Questions tagged [magnetostatics]

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34
votes
4answers
6k views

Why is this vector field curl-free?

The curl in cylindrical coordinates is defined: $$\nabla \times \vec{A}=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\hat {\...
12
votes
2answers
3k views

Deriving Biot-Savart Law from Maxwell's Equations

As an exercise, I've been trying to derive the Biot-Savart law from the second set of Maxwell's equations for steady-state current $$\begin{align}&\nabla\cdot\mathbf{B}=0&&\nabla\times\...
12
votes
3answers
583 views

Magnetic scalar potential far above a magnetic film

The situation I am looking at is a magneto-static problem of a finite magnetic film with magnetization $\bf{M}$. I would like to find the the magnetic field far above the plate. My expectation is that ...
9
votes
1answer
3k views

Is Biot-Savart Law valid for time-varying currents unlike Ampere's law?

I have just finished learning the basics of magnetism, and it should be noted that I am not very familiar with Maxwell's equations. Note: In the question, when I say "Ampere's Law", I am referring ...
8
votes
3answers
233 views

Are there cases where $\nabla\cdot\iiint\frac{\mathbf{J}(\mathbf{x}')}{\left|\mathbf{x}-\mathbf{x'}\right|}\mathrm{d}V' \neq 0$?

In Jackson's Classical Electrodynamics, Section 5.4 (Vector Potential), the author seems to assume that because $\nabla\cdot\mathbf{J} = 0$, the following holds for the current density (where the ...
7
votes
1answer
449 views

For a constant magnetic field, is there a gauge with both canonical momenta conserved?

To describe a constant magnetic field $\mathbf B=(0,0,B)$ (ignoring the motion along the $z$ dimension) within hamiltonian (or quantum) mechanics, one needs to choose a gauge. One common gauge is the ...
7
votes
1answer
181 views

On the applicability of Coulomb's law and the Biot-Savart law

Jefimenko's equations are $$\textbf{E}(\textbf{r}, t_r) = \frac{1}{4\pi\epsilon_0}\int \left[\rho\left(\textbf{r}', t_r\right)\frac{\textbf{r} - \textbf{r}'}{\left|\textbf{r} - \textbf{r}'\right|^3} + ...
6
votes
6answers
8k views

Why is curl of current density $\nabla \times \vec{J}$ equal zero?

I am revisiting the derivation for $\nabla \cdot \vec{B} = 0$ in magnetostatics for the field $\vec{B}(\vec{r})$ of a charge $q$ at position $\vec{0}$ with velocity $\vec{v}$. It proceeds like \begin{...
6
votes
2answers
541 views

What are the possible magnetic fields with constant magnitude?

A now-deleted answer to this recent question prompted me to wonder about this and I can't find a clear answer in the top layer of google results, so I thought I'd ask here. What are the possible ...
6
votes
4answers
2k views

How would you define electrostatics and magnetostatics starting from Maxwell's equations?

I'm reading Griffith's text, and he starts by defining Electrostatics as requiring the source charges don't move. I've seen a few slightly different definitions of electrostatics and magnetostatics. ...
6
votes
0answers
129 views

Total current for an arbitrary current density

Imagine a localized region $\mathcal{R}$ which contains a current density $\mathbf{j}$, which we take to be divergence-less, $\mathbf{\nabla\cdot j} = 0$. What is the total current associated with ...
5
votes
2answers
852 views

Cylinder , charge on surface, why is B inside zero?

Suppose we have a cylindrical wire of radius a carrying a current I. If the current is uniformly distributed over the surface of the wire, the magnetic field inside is zero. We can prove that easily ...
5
votes
1answer
863 views

What is the correct expression for the magnetic energy density inside matter?

I'll use a magnetized sphere as an example, of radius $R$, with a magnetization density $\vec{M}$. The magnetic moment of the sphere is $\vec{\mu} = \vec{M} \, V$. The magnetic field inside and ...
5
votes
1answer
188 views

Question about the definition of magnetostatics

From my understanding, magnetostatics is defined to be the regime in which the magnetic field is constant in time. However, Griffiths defines magnetostatics to be the regime in which currents are "...
5
votes
2answers
2k views

Derivation of Ampere's Law in Jackson

The derivation of Ampere's Law in Jackson E&M from the Biot Savart law is for the most part fairly traditional, using the $\nabla\times(\nabla\times A)$ identity on the vector potential: $$\nabla\...
5
votes
1answer
2k views

What is the importance of vector potential not being unique?

For a magnetic field we can have different solutions of its vector potential. What is the physical aspect of this fact? I mean, why the nature allows us not to have an unique vector potential of a ...
5
votes
1answer
991 views

Current geometry and Ampere's law

Under the right circumstances, Ampere's law $\oint \vec H\cdot d\vec \ell=I_{encl}$ can be used to deduce the field $\vec H$ at a point from the current enclosed by the circuit which produces $\vec H$....
5
votes
1answer
135 views

Deduction of $\mathbf H =\dfrac{\mathbf B}{\mu_0}-\mathbf M$

merry Christmas to all the users! I want to get to $\mathbf H =\dfrac{\mathbf B}{\mu_0}-\mathbf M$ from the superposition principle, like some texts have done in electrostatics with $\mathbf D=\...
4
votes
1answer
324 views

When does $\nabla \times B =0$?

In my lab, I use electromagnets to apply a magnetic gradient force to lots of very small (superparamagnetic) nanoparticles embedded in an elastic medium. I believe that these can be treated as ...
4
votes
1answer
128 views

Equation for the field of a magnetic dipole

In my electrodynamics class, my professor derived the equation for the field of the magnetic dipole $$\vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\frac{1}{r^3}[3(\vec{m}\cdot\hat{r})\hat{r}-\vec{m}]+\frac{2\...
4
votes
2answers
4k views

Ampère's law vs Biot Savart law

So I feel like I understand how both these laws work however it seems like Ampère's law will find the strength of the magnetic field at a point (the point is taken as $z$ in this example and the red ...
4
votes
2answers
57 views

Neat expression for the dipole potential in Fourier space?

In textbook electromagnetism we are used to seeing neat, coordinate-free, expressions for the scalar potential from an electric dipole (using Gaussian units) $$\phi(\mathbf{r}) = \frac{\mathbf{p} \...
4
votes
0answers
74 views

Work done by a magnetostatic field [duplicate]

How does a (static) magnetic field accelerate objects (like an iron nail) if we are taught that the magnetic force does always zero work?
3
votes
2answers
359 views

Dipolar magnetic field pressure

It is relatively easy to derive the potential energy stored into the magnetic field of an uniformly magnetized sphere of radius $R$ and total magnetic moment $\mu$ : \begin{equation}\tag{1} U_{\text{...
3
votes
1answer
1k views

Finding the vector potential of a spinning spherical shell with uniform surface charge?

I have problem solving the following magneto-static problem. I would greatly appreciate help and guidance. This is how the problem is stated: A spherical shell of radius $R$, carrying a uniform ...
3
votes
3answers
2k views

Direction of electric dipole moment and magnetic dipole moment

For electric dipole: The direction of the dipole moment is from negative to positive charge, but that of the electrostatic field is from positive to negative. But in the case of magnetostatics, the ...
3
votes
2answers
615 views

Challenging Magnetostatics Problem - the “blind spot” of a magnetic dipole

I'm reviewing for an electromag exam and I stumbled upon a problem that's really hard to figure out. Here it is: A small magnetic dipole with moment $\vec m = m_o \hat z$ is in a region with uniform ...
3
votes
1answer
4k views

Ampere's circuital law for finite current carrying wire

When I was studying about Ampere's circuital law. Then there comes a question in my mind that "whether this law is applicable for finite current carrying wire or not"
3
votes
2answers
531 views

Why does a solenoid's field look like this?

My book gives the above diagram but doesn't provide an explanation why the field looks like that. It simply says that the fields mostly cancel leaving the field above. Could someone walk me through ...
3
votes
1answer
64 views

Current density $\mathbf{J}$ of particle with magnetic dipole moment $\mathbf{m}$ [closed]

I'm solving some excercises on magnetostatics, and encounterded this on which i'm having some trouble. Given a particle of magnetic dipole moment $\mathbf{m}$, show that its current density is given ...
3
votes
2answers
3k views

What is the permeability of a permanent magnet?

The following is a diagram that shows the B- and H-fields of a permanent magnet. Inside the magnet, the H-field is in the opposite direction to the B-field, because of the magnetisation M-field and ...
3
votes
2answers
604 views

Curl of a simple magnetic field and resulting current distribution

I've been doing some thinking lately and here is my question: If one imagines that there is an auxiliary magnetic field $H$ whose spatial dependence is given by equation: $$H(x,y,z)=y\hat{i}$$ ...
3
votes
1answer
142 views

Building huge magnetic field gradient

Looking at how MRI works, I came across the fact that the spatial resolution depends on the magnetic field gradient, this gradient being created by "gradient coils". I was not able to find what the ...
3
votes
1answer
517 views

Assumptions when calculating $\vec{B}$ using Ampère's (circuital) law

When considering the same setup as in this question, i.e. a straight, infinitely long wire carrying the current $I$, Ampère's circuital law $$\oint_C \vec{B} \cdot \mathrm{d}\vec{r} = \mu_0 I_\text{...
3
votes
1answer
74 views

How is it possible that one face of my electromagnet produces stronger magnetic field than the other?

I work in an experimental research lab. I made an electromagnet by having the machine shop cut through a toroid core and hand-winding wire around it. The objective is to have a region of uniform ...
3
votes
0answers
1k views

Force between two current carrying wires: the general case

Assume two straight current carrying parallel wires with currents ($I$ and $I'$) flowing in the same direction, at a distance $R$ from each other. From Ampère's law (and from Biot-Savart as well) it ...
3
votes
0answers
231 views

Where can I find actual data for the hysteresis of a magnetic material?

When a magnet is exposed to an external magnetic field it can be further magnetized or demagnetized. This amount can be found by looking at the hysteresis curve for that magnet. I do not have the ...
2
votes
6answers
859 views

Forms of Maxwell's equations

In my physics class, I was taught two forms of one of Maxwell's equations: Ampere's law $$\vec{\nabla} \times \vec{B} = \mu J$$ and Maxwell-Ampere's law $$c^2\vec{\nabla} \times \vec{B} = \dfrac{\...
2
votes
4answers
4k views

Why is the magnetic field due to an infinite current sheet constant throughout all of space?

By Ampere's Law, using a rectangle as an Amperian loop, I know that it can be derived that the magnetic field is constant throughout all of space (i.e. it does not depend on the distance r from the ...
2
votes
2answers
1k views

Why is there a magnetic field around a magnet if there are no charges moving? [duplicate]

From what I understand, the magnetic force is just a relativistic effect of the electric force, and I understand how this is can be the case when considering the magnetic field generated by a current- ...
2
votes
2answers
310 views

Derivation of Ampere's Law from Biot-Savart

Our aim is to derive $\nabla\times \mathbf B=\mu_0(\mathbf J+\epsilon_0\frac{\partial E}{\partial t})$. To begin with, let $ \mathbf A=\frac{\mu_0}{4\pi}\int_{\mathbb R^3}\frac{\mathbf J(\mathbf r')}{...
2
votes
2answers
3k views

Solution of simple problems using only Maxwell equations in differential form

Solve simple electrostatic or magnetostatic problems using only Maxwell equations. For example: In every book there is an excercise to find a magnetic field outside a thin wire of radius $a$ with ...
2
votes
3answers
80 views

What determines whether we use a vector or scalar potential?

I understand that electrostatic potential is scalar because the curl of the field is zero, and this implies the electrostatic field is the gradient of the scalar potential to satisfy this. Similarly ...
2
votes
2answers
1k views

Ampère's law applied on a “short” current-carrying wire

Why doesn't Ampère's law hold for short current carrying wires? Of course, such wires should be part of a closed circuit, but that's a physical fact, and there is a numerous amount of ways to close it....
2
votes
1answer
2k views

How does the magnetic field generated from a rectangular cross-sectional current-carrying conductor differ from a circular cross-sectional conductor?

I can find much information of cylindrical conductors (ie. regular wires), where $B=\frac{\mu_0 i}{2 \pi r}$ and $r$ represents the radius (or distance) from the centre of the conductor, however I ...
2
votes
1answer
138 views

What is the physical basis of the $\rho \bf{u} \times \bf{\omega}$ force on a fluid vortex line?

In fluid dynamics, the force density on a vortex line is $\bf{f} = \rho \bf{u} \times \bf{\omega}$. In Faber, Fluid Dynamics for Physicists, ch. 4, this is "derived" by analogy with magnetostatics, ...
2
votes
1answer
196 views

Frozen in Magnetic Field Lines

A perfectly conducting fluid undergoes an axisymmetric motion and contains an azimuthal magnetic field $\textbf{B}_θ$ . Show that $ \dfrac{B_θ}{r}$ is conserved by each fluid element. Question from ...
2
votes
1answer
276 views

Can the magnetic force between two magnets be explained classically via magnetization current?

Usually in Electrodynamics courses the magnetic forces are analysed via wire-wire interaction. I don't remember being shown a classical explanation for magnetic forces between magnets. Since it is a ...
2
votes
1answer
150 views

Method of images in cylindrical coordinates for a circular ring of current

I am trying to determine if the following problem has an analytical solution using the method of images. The problem is an infinitely long (in $z$) cylinder of a material with permeability $\mu_2$ of ...
2
votes
1answer
103 views

Are there any south and north poles developed when we flow electric current through a Toroidal Solenoid?

I've read that when current flows in a solenoid it behaves like a bar magnet having both north and south poles. I was wondering how can we visualise north and south poles in a **Toroidal Solenoid".

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