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} .}

View More On Wikipedia.org
Recent contents Top users
  1. Z

    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...
  2. O

    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...
  3. T

    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...
  4. C

    Square root of volume in fourier expansion of the vector potential

    Hi. I just wondered why we use a 1/\sqrt{V} in the Fourier expansion of the vector potential. A regular 3 dimensional Fourier expansion is just f(\vec r) = \sum_{\vec k} c_\vec{k} e^{i \vec k \cdot \vec r} but as the solution to the equation (\frac{\partial ^2}{\partial t^2} -...
  5. L

    (Electric) Scalar and vector potential

    Homework Statement In the problem, the electric scalar and vector potentials are, \phi=0, \vec{A}=A_0 e^{i(k_1 x-2k_2y-wt)}\vec{u_y} I have to find E, B and S. Then, I have to calculate \phi ' that satisfies div\vec{A}+\frac{\partial \phi '}{\partial t}=0 Then calculate E and B...
  6. PerpStudent

    Geometric perspective of the vector potential

    I'm struggling with trying to visualize the vector potential as in the identity: B = ∇⨯A For starters, how does A relate to, say, a uniform magnetic field, which is quite easy to visualize. Then, how about the magnetic field around a bar magnet -- where is A? Any help would be appreciated.
  7. Z

    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...
  8. B

    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...
  9. M

    Vector Potential Stress-Energy Tensor

    The vanishing divergence(s) of the stress-energy tensor, which proves/demands (not sure which) the conservation laws for mass-energy and momentum, would seem to suggest to a naive person (me) that there might be some sort of "vector potential" associated with the stress-energy tensor, similar to...
  10. K

    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...
  11. M

    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...
  12. M

    Vector potential and energy calculations in magnetostatics

    I have some trouble with the calculation of energy in magnetostatics, using the vector potential A. From the classic formula that uses B*H, I find the expression (in magnetostatics) in terms of A and J (current density): \begin{align}W &=\frac{1}{2}\int_V{\vec{B}\cdot\vec{H}{\rm d}V}\\...
  13. I

    How can we find the vector potential for a given vector field?

    In problem of finding the vector potential of a vector F = yz i + xz k + xy j, the solution gives in Griffith's solution manual is http://img843.imageshack.us/img843/2725/vectorpotential.jpg Uploaded with ImageShack.us But I don't understand how we can integrate \frac{\partial Az}{\partial...
  14. G

    What Is the Physical Meaning of Vector Potential in Electromagnetism?

    Homework Statement The concept of a scalar potential is reasonably straight forward. It is the energy needed to move to a point from some arbitrary reference point, the reference point being the origin for most mechanical problems and infinity for most electromagnetic problems.And of course...
  15. M

    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...
  16. J

    Finding the magnetic field based on vector potential

    Homework Statement If you have |\psi>=cos(\theta/2)ei*l*\varphi/2+sin(\theta/2)e-i*l*\varphi/2ei\varphi and A=<\psi|\partial\varphi|\psi>\hat{r} find B in polar coordinates Homework Equations B=\nablaxAThe Attempt at a Solution So far I got...
  17. jfy4

    Decomposition of vector potential

    Hi, The vector potential for elctrodynamics, A_{\mu}, can be decomposed A_{\mu}\in\mathbf{0}\oplus\mathbf{1} but the \mathbf{0} part we ignore. My question is: do we ignore the scalar part because of experiment, or is it ignored for mathematical reasons I am naive about? Thanks,
  18. R

    Vector Potential applications in Aharonov-Bohm effect

    The Aharonov-Bohm effects show how a electro-magnetic field could affect a region of space in which the field had been shielded, although its vector potential did exist there and could interact with the wave function of say the electron. What practical application(s) (so far) can be derived...
  19. M

    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
  20. C

    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...
  21. D

    Fourier components of vector potential

    Hi all, I was reading the http://en.wikipedia.org/wiki/Quantization_of_the_electromagnetic_field#Electromagnetic_field_and_vector_potential" and I am a little bit confused. In this equation defining the vector potential \mathbf{A}(\mathbf{r}, t) = \sum_\mathbf{k}\sum_{\mu=-1,1} \left(...
  22. G

    What is the physical interpretaion of the vector potential.

    I was reading the text of electricity and magnetism by griffiths. Here I read a term called magnetic potential but I did not completely understood the physical essence of the term, neither it is explained in the book. It should have some physical interpretation as it is named a potential. In...
  23. S

    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}}}, &...
  24. I

    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...
  25. S

    Vector potential and constant magnetic flux density

    Hi, I would like to verify analytically that a vector potential of the form A=1/2(-yB0,xB0,0) produces a constant magnetic flux density of magnitude B0 in the z direction. (I guess I'd have to use the relation B=\forall\wedgeA...)
  26. S

    A beutifully simple interpretation of Vector Potential

    This is not my theory, or even new, rather pertaining to established physical knowledge, but I simply find it fascinating. It pertains to several areas of physics, and/or variational mathematics, so I've posted it here in the General Physics area. My reasons for posting is because it is one...
  27. H

    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...
  28. A

    Feynman's vector potential, a physical meaning clue

    Hi everybody! I'd like to understand the physical meaning of the Feynman's vector potential definition: $ A_{m}^{(b)}(x) = e_b \int \delta (xb_{\mu}xb^{\mu})db_m(b), \qquad m=0,1,2,3 $ (component m of the vector potential of the particle b at the point x) Here - the integration is done over...
  29. C

    Vector Potential: Reality or Tool?

    Has vector potential in classical electrodynamics a physical reality or it's just a mathematical tool?
  30. O

    Current density and vector potential

    Homework Statement What current density would produce the vector potential A(r)=(-kmu/2pi)In(r/a) (in the z direction) where k is a constant, in cylindrical coordinates? Homework Equations The Attempt at a Solution i have done this three times and i get zero current...
  31. P

    How do I determine the minus sign in gauge transformations?

    \vec{B}=rot\vec{A} \vec{E}=-\frac{\partial\vec{A}}{\partial t}-grad\varphi If I define \varphi=\widetilde{\varphi}-\frac{\partial f}{\partial t} \vec{A}=\widetilde{\vec{A}}+gradf where f=f(x,y,z,t) I will get \vec{B}=rot\vec{A}=rot\vec{\widetilde{\vec{A}}}...
  32. T

    Probability current of dirac equation with vector potential

    Homework Statement Given the probability/energyprobability current of the dirac equation j^\mu=\Psi^{+}\gamma^{0}\gamma^{\mu}\Psi with continuity equation \partial_\mu j^\mu = 0 I need to find the current when there is an additional vector potential, introduced via minimal substitution...
  33. pellman

    Lagrangian for E and B fields, not vector potential?

    Anyone know of a Lagrangian given in terms of E and B (or equivalently the tensor F) that yields Maxwell equations? A link or reference would be appreciated. I can write down such a Lagrangian which yields the two second-order Maxwell equations, but not the usual four 1st order equations...
  34. pellman

    EM vector potential - covariant or contravariant?

    Are potentials appearing in the Maxwell equations the components of a contravariant vector or a covariant vector? Let us be specific. metric is (+,-,-,-) . Let us write the potentials which appear in the Maxwell equations as \Phi and \vec{A}=(A_x,A_y,A_z) Is it then the case that...
  35. K

    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...
  36. V

    Magnetic Field from quantized vector potential

    Homework Statement Please help me find curl of A(vector potential) to find the magnetic field in the case of quantizes EnM fields.Homework Equations \vec{A}=\sum_{k,\lambda } e^{ikr}\sqrt{2\hbar / \omega_k}}\sqrt{\pi c^2/V}}(b_{\lambda,k} \hat{\epsilon}_{\lambda}(k)+b^{\dagger}_{\lambda,-k}...
  37. M

    Vector Potential of Long Solenoid

    Let me preface by saying that I am a freshman in an introductory level Electricity and Magnetism course. My professor has assigned this problem, as he briefly introduced the idea of vector potentials, along with curl and divergence operators. I am VERY much lacking in knowledge of any of these...
  38. T

    Vector potential in cavity of arbitrary shape

    we know that vector potential in a resonator satisfies the equation \intA(\lambda)A^{*}(\lambda^{'})dV=4\pic^{2}\delta_{\lambda\lambda^{'} So how about in cavity of arbitrary shape? Does this equation still valid? Thanks!
  39. A

    Vector potential due to oscillating dipole

    Homework Statement Calculate the magnetic vector potential A at a point p located at a distance r from the axis of an oscillating dipole of length s. It is assumed that r\gg s and that the current is the same throughout s. Homework Equations r=\sqrt{(x^2+(z-z')^2)}, where x,z is the...
  40. A

    Axially symmetric B field vector potential?

    Suppose you have an axially symmetric magnetic field for which the azimuthal component B_\phi = 0. This is all you know. What are some possible vector potentials \vec A (such that \vec B = \nabla \times \vec A) that would produce this field? (So we're working in cylindrical coordinates.) The...
  41. M

    Sufficient Condition for Existence of Vector Potential

    Hello! Can you tell me the sufficient condition for the existence of the vector potential? Thank you very much!
  42. J

    The vector potential and the Hamiltonian?

    Hi, I have a problem involving the Hamiltonian of a particle of mass m, charge q, position r, momentum p, in an external field defined by vector potential A and scalar potential X. Here's the Hamiltonian: H(r,p) = (1/2m)[p - qA(r,t)]2 + qX(r,t) = (1/2m)(pjpj - 2qpjAj + q2AjAj) + qX The...
  43. A

    How do you know the vector potential inside a solenoid only depends on r?

    How can you argue, by symmetry, that the vector potential inside a solenoid depends only on \rho, the perpendicular distance from the axis of the solenoid? And how can you argue that there is only a \hat{\phi} (azimuthal) component of the vector potential, such that \vec{A} takes the form...
  44. S

    Numerical solution of vector potential

    I have a field ,B, I need to find the other field,A, such that -> __ -> B = \/ x A I need numerical solution, given B sampled on a 3D computational grid (finite difference hexahedra) find A. What numerical methods could be used?
  45. F

    Urgent: Understanding Vector Potential from a Straight Wire

    urgent EMM question Homework Statement find the vector potential a distance s from an infinite straight wire carrying current I, check that dell dot A=0 and dellxA=B... ok i know the answer is A=A(s) z, in cylindricals. A(s)=(mu*I/2*pi)ln(s/a)z when a is a constant. can someone please...
  46. Z

    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...
  47. snoopies622

    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...
  48. Matterwave

    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...
  49. T

    P and T transformations of EM vector potential

    Help! What are the P and T transformation laws for the electromagnetic vector potential, A_\mu? and how are these consistent with the transformation laws of the electric and magnetic vectors that I am familiar with? under P: E is odd, B is even under T: E is even, B is odd When I try to...
  50. muscaria

    Gauge invariance of the vector potential

    The vector potential can be expressed in the following way: ∇^2 Ay-∂/∂y (∇∙A)=-μJy (Here only taking y components) Vector A is not determined uniquely. We may add derivatives of an arbitrary function (gradient) to the components of A, and the magnetic field does not change (curl of...
Back
Top