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






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

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  1. L

    I How to use geometrical symmetries -- general advice? (Vector potential)

    The following is an example from my script. I always have trouble identifying useful symmetries. Can someone explain to me why (for example) the vector potential doesn't have a ##z## dependence? I understand that there is no ##\varphi## dependency. I don't understand why the field of ##\vec{A}##...
  2. K

    A Confused about multipole expansion of vector potential

    Hello! I found an expression in this paper (eq. 1) for the multipole expansion of the vector potential. I am not sure I understand what form do the vector spherical harmonics (VSH) have. Also, for example, the usual hyperfine interaction operator is given by...
  3. K

    4- vector potential transformation

    What is 4- vector potential transformation under Gauge fixing ?
  4. G

    I 4-Current vector potential transformation under Gauge fixing

    I am given an initial vector potential let's say: \begin{equation} \vec{A} = \begin{pmatrix} g(t,x)\\ 0\\ 0\\ g(t,x)\\ \end{pmatrix} \end{equation} And I would like to know how it will transform under the Lorenz Gauge transformation. I know that the Lorenz Gauge satisfy...
  5. P

    I Vector Potential Multipole Expansion

    when you do a multipole expansion of the vector potential you get a monopole, dipole, quadrupole and so on terms. The monopole term for a current loop is μI/4πr*∫dl’ which goes to 0 as the integral is over a closed loop. I am kinda confused on that as evaulating the integral gives the arc length...
  6. A

    I Explicit non-local form for the vector potential?

    Hello everyone, I was looking at the light matter interaction Hamiltonian and I worked out a simple calculation where I was surprised to see that I had to introduce an explicitly non-local vector potential if I want to go further: $$\langle\psi|...
  7. arjun_ar

    Calculate the magnetic field from the vector potential

    I am trying to derive radial and axial magnetic fields of a current carrying loop from its magnetic vector potential. So far, I have succeeded in deriving the radial field but axial field derivation gives me trouble. My derivation of radial field (eq 1) can be found here. Can anyone point out...
  8. LCSphysicist

    The potential electric and vector potential of a moving charge

    I could try to apply the Liénard-WIechert equations immediatally, but i am not sure if i understand it appropriately, so i tried to find by myself, and would like to know if you agree with me. When the information arrives in ##P##, the particle will be at ##r##, such that this condition need to...
  9. Salmone

    I Derivative of the retarded vector potential

    In a problem of an oscillating electric dipole, under appropriate conditions, one can find, for the potential vector calculated at the point ##\vec{r}##, the expression ##\vec{A}=\hat{k}\frac{\mu_0I_0d}{4\pi}\frac{cos(\omega(t-r/c))}{r}## where: ##\hat{k}## is the direction of the ##z-axis##...
  10. Haorong Wu

    I Coefficients in expansions of a vector potential

    I have seen two expansions of a vector potential, $$\mathbf A=\sum_\sigma \int \frac{d^3k}{(16 \pi^3 |\mathbf k|)^{1/2}} [\epsilon_\sigma(\mathbf k) \alpha_\sigma (\mathbf k) e^{i \mathbf k \cdot \mathbf x}+c.c.],$$ and $$\mathbf A=\sum_\sigma \int \frac{d^3k}{ (2 \pi)^3(2 |\mathbf k|)^{1/2}}...
  11. Garlic

    I Cyclic rotation of the cross product involving derivation

    Dear PF, so we know that cross product of two vectors can be permutated like this: ## \vec{ \alpha } \times \vec{ \beta }=-\vec{ \alpha} \times \vec{ \beta} ## But in a specific case, like ## \vec{p} \times \vec{A} = \frac{ \hbar }{ i } \vec{ \nabla } \times \vec{A} ## the cyclic permutation of...
  12. Gustav

    Solving Vector Potential: Calculating Magnetic Field & Difficulties

    My solution for the vector potential ##A=2Cln\frac{x^2+y^2}{z^2} \hat{z}## is: a) I used the following formula to calculate the magnetic field $$ \mathbf{B} = \nabla \times \mathbf{A} = \left( \frac{dA_z}{dy} - 0 \right) \hat{x} + \left( 0 - \frac{dA_z}{dx} \right)\hat{y} + 0 \hat{z} =...
  13. D

    Deriving electric and vector potential

    1- Write down the complete MAXWELL equations in differential form and the material equations. 2- An infinitely extensive area is homogeneously filled with a material with a location-dependent permittivity. There are charges in the area. Give the Maxwell equations and material equations of...
  14. patric44

    Finding the magnetic field B given the vector potential A ?

    hi guys this seems like a simple problem but i am stuck reaching the final form as requested , the question is given the magnetic vector potential $$\vec{A} = \frac{\hat{\rho}}{\rho}\beta e^{[-kz+\frac{i\omega}{c}(nz-ct)]}$$ prove that $$B = (n/c + ik/\omega)(\hat{z}×\vec{E})$$ simple enough i...
  15. Tony Hau

    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, $$\nabla \times \vec A = \vec B$$...
  16. brotherbobby

    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...
  17. Adesh

    I'm not getting the curl of vector potential equal to magnetic field

    In this image of Introduction to Electrodynamics by Griffiths . we have calculated the vector potential as ##\mathbf A = \frac{\mu_0 ~n~I}{2}s \hat{\phi}##. I tried taking its curl but didn't get ##\mathbf B = \mu_0~n~I \hat{z}##. In this thread, I have calculated it like this ...
  18. L

    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?
  19. T

    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}'...
  20. Decimal

    Magnetic vector potential of a moving current sheet

    Hello, I start by applying the integral for the vector potential ##\vec{A}## using cylindrical coordinates. I define ##r## as the distance to the ##z##-axis. This gives me the following integral,$$\vec{A} = \frac{\mu_0}{4\pi} \sigma_0 v 2 \pi \hat{x} \int_0^{\sqrt{(ct)^2-z^2}}...
  21. R

    Vector potential of current flowing to a point from all directions

    I am having problem with part (b) finding the vector potential. More specifically when writing out the volume integral, $$A = \frac{\mu_0}{4\pi r}\frac{dq}{dt}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{?}\frac{1}{4\pi r'^2} r'^2sin\theta dr'd\theta d\phi$$ How do I integrate ##r'##? The solution...
  22. T

    Expressing the magnetic vector potential in terms of its curl

    We have the 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}'...
  23. TheBigDig

    Magnetic field of vector potential

    So I was able to do out the curl in the i and j direction and got 3xz/r5 and 3yz/r5 as expected. However, when I do out the last curl, I do not get 3z2-3r2. I get the following \frac{\partial}{\partial x} \frac{x}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{-2x^2+y^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}...
  24. W

    Working with Electric Field E, not Vector Potential A

    We commonly have E and B defined as: But how can I work in electric field E, instead of vector potential A?
  25. MichPod

    Does "magnetic charge" fit into vector potential?

    Disclaimer: I am not a physicist, just trying to learn some parts of it in my free time. And I do not mean to propose any kind of "new-theory" with my question. I always thought that Maxwell equations in their differential form for B and E may be reformulated/updated to include a magnetic...
  26. S

    Vector potential and and Intensity of Radiation

    Homework Statement Let a charge oscillate on a straight line between -a to +a with a frequency ω and according to the law: κ (x.t) = κ° sin(πx/a) e^(-iωt) I have to find the following: 1. Vector potential in the dipole approximation 2. Integral of the intensity of radiation Homework...
  27. It's me

    Dirac hydrogen atom vs spin symmetry

    Homework Statement Exact spin symmetry in the Dirac equation occurs when there is both a scalar and a vector potential, and they are equal to each other. What physical effect is absent in this case, that does exist in the Dirac solution for the hydrogen atom (vector potential = Coulomb and...
  28. S

    Vector potential of an infinitely long cylinder

    I have seen the other threads on an infinitely long wires vector potential.Its obvious that really small wires are just infinitely long cylinders: ∇xA=B ∫∇xA.da=∫B.da ∫A.dl = ∫B.da = φ(flux) For an infinite cylinder A.2πri=B.2πrih A=Bh A=μ0*I*h/(2π*r) Now for a cylinder of radius limr->0 =>...
  29. Another

    Vector potential in spherical coordinates

    in this problem i can solve v = ω x r = <0, -ωrsinψ, 0> in cartesian coordinates but i don't understand A in sphericle coordinates why? (inside) A = ⅓μ0Rσ(ω x r) = ⅓μ0Rσωrsin(θ) θ^ how to convert coordinate ?
  30. S

    Is the Vector Potential of a Toroid Nonzero Outside the Toroid?

    I am sure that the vector potential of a toroid isn't 0 even though its magnetic field is , does anyone have a derivation for the its the vector potential at a point P(x,y,z) outside the toroid ? i expect that since its curl is 0 we have a general form of : A = f(!x)ex + g(!y)ey+m(!z)ez Such...
  31. V

    Vector potential of a plate with a uniform, time-harmonic current density

    Hi all, i tried to do this question but got stuck on the last point . Can anyone help me please? The general form of vector potential: I got the answer for A1 vector potential but don't know what assumptions i need to get the expression for the A2. Does anyone know how one can derive it...
  32. Spinnor

    I Vector potential A_mu from scalar function theta(x_mu)?

    Suppose we have a scalar function θ(x,y,z,t) of space and time where theta is some angle (0≤θ≤2π) that represents the compact coordinate of a 3 dimensional space (x,y,z) filling membrane at the space time point (x,y,z,t) in a compact space dimension w. Suppose that charge density "pushes" on the...
  33. S

    Graphics illustrating vector potential for plane EM wave

    I'm looking for a diagram or animation that shows the vector potential A (in the form of arrows or whatever) superimposed on the E and B fields of a plane EM wave. Since A is not unique, maybe two or three versions of the diagram (including one with Coulomb guage). An animation with a slider to...
  34. allison west

    Vector Potential of a Rotating Mangetic Dipole

    Homework Statement A rotating magnetic dipole is built by two oscillating magnetic dipole moments, one along the y-axis and one along the x-axis. Find the vector potential at a point: (0, 0, ##z_0##) along the z-axis. Then find the magnetic field at ##z_0## . As the magnetic field is a function...
  35. DoobleD

    When does the vector potential A affect the E field?

    This is probably fairly simple, but I have a hard time to grasp the concept of the vector potential A in electromgnetism. Especially, in the following equation for the electric field : \vec{E} = - \nabla V - \frac{\partial \vec{A}}{\partial t} When does the second term is not 0 ?
  36. B

    Zangwill, problem 10.19 - A Matching condition for the vector potential A

    Homework Statement Show that the normal derivative of the coulomb gauge vector suffers a jump discontinuity at a surface endowed with a current density K(\vec r_s ) Homework Equations The vector potential A is given by: A=\frac{\mu_0}{4\pi}\int{\frac{J(x')}{|x-x'|}d^3x} The magnetic...
  37. I

    Vector differential displacement - Magnetic vector potential

    Homework Statement http://imgur.com/a/k7fwG Find the vector magnetic potential at point P1. Homework Equations Vector magnetic potential given by: $$ d \bar{A} = \frac{\mu I d\bar{l'}}{4 \pi | \bar{r} - \bar{r'} | } $$ The Attempt at a Solution I split up the problem in 3 parts, first...
  38. J

    Vector potential inside an infinite wire

    Homework Statement The problem statement is simply to find the vector potential inside and outside an infinite wire of radius R, current I and constant current density j using the Poisson equation. Homework Equations The Poisson law can be written A = μ0 /4π *∫(I/r*dl) or A = μ0 /4π *∫(i/r*dV)...
  39. Q

    I Christoffel Symbol vs. Vector Potential

    As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative. The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain...
  40. garylau

    What is the meaning of Ai Aii Bi Bii in Vector potential?

    Sorry may i ask a question here~ i don't understand how did GRIFFITHS prove the statement from(C) to (B) What is his logic in this case? and What is Ai Aii Bi Bii in the integral? thank you
  41. cetmoimonsieur

    Proving Vector Potential Pump Amplitude-Intensity Relationship

    Homework Statement Prove that the relationship between the dimensionless amplitude a of a wave and the intensity I of the wave is given by: Homework Equations B = curl (A) B: Magnetic Field A = [mass of electron * c / e ]*a*e^[i*(kx-wt)] where k is the wavenumber and is given by: 2pi /...
  42. 1

    Finding the magnetic field of a loop at far distances

    Homework Statement Loop of current ##I## sitting in the xy plane. Current goes in counter clockwise direction as seen from positive z axis. Find: a) the magnetic dipole moment b) the approximate magnetic field at points far from the origin c) show that, for points on the z axis, your answer is...
  43. S

    Vector potential from magnetic field

    <<Mentor note: Moved from non-homework forum>> If a uniform magnetic field ##{\bf{B}}=B_{z}{\bf{\hat{z}}}## exists in a hollow cylinder (with the top and bottom open) with a radius ##R## and axis pointing in the ##z##-direction, then the vector potential...
  44. M

    A Quantized E field, Coulomb Gauge with Interactions

    The common presentation for free field quantization proceeds with the Lorentz and Coulomb (##\phi = 0, \,\nabla \cdot \mathbf{A} = 0 ##) constraints. Then ##A## can be defined $$\mathbf{A} \propto \iint \frac{d^3 p}{\sqrt{2\omega_p}}\sum_{\lambda} \Big(e^{i\mathbf{p}\cdot...
  45. DavideGenoa

    I Magnetic vector potential of infinite straight wire

    The magnetic field generated by an infinitely long straight wire represented by the straight line ##\gamma## having direction ##\mathbf{k}## and passing through the point ##\boldsymbol{x}_0##, carrying a current having intensity ##I##, if am not wrong is, for any point ##\boldsymbol{x}\notin...
  46. brad2292

    How do we formulate the electromagnetic Lagrangian?

    I'm trying to understand how we set up the lagrangian for a charged particle in an electromagnetic field. I know that the lagrangian is given by $$L = \frac{m}{2}\mathbf{\dot{r}}\cdot \mathbf{\dot{r}} -q\phi +q\mathbf{\dot{r}}\cdot \mathbf{A} $$ I can use this to derive the Lorentz force law...
  47. I

    A Lagrangian of a monopole (Einstein notation is used)

    Hi everyone, I am trying to calculate the equation of motion of a charged particle in the field of a monopole. The magnetic field of a monopole of strength g is given by: \textbf{B} = g \frac{\textbf{r}}{r^3} And the Lagrangian by: \mathcal{L} = \frac{m\dot{\textbf{r}}^2}{2} +...
  48. P

    Multipole expansion of Vector Potential (A)

    Homework Statement So my teacher, as we made the multipole expansion of Vector Potential (\vec A) decided to proof that the monopole term is zero doing something like this: ∫∇'⋅ (J.r'i)dV' = ∮r'iJ ndS' = 0 The first integral, "opening" the nabla: J⋅(∇r'i) + r'i(∇⋅J) this must be equals 0 J =...
  49. DrPapper

    Vector Potential: Infinite Wire and Infinite Solenoid

    Homework Statement Homework Equations Provided in the questions I believe. Here's the triangle from question two. The Attempt at a Solution QUESTION SET 1 TOP OF PICTURE A.) I didn't know how to just "guess" what the constant should be so I actually worked it out. I found the constant...