What is Vector potential: Definition and 183 Discussions

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





v

=

×

A

.


{\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}

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  1. J

    Finding the magnetic vector potential

    I'm supposed to find the magnetic field, the scalar electric potential and magnetic vector potential for the following electromagnetic wave: \vec{E} = E_0 cos (kz - \omega t) \left \{ \hat{x} + \hat{y} \right \} Alright, the magnetic field goes as \vec{B} = \frac{1}{c} \hat{k} \times \vec{E}...
  2. muscaria

    Longitudinal waves and vector potentials.

    Hello, I was wondering if anybody knew of any material (books, papers etc..) which considers a possible connection between longitudinal waves and vector potentials, at least mathematically. I have been scouting about, but failed to find anything substantial. I understand that there seems to be...
  3. T

    Layman explanation of some simple EM equations

    So its been a while since I studied maxwells equations, anyway: So From my ignorant perspective, trying to derive conceptual meaning from these, I can see that the time dependant study there is some conductivity x the partial differential of the magnetic vector potential plus the cross product...
  4. Amith2006

    Fourier transform of vector potential

    Homework Statement I have question on doing the following indefinite integral: $$\int{d^3x(\nabla^2A^{\mu}(x))e^{iq.x}}$$ Homework Equations This is part of derivation for calculating the Rutherford scattering cross section from Quarks and Leptons by Halzen and Martin. This books gives the...
  5. DrClaude

    Understanding Magnetic Field in Length Gauge

    I've having trouble understanding one of the consequences of using the length gauge. The length gauge is obtained by the gauge transformation ##\mathbf{A} \rightarrow \mathbf{A} + \nabla \chi## with ##\chi = - \mathbf{r} \cdot \mathbf{A}##. Starting from the Coulomb gauge, we have $$...
  6. G

    Proof divergence of vector potential = 0

    Homework Statement I need to show that $$\del*\vec{A(\vec{r})}=\frac{\mu}{4\pi}\int{\frac{\vec{J{vec\r'}}}{\vec{R}}}d\tau=0$$ where A is the vector potential and R refers to "script r" or (r-r') where r is source point of charge and r' is the measurement point. tau refers to a volume integral...
  7. G

    How Fourier components of vector potential becomes operators

    Hello. I'm studying quantization of electromagnetic field (to see photon!) and on the way to reach harmonic oscillator Hamiltonian as a final stage, sudden transition that the Fourier components of vector potential A become quantum operators is observed. (See...
  8. M

    Conceptual origin of the magnetic vector potential....?

    In Griffiths, it seems that the conceptual introduction of the magnetic vector potential to electrodynamics was justified based on the fact that the divergence of a curl is zero; so we can define a magnetic field as the curl of another vector A and still maintain consistency with Maxwell's...
  9. P

    Curl of the vector potential produced by a solenoid

    Homework Statement / Homework Equations[/B] I was looking at Example 5.12 in Griffiths (http://screencast.com/t/gGrZEPBpk0) and I can't manage to work out how to verify that the curl of the vector potential, A, is equal to the magnetic field, B. I believe my problem lies in confusion about how...
  10. fricke

    Why Does Equation (7) in Magnetic Vector Potential Have a Negative Sign?

    Not a homework question! I am doing exercises for upcoming final exam. So, I get stuck at question 5.27 (Griffith 4th edition textbook). Question: Find the vector potential above and below an infinite uniform surface current with constant current sheet, K flowing at positive x direction. I...
  11. fricke

    Uniqueness of magnetic vector potential

    I able to prove magnetic field is uniquely determined but I am confused how to prove that magnetic vector potential is also unique. Can I say that magnetic vector potential is uniquely determined since magnetic field has unique solution? Thanks.
  12. M

    EM: Vector potential vs. Field tensor: Which is fundamental?

    In my lecture we were discussing the Lagrangian construction of Electromagnetism. We built it from the vector potential ##A^\mu##. We introduced the field tensor ##F^{\mu \nu}##. We could write the Langrangian in a very short fashion as ##-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}## In the end we...
  13. A

    Vector potential of the Magnetic monopole

    It is argued here that the use of vector potential to describe the magnetic field of a monopole is inherently wrong http://arxiv.org/ftp/physics/papers/0701/0701232.pdf It will indicate that the affirmation that charge quantization will be proved if a magnetic monopoles exists is wrong. The...
  14. T

    How to calculate magnetic vector potential?

    Source WIki: An axially symmetric toroidal inductor with no circumferential current totally confines the B field within the windings, the A field (magnetic vector potential) is not confined. Arrow #1 in the picture depicts the vector potential on the axis of symmetry. Radial current sections a...
  15. KleZMeR

    Vector Potential of Hertzian with Two Positive Charges

    Homework Statement Find the vector potential from two oscillating point charges on the z axis. Homework Equations z_+ = -z_0cos(\omega t) q1_+ = q2_+ = q The Attempt at a Solution I know how to solve this problem when the two charges are equal and opposite, but now the two oscillating...
  16. S

    Vector Potential A: Discontinuity at the surface current

    Homework Statement Prove Eqn. 1 (below) using Eqns. 2-4. [Suggestion: I'd set up Cartesian coordinates at the surface, with z perpendicular to the surface and x parallel to the current.] Homework Equations I used ϑ for partial derivatives. Eqn. 1: ϑAabove/ϑn - ϑAbelow/ϑn = -μ0K Eqn. 2: ∇ ⋅ A...
  17. S

    Calculate a vector potential of a simple loop

    Homework Statement Calculate the vector potential of a loop with current ##I##, raidus ##a##. Calculate it for anywhere in space and use approximation where ##r>>a##. Homework Equations ##\vec A=\frac{\mu _0I}{4\pi }\oint\frac{d\vec l}{|\vec r-\vec{r(l)}|}## The Attempt at a Solution Ok...
  18. P

    Normal derivative of vector potential discontinuity

    Homework Statement In Griffiths, the following boundary condition is given without proof: ∂Aabove/∂n-∂Abelow/∂n=-μ0K for the change in the magnetic vector potential A across a surface with surface current density K, where n is the normal direction to the surface. A later problem asks for a...
  19. E

    Understanding the Significance of Vector Potential and Gauge Choices in Physics

    Does the vector potential have physical significance or is it just a mathematical tool? What is your interpretation of Aharanov-Bohm effect? Are "gauge choices" really a choice or a restriction? It seems like gauge choices are really gauge conditions, and in some sense, a restriction. Thoughts...
  20. ShayanJ

    Vector potential of a current carrying circular loop

    Consider a circular loop of radius R that carrys a uniform current I. We know(by Biot-Savart law) that the magnetic field it produces on its axis is given by \vec{B}=\frac{\mu_0 I R^2 \hat z }{2(z^2+R^2)^\frac 3 2} . But let's calculate its vector potential: \vec{A}=\frac{\mu_0}{4\pi} \int...
  21. V

    Hydrogen Atom in homogeneous magnetic vector potential

    Hey! I did an quantum mechanical analysis of a Hydrogen Atom in a homogeneous magnetic vector potential (I know that it might be impossible to create this kind of field) out of curiousity. I showed it to some professors of mine, but they all said that they don't have time. So I decided to post...
  22. P

    Magnetic Vector Potential prob 5.32 Griffiths

    Please refer to the problem towards the end of page in the following link. It's related to discontinuity in normal derivative of magnetic vector potential across a current carrying surface. Prob 5.32 in Griffths. http://physicspages.com/2013/04/08/magnetostatic-boundary-conditions/ The...
  23. M

    Can Introductory Vector Calculus Prove C = Curl A?

    Homework Statement Given that the divergence of a vector C = 0, show that there exists a vector A such that C = curl A. Homework Equations See above. The Attempt at a Solution No clue. Can this be proved with introductory vector calculus? That's all I know, including many of the...
  24. E

    Vector potential components in small loop antenna

    Consider a small, thin loop in the (x,y) plane centered in the origin and with radius a. We are interested in the vector potential \mathbf{A} generated by the loop at a point P(r, \theta, \phi), with 2 \pi a \ll r, so at a great distance (moreover, a \ll \lambda). We need two coordinates...
  25. saybrook1

    Understanding the Curl of a Vector Potential in Spherical Coordinates

    Homework Statement For some reason I can't find anywhere online that gives a good example of the curl of a vector function in spherical coordinates. I need to compute ∇ X A where A = \frac{ksinθ}{r^{2}}\widehat{ϕ} If anyone can point me in the right direction of a good video or text...
  26. V

    Magnetic Vector Potential of Coil

    Hi Basically I want to examine the effect of a magnetic vector potential created by a coil on the spin of an electron in a Coulomb potential. The Hamiltonian of a charged particle in a Vector Potential is well known. But I have a problem in calculating the Magnetic Vector Potential of a...
  27. ShayanJ

    Spherical capacitor and vector potential

    There is a spherical capacitor with inner and outer radii of a and b respectively,has a dielectric material of small conductivity \sigma between its concentric conducting spheres.Calculate vector potential and magnetic field arising from this configuration. This is how I did it...
  28. H

    Dirac Expression for Vector Potential of a Magnetic Monopole Problem

    Hi, Homework Statement Consider the vector potential, \vec{A}(\vec{x}), below. The problem is to calculate \vec{A}(\vec{x}) explictly, and show that it has components A_{r}, A_{\theta} and A_{\phi} Homework Equations \vec{A}(\vec{x}) = \frac{g}{4\pi} \int_{-\infty}^{0} \frac{dz'...
  29. N

    Magnetic scalar and vector potential

    Mathematically, Scalar V_m Magnetic Potential is given by \overline{H}=∇V_m and Vector Magnetic Potential \overline{A} is given by \overline{B}=\overline{∇}X\overline{A} Is there any way I can explain it or define it in words?
  30. O

    Satisfying Coulomb Gauge: What are the Conditions for Vector Potential?

    Homework Statement We study the free electromagnetic field in a charge and current free cubic box with with edge length L and volume V. The vector potential in such a system is given via Fourier series: Homework Equations \vec{A}(\vec{r}, t) = \sum\limits_{k} \vec{A}_k(t) e^{i...
  31. S

    Magnetic Vector Potential and conductor

    Homework Statement Show that, inside a straight current-carrying conductor of radius R, the vector potential is: $$ \vec{A} = \frac{\mu_{0}I}{4\pi}(1-\frac{s^2}{R^2}) $$ so that ##\vec{A}## is set equal to zero at s = R Homework Equations ## \vec{A} =...
  32. PsychonautQQ

    How Do You Calculate the Flux of a Vector Field Through a Hemisphere?

    Homework Statement F has a vector potential A = <x,y,x^2+y^2>. Find the flux of F through the upper hemisphere x^2+y^2+z^2=1 z≥0 oriented with upward pointing normal vector The Attempt at a Solution So if F has a vector potential A, would you take the gradient of A to get F? in...
  33. B

    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...
  34. D

    Time derivative of vector potential - indices help

    Hey guys, So I'm reading something about vector potentials and I've come across this one line which is really annyoing me. Here's how it goes \frac{d}{dt}\mathbf{A}=\frac{\partial \mathbf{A}}{\partial t}+\frac{\partial \mathbf{r}}{\partial t}\cdot \frac{\partial }{\partial...
  35. A

    Vector potential due to sudden burst of current in a wire

    There is a wire oriented in the z direction with no current in it for t<0. At t=0, there is a burst of current: I(t) = qδ(t). What is the vector potential \vec{A}? (\vec{B}=∇×\vec{A}) My solution: let point P be located distance s from the wire, z be length along wire, and Ω be distance...
  36. T

    Magnetic field from vector potential function using tensor notation

    Homework Statement We will see (in Chap. 5) that the magnetic field can be derived from a vector potential function as follows: B = ∇×A Show that, in the special case of a uniform magnetic field B_{0} , one possible vector potential function is A = \frac{1}{2}B_{0}×r MUST USE TENSOR NOTATIONm...
  37. L

    Gauge conditions concerning vector potential and potential

    Hi everyone Homework Statement Give is a generall gauge transformation \Phi \rightarrow \Phi ' =\Phi -\frac {\partial \chi}{\partial t} and \vec A \rightarrow \vec A' = \vec A + \nabla \chi first task for now is the following: How do I have to choose Chi in order to fulfill the...
  38. D

    Finding scalar potential from E-field and vector potential

    Homework Statement Hi. This one I really am lost on :/ In my mind it seems rather easy, but I still can't figure it out. I have been given the E-field: \mathbf{E}\left( t,\,\,\vec{r} \right)=\frac{\kappa }{{{\varepsilon }_{0}}}\left[ \begin{matrix} ctx+{{x}^{2}}-{{y}^{2}} \\...
  39. P

    Vector Potential: -1/2 in Coulomb Gauge

    This is not a homework question but I think this is the best place to ask it. I was reading a book on quantum mechanics and I came across this expression using the Coulomb gauge in a constant magnetic field, \left(\vec{\nabla}\times\vec{A}\right)_{i} =...
  40. B

    Finding Vector potential due to an infinite cylinder?

    Hi there, 2nd year student, absolutely stumped on this don't even know where to begin. [I]"Determine the vector potential due to an infinite cylinder of radius, R, carrying a uniform current density, j. Use this to describe the magnetic field inside a current carrying wire I am using...
  41. S

    Vector Potential of Spinning Sphere

    Homework Statement I am just wondering, in Griffiths text, he solves this problem for a spinning shell. He states that the problem is easier if you tilt the sphere so it is spinning in the xz plane. Homework Equations When solving for the current density, Griffiths writes, $$...
  42. C

    How do I determine the correct vector potential for a given electric field?

    Hi PF members, I have a question about how to find the vector potential from a given electric field. For example, \textbf{E}=-\nabla\phi-\partial\textbf{A}/\partial t and \textbf{B}=∇\times\textbf{A} Given \textbf{E}=E_{0}\hat{x}, electrostatic potential may be 0 and...
  43. H

    Vector Potential and Zero Divergence

    I'm trying to understand when a vector field is equal to the curl of a vector potential. Why is it possible that there is always a vector potential with zero divergence? Relevent Equation: B=∇χA I'm trying to understand the proof that the above vector potential A can be one with zero divergence.
  44. M

    Meaning of Square of four vector potential

    The square of the four vector potential. (\phi,A)^2=\phi ^2 - A^2 What's the physical meaning of this lorentz invariant?
  45. sergiokapone

    The fate of the vector potential in the case of the existence of a mag

    The vector potential in classical electrodynamics can be introduced due to the fact that the magnetic field is the vortex: div \vec B = 0 → \vec B = rot \vec A In the four-dimensional form (including gauge) Maxwell's equations look particularly beautiful: \partial_{\mu}\partial^{\mu} A^{\nu} = j...
  46. S

    Complex vector potential of solid sphere and Heaviside function

    Homework Statement I have done this problem for the case of a spherical shell, however, I am not understanding how to go about this for a solid sphere. Homework Equations \vec{A} = \frac{1}{4 \pi} \int_{\phi' = 0}^{2 \pi} \int_{-1}^1 \int_0^R \rho_o \Theta(R-r) \sum_{l=0}^\infty...
  47. Spinnor

    Picture of electro-magnetic vector potential in 1+1 dimension spacetime?

    "Picture" of electro-magnetic vector potential in 1+1 dimension spacetime? I'm trying to model the electro-magnetic vector potential, does the following come close for 1 + 1 dimension spacetime? See sketches below. Consider an elastic string, under tension, between two fixed points A and B...
  48. N

    Problem related to the vector potential of B

    Homework Statement A particle of mass m and charge q moves in a region with a magnetic field B(r) which is time independent. What is \frac{dA}{dt} as seen by the particle. Homework Equations The Attempt at a Solution Since the B field is not varying with time, I know that the change in A will...
  49. F

    Vector potential with current density

    Homework Statement Hey, I got the current density \vec{j}=\frac{Q}{4\pi R^2}\delta(r-R)\vec{\omega}\times\vec{r} and now I should calculate the vector potential: \vec{A}(\vec{r})=\frac{1}{4\pi}\int\frac{j(\vec{r})}{|r-r'|}. Homework Equations The Attempt at a Solution here my attempt...
  50. Spinnor

    Can the electromagnetic vector potential be written in terms of a complex field?

    Is there a complex field that when properly interpreted yields the four components of electromagnetic vector potential, A_0, A_1, A_2, and A_3? Somewhat along the lines of the complex field ψ yielding information about a particles energy, momentum, and position probability. Thanks for any...
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