Vector potential Definition and 164 Threads

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 $$...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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'...

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?

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...

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} =...

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...

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...

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...

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...

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...

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...

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}} \\...

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} =...

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...

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, $$...

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...

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.

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

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...

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...

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...

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...

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...

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...

∇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...

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...

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} -...

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...

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.

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...