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.
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...
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...
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...
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, $$\nabla \times \vec A = \vec B$$...
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...
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?
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}'...
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,$$\vec{A} = \frac{\mu_0}{4\pi} \sigma_0 v 2 \pi \hat{x} \int_0^{\sqrt{(ct)^2-z^2}}...
We have the retarded magnetic vector potential ##\mathbf{A}(\mathbf{r},t) = \dfrac{\mu_0}{4\pi} \int \dfrac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 \mathbf{r}'##
And its curl ##\mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{J}(\mathbf{r}'...
I am trying to understand the magnetic dipole field via loop of wire.
The above pictures show how this problem is typically setup and how the field lines are typically shown.
The math is messy but every textbook yields the following:
β = ∇xA = (m / (4⋅π⋅R3)) ⋅ (2⋅cos(θ) r + sin(θ) θ)
The...
Homework Statement
http://imgur.com/a/k7fwG
Find the vector magnetic potential at point P1.
Homework Equations
Vector magnetic potential given by:
$$
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...
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...
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}...
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...
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.
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...
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...
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...
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
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...
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...
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...
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...
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...
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...
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...
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
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...
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}}}, &...
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...
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...
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...
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...
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...
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...
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...
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...
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}...
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...
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
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}...
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.