52 votes
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Why is this vector field curl-free?

The vector $\hat \varphi$ is not defined at the origin, because the coordinate transformation $$(x,y) \mapsto (r,\varphi) = \left(\sqrt{x^2 + y^2}, \arctan(y/x)\right)$$ is singular there. Hence your ...
Robin Ekman's user avatar
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25 votes

Why is this vector field curl-free?

There already are very good answers so I would just like to give some physical intuition why this vector field is curl-free even though it has non zero circulation. We can make an analogy of the curl ...
Diracology's user avatar
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13 votes

Why is this vector field curl-free?

That formula is valid outside the wire, where $J=0$. Maxwell 's equation says that $\nabla \times B = 0$ there. However there is no scalar field whose gradient is $B$ around the wire. This is the ...
Valter Moretti's user avatar
13 votes

Are the field lines on a bar magnet diagram contour lines?

No, the field lines are not contours. Contours usually connect places of equal magnitude, height, for example, on a map of a hill. The field lines on the diagram aren't connecting places of equal ...
John Hunter's user avatar
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10 votes
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Why does the electric field have non-zero curl for magnetic monopoles?

With the standard caveat that all theory is predicated on experimental validation, I would strongly expect magnetic charge to obey a continuity equation. If you make the modifications you suggest ...
J. Murray's user avatar
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9 votes

Why is this vector field curl-free?

OP's magnetic field $$\vec{B}~=~\frac{k }{\rho}\hat{\boldsymbol \varphi}, \qquad \rho~\neq~ 0,\tag{1}$$ in cylindrical coordinates $(\rho,\varphi,z)$ obeys (in a distributional sense) Ampere's ...
Qmechanic's user avatar
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9 votes
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When does $\nabla \times B =0$?

Ampere's law says that $$\nabla \times \boldsymbol{B} = \mu_0 \boldsymbol{J} + \epsilon_0\mu_0 \frac{\partial}{\partial t} \boldsymbol{E}$$ so "far away from sources" means that the current density $\...
J. Murray's user avatar
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8 votes
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Cylinder , charge on surface, why is B inside zero?

This diagram might give you the insight you are looking for: I visualize two segments of current (into the plane of the image) on opposite ends of an arbitrary (off axis) point, where both cover the ...
Floris's user avatar
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8 votes

Forms of Maxwell's equations

When $\frac{\partial E}{\partial t}=0$, then the 1st equation is valid. That is, it is for magnetostatics, where currents (and fields) are not time-varying.
JEB's user avatar
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8 votes
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Question about the definition of magnetostatics

Like all questions about definitions, the answer is not fundamentally "interesting" because, well, it all just boils down to your choice of definitions. But I would define "magnetostatics" to be the ...
tparker's user avatar
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8 votes
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Equation for the field of a magnetic dipole

On pages 187 and 188 Jackson explains the reason for this singular term. If you take a dipole whose magnetization is distributed uniformly in a sphere of radius $R$ then one can show that $\int_{r<...
hyportnex's user avatar
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8 votes
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Metals and Magnets

Newton's Third Law tells us that if object A exerts a force on object B then object B will exert an equal and opposite force on object A (this is required by the principle of conservation of momentum)....
gandalf61's user avatar
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7 votes
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Is Biot-Savart Law valid for time-varying currents unlike Ampere's law?

The Biot Savart law is equivalent to Ampere's law without the Maxwell term under the assumption that the charge density has no time dependence. So if we have the usual situation where there are ...
octonion's user avatar
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7 votes
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What is the correct expression for the magnetic energy density inside matter?

The energy of the magnetic field is the work required to establish a general steady-state distribution of currents and fields. This work is, in infinitesimal form, $$ \label{0}\tag{0} \delta W = \...
valerio's user avatar
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7 votes
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Is the current density constant in magnetostatics?

While at first glance this might seem a trivial question, I do not believe that it is. It depends on what exactly one defines the regime of magnetostatics to be. If you define magnetostatics to be the ...
Philip's user avatar
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7 votes
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Example of information travelling faster than the speed of light?

Therefore, we can use the law of Biot-Savart No, you cannot use the Biot Savart law in this situation. The Biot Savart law is derived from the magnetostatic assumption. That is, the assumption that $...
Dale's user avatar
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7 votes

Metals and Magnets

To enlarge slightly upon gandalf61's answer, when a piece of ferromagnetic metal is placed near a magnet, it (temporarily) becomes a magnet too, and so the two magnets then attract each other.
niels nielsen's user avatar
6 votes

Derivation of Ampere's Law in Jackson

In Jackson's textbook, paragraph $\S 5.3$ starts with equation (5.14) given here as (001) \begin{equation} \mathbf{B}\left(\mathbf{x}\right)=\dfrac{\mu_{o}}{4\pi}\int\:\mathbf{J}\left(\mathbf{x}'\...
Frobenius's user avatar
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6 votes
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What is the importance of vector potential not being unique?

There is no "physical aspect of this fact". The physical variables are the electric and the magnetic field, not the potentials. Introducing the potential is aesthetically and technically pleasing, but ...
ACuriousMind's user avatar
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6 votes
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Deriving Biot-Savart Law from Maxwell's Equations

As far as I can remember, the formula you obtain is right. You can make this "problematic" integral disappear by using the following identity, that we will call "curl theorem" : $$\int\vec{\nabla}\...
Frotaur's user avatar
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6 votes

What are the possible magnetic fields with constant magnitude?

Partial answer: if there are no currents, all such magnetic fields must be constant. In the absence of currents, we have $$\nabla \cdot \mathbf{B} = 0, \quad \nabla \times \mathbf{B} = 0.$$ The curl-...
knzhou's user avatar
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6 votes
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What exactly is enclosed current?

You have highlighted the fact that you can choose *any (well belaved) surface as long as it is bounded by the Amperian loop which means that $\displaystyle \mu_0 \iint_{S_1} \mathbf{J} \cdot d\mathbf{...
Farcher's user avatar
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6 votes

Shouldn’t the magnetic field of a solenoid be affected by length?

The field at the center does depend on the total number of turns, but that is only relevant for a solenoid of finite length. Meanwhile, Ampère's Law only applies to a solenoid of infinite length. ...
Adam Herbst's user avatar
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6 votes

Are the field lines on a bar magnet diagram contour lines?

No. Contour lines depict scalar fields, whereas the information in a magnetic field cannot generally be represented in terms of a scalar field. Contour lines are a way of representing a scalar field (...
ComptonScattering's user avatar
6 votes

Induction in a magnetostatic scenario

Maxwell's equations $$\begin{align} \vec{\nabla} \times \vec{E} &= -\frac{1}{c} \frac{\partial \vec{B}}{\partial t}, \qquad \qquad \vec{\nabla} \cdot \vec{E}= 4 \pi \rho, \\ \vec{\nabla} \times \...
Hyperon's user avatar
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5 votes

Ampère's law vs Biot Savart law

When you ask why does Ampère's law often apply to an infinitely long wire and the Biot Savart Law apply to a short wire? you're completely mistaken: both Ampère's law and the Biot-Savart law ...
Emilio Pisanty's user avatar
5 votes
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Why is the magnetic field due to an infinite current sheet constant throughout all of space?

The central question you must ask yourself is: how do you tell how far away you are from a featureless infinite wall? No matter what distance you are from it, the wall looks exactly the same: ...
probably_someone's user avatar
5 votes

Why don't we use Ampere's law to find the magnetic field due to a wire of finite length at its perpendicular bisector?

Ampere's law always applies everywhere (in static conditions); it is one of the Maxwell equations. The only thing is that you have to apply it correctly. It only ever tells you about one component of $...
Andrew Steane's user avatar
5 votes

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

Why do you claim that $$ \int_V \nabla \cdot \frac{{\bf J}({\bf x}')}{|{\bf x}-{\bf x}'|} {\rm d}V' = -\int_V \nabla' \cdot \frac{{\bf J}({\bf x}')}{|{\bf x}-{\bf x}'|} {\rm d}V'$$? This doesn't seem ...
Adam Latosiński's user avatar
5 votes
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Derivation of Ampere's Law from Biot-Savart

The Biot-Savart law says that under magnetostatic conditions ($\frac{\partial}{\partial t}\rightarrow 0$), $$\mathbf B(\mathbf r) = \frac{\mu_0}{4\pi}\int \frac{\mathbf J(\mathbf r') \times (\mathbf ...
J. Murray's user avatar
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