix angle, but is that different than the pitch angle? Can anyone draw me a picture of where the pitch angle is? I assumed...
4. Problems on magnetic vector potential

The direction of the magnetic potential, ##\vec A##, must be in the direction of the current, which is in ##\hat z## direction in cylindrical coordinates. It is obvious that the potential only varies with ##s##. Therefore, $$\vec A = A(s) \hat z$$ Therefore, $$\nabla \times \vec A = \vec B$$...
5. Vector potential ##\vec A## in terms of magnetic field ##\vec B##

My solution is making an analogy of the ##\text{Relevant equations}## as shown above, starting from the equation ##\vec \omega = \frac{1}{2} \vec \nabla \times \vec v##. We have ##\vec B = \vec \nabla \times \vec A = \frac{1}{2} \vec \nabla \times 2\vec A \Rightarrow 2\vec A = \vec B \times...
6. Gauge choice for a magnetic vector potential

How do we verify whether a condition on the magnetic vector potential A constitutes a possible gauge choice ? Specifically, could a relation in the form A x F(r,t) be a gauge , where F is an arbitrary vector field?
7. Expressing the magnetic vector potential A-field in terms of the B-field

We have a retarded magnetic vector potential ##\mathbf{A}(\mathbf{r},t) = \dfrac{\mu_0}{4\pi} \int \dfrac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 \mathbf{r}'## And its curl, ##\mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{J}(\mathbf{r}'...

22. Determining the direction of magnetic vector potential

Homework Statement Im doing this practice question and I am to determine the magnetic vector potential A for a cylindrical wire with a uniform current density J. i have already determined B both inside and outside the wire no problem.My issue is in the solution given my professor states that...
23. Magnetic vector potential in the case of specific current distribution

1. Use equation for the magnetic vector potential in the case of specific current distribution and show by direct differentiation that ∇\bulletA=0 A(r)= µ_{0}/4\pi \int J(r')/|r-r'| dv' Homework Equations ∇\times B(r)= µ0J(r) The Attempt at a Solution We know that: curl of...
24. Proof that ∇ * A (magnetic vector potential) = 0

∇ΔHomework Statement Show that \nabla \cdot A = 0 Where A is formally defined as A(r) = \frac{\mu }{4\pi }\int \frac{J(r')\text{ }}{r} \, dv' I understand that we can distribute ∇ into the integral, and from there we can do a little bit of algebra to get the terms inside the...
25. Magnetic vector potential for antiparallel currents

Two infinitely long wires separated by distance d. Currents: I1 = -I2. Find potential vector as a function of r1 and r2 at a point P (r1 and r2 distances to P from wire one and wire two). Del cross A= B B = (mu I)/(2pi r) Using Ampere's, I get an expression for the magnetic field that...
26. Magnetic vector potential of finite wire

Homework Statement The problem statement is attached. The Attempt at a Solution I know how to solve the problem. However, my teachers solutions notes and my book's do it differently, and I want to ask what the difference is, so I have attached them both. My book does it the way I did it. My...
27. Magnetic Vector Potential help

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...
28. LT of the magnetic vector potential when the scalar potential=0

Special relativity predicts that electric fields transform into magnetic fields via Lorentz transformations and that the vice versa also occurs. It also has been argued, since experiments verifying the quantum mechanical phenomenon of the Aharonov–Bohm effect, that the vector potentials are more...
29. I have a question about the direction of the magnetic vector potential

Suppose my reference system is x coming out of the page toward you, y is in the plane of the page going left and right and z is in the plane of the page going up and down. Further suppose that the magnetic field is parallel to the x-axis and the electric field is parallel to the z axis. Finally...
30. Solving for magnetic vector potential (An integral issue)

Homework Statement There is a cylinder of conducting ionized gas that occupies rho < a. For the given B, show that a suitable A can be found with only one non-zero component, Aphi, find Aphi which is also continuous at rho=a. (Part A was solving for a few relavant things) Homework Equations...
31. How Can the Magnetic Vector Potential be Numerically Solved in the X-Y Plane?

hi everybody i want to solve the wave equation of the magnetic vector potential numerically in x-y plane grid, curl curl A= µ J anyone can help me thanks in advance
32. 3D Schrödinger Equation for a magnetic vector potential in cylinder coordinates

Hi there, during my work on my PhD thesis as an experimental physicist I ended up with a very theoretical problem: What does the wavefunction of an electron traveling through a magnetic vector potential look like? I chose a cylindrical coordinate system with a magnetic vector potential A...
33. Magnetic vector potential from an infinite current sheet with thickness t

Homework Statement An infinite sheet of copper conductor, thickness t, lies in the xz-plane. The sides of the sheet intersect the y-axis at y=\pm\frac{t}{2}. The current density in the sheet is given by: {\bf{j}}({\bf{r}}) = \begin{cases} j_0\left(\frac{y}{t}\right)^2{\bf{\hat{x}}}, &...
34. Boundary condition for magnetic vector potential

Hi I am studying magnetic vector potential from Griffiths book. The eq 5.76 in his book gives the boundary condition for the magnetic vector potential. \frac{\partial \vec{A_2} }{\partial n}- \frac{\partial \vec{A_1} }{\partial n}=-\mu_o \vec{K} where n is the vector perpendicular to the...
35. Intuitive/conceptual understanding of magnetic vector potential

Greetings everyone, I have been reading up on the magnetic vector potential, and I understand the vector calculus behind its definition and use. However, I am seeking an intuitive way to understand what it is conceptually, not just mathematically. I am assuming that any conceptual...
36. Magnetostatics: Magnetic Vector Potential

Homework Statement Give an expression for the magnetic field and show that a magnetic vector exists such as \vec{A}(P) = A(r)\hat{z} and \vec{B}(P) = \vec{\nabla} \times \vec{A} For the infinite wire shown in figure 1. Here is a link to the figure and problem statement. The problem is the...
37. How Does the Magnetic Field Behave Along the Z-Axis of a Current Loop?

Homework Statement A current loop (of circular shape) is located at the X-Y plane. What is the magnetic field on the Z axis? (Use the magnetic moment and magnetic potential) Homework Equations M=(Ia)/c \widehat{z} A=(MXr)/r^{3} B=curl(A) The Attempt at a Solution I got that the...
38. About the magnetic vector potential

While I understand the mathematical definition of the magnetic vector potential field A ( \bf {B} = \nabla \times \bf {A} ), I don't have an intuitive grasp of its physical meaning. For the (scalar) electric potential the matter seems rather simple. The dimensions of φ are energy per unit...
39. How is Magnetic Vector Potential Calculated for a Current Loop on the X-axis?

Homework Statement Find the vector potential \vec{A}(x,0,0) (i.e. on the x-axis) for a current loop of radius a, carrying a current I in the \phi direction. Homework Equations \vec{A} = \frac{\mu_0}{4\pi}\int_{V'}{\frac{\vec{J}dV'}{R}} Where R is the distance from the source point to the...
40. Magnetic Vector Potential Drawing

Why is it that I can't find a drawing on the internet of what the magnetic vector potential looks like around a current carrying wire?
41. How Does the Magnetic Vector Potential Change with Infinite Wire Length?

Homework Statement A finite wire lies on the z axis and extends from the point z=-V to z=+V. the equation from griffiths introduction to electrodynamics (equation 5.62) can be used to determine the vector potential at a point in the xy plane a distance s from the wire to be A=mu(naught)I/4pi...
42. What is the Surface Current and Magnetic Field for a Rotating Charged Disc?

Homework Statement There is a disc with radius R which has a uniformly-distributed total charge Q, rotating with a constant angular velocity w. (a) in a coordinate system arranged so that the disc lies in the xy plane with its center at the origin, and so that the angular momentum point in...
43. Deriving Jeans' Result from Magnetic Vector Potential

I'm trying to get from the magnetic vector potential \vec{A}(\vec{x},t) = \frac{1}{\sqrt{\mathcal{V}}}\sum_{\vec{k},\alpha=1,2}(c_{\vec{k}\alpha}(t) \vec{u}_{\vec{k}\alpha}(\vec{x}) + c.c.) where c_{\vec{k}\alpha}(t) = c_{\vec{k}\alpha}(0) e^{-i\omega_{\vec{k}\alpha}t}...
44. Is the Magnetic Vector Potential Dependent on a Constant Vector?

Homework Statement It's problem 2.b) on this page: http://www.phys.washington.edu/users/schick/322A/322-08ps3.pdf The Attempt at a Solution So, what it looks like to me, since the only terms in the actual vector potential are all multiplied, Lambda would have to be a constant, so...
45. Why Is the Magnetic Vector Potential Azimuthal Inside an Infinite Solenoid?

Why is the magnetic vector potential of a point inside a infinite soleniod azimuthal assuming the axis of solenoid is the z axis. Problem is the formulae A(\vec r)=\int_{v} \frac {\vec J(\vec r^{'})}{r} d\tau doesn't hold any more due to the infinite extent of the current
46. Magnetic vector potential of infinite wire

I don't know if this thread belongs in Introductory Physics or here, so please feel free to move it if you wish. The question is Find the vector potential a distance r from an infinite straight wire carrying a current I I know that the vector potential can be given by \frac{\mu_0}{4 \pi}...
47. How would one measure magnetic vector potential?

How would one measure magnetic vector potential? I know there is a formula, but is there an experiment that proves it?
48. Can the B field be obtained from the magnetic potential using matrix inversion?

How do you get the B field from the magnetic potential? I tried converting the curl into matrix format, but the corresponding matrix can't be inverted.