52
votes
Accepted
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 ...
- 14.9k
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 ...
- 18k
15
votes
Wrong representation of the field of an axially magnetized ring magnet floating on the internet?
Figure 2 is the correct depiction of the magnetic field, Figure 1 is incorrect. If you look at the field lines of a regular bar magnet, and compare them to one side of the cross-section in Figure 2, ...
- 593
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 ...
- 78.1k
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 ...
- 13.8k
12
votes
Do moving charges still generate an electrostatic field?
When we speak of a current-carrying wire, we usually mean a wire that is electrically neutral (as most wires are). There are as many positive charges as there are negative charges. The net charge is ...
- 12.9k
10
votes
Accepted
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 ...
- 71.5k
10
votes
Wrong representation of the field of an axially magnetized ring magnet floating on the internet?
In a comment you write,
In my opinion, a superposition analysis of all the vectors will result in the [correct] field …
where your assumption about the correct field is stated in the question. You'...
rob♦
- 94.2k
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 ...
- 213k
9
votes
Accepted
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 $\...
- 71.5k
8
votes
Accepted
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 ...
- 119k
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.
- 39.7k
8
votes
Accepted
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 ...
- 49.4k
8
votes
Accepted
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<...
- 20.4k
8
votes
Accepted
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)....
- 60.5k
8
votes
Do moving charges still generate an electrostatic field?
The phrasing of your professor's question is unclear, as it talks about an infinite line of current, which is, ofc, charged. So the implication in the question is:
There are two linear charge ...
- 39.7k
8
votes
From what equations is magnetic field uniquely determined for a given current distribution?
The domain is important. The solution is given by Biot-Savart if it is the entire $\mathbb R^3$ space, with some assumptions on the regularity and decay of $B$ (e.g. $L^2$). In this case, you can ...
- 15.1k
7
votes
Accepted
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,
$$
\label{0}\tag{0}
\delta W = \...
- 16.5k
7
votes
Accepted
Is the current density constant in magnetostatics?
While at first glance this might seem a trivial question, I do not believe that it is. It depends on what exactly one defines the regime of magnetostatics to be. If you define magnetostatics to be the ...
- 11.5k
7
votes
Accepted
Example of information travelling faster than the speed of light?
Therefore, we can use the law of Biot-Savart
No, you cannot use the Biot Savart law in this situation. The Biot Savart law is derived from the magnetostatic assumption. That is, the assumption that $...
- 109k
7
votes
Metals and Magnets
To enlarge slightly upon gandalf61's answer, when a piece of ferromagnetic metal is placed near a magnet, it (temporarily) becomes a magnet too, and so the two magnets then attract each other.
- 97.8k
6
votes
Accepted
Deriving Biot-Savart Law from Maxwell's Equations
As far as I can remember, the formula you obtain is right. You can make this "problematic" integral disappear by using the following identity, that we will call "curl theorem" :
$$\int\vec{\nabla}\...
- 2,302
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-...
- 105k
6
votes
Accepted
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{...
- 99.9k
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. ...
- 2,587
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 (...
- 1,105
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 \...
- 7,328
6
votes
Wrong representation of the field of an axially magnetized ring magnet floating on the internet?
Just in case, I'll outline a quantitative approach to the problem. Just for the record, as has been already mentioned many times, fig. 2 is the correct picture.
I will consider a uniformly, axially ...
- 15.1k
6
votes
Do moving charges still generate an electrostatic field?
Yes, a single moving charge does produce an electric field as well as a magnetic field (although it isn’t really correct to describe that electric field as “electrostatic”). But a typical current-...
- 49.4k
5
votes
Accepted
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: ...
- 35.8k
Only top scored, non community-wiki answers of a minimum length are eligible