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Question

Are moving charges and time-varying electric fields really distinct causes of magnetic fields??


Various EM Phenomena

Two larger purposes are providing some Background to make sure I follow Maxwells equations, and gaining any intuition about these four or five seemingly disjoint laws, please flag anything that’s off at all:

1.Electric charges exist, and attract/repel. We can call the would-be force per unit-charge from them the “electric field”.

No analogous “magnetic charges” exist. (Usually said as, “no magnetic monopoles”)

2.In addition to charges, electric fields can also be made from the magnetic field changing at that point.

3.Magnetic fields put force on moving charges

Q1: Correct so far?


The Important One(s)

Back to the main question:

4.(Or 4 and 5.) Magnetic fields are made by changing electric field, only. One example of this is a moving charge (which obviously causes a changing electric field). One moving point charge creates a magnetic field that can be determined just the same by looking at how that charge’s electric field is changing. But with current, the apparent steady, unchanging electric field around a current is actually new fields replacing the ones moving away. So it actually is moving fields superimposed, and hence makes a magnetic field.

The question is basically, is that ⬆️ paragraph correct? (Q2)


How they relate?

Feel free to add how they may be less total independent phenomena than the 5 listed (4 if Im right). Maybe 1 and 2 come from stationary electric charge being like a moving magnetic field.

Q3: Other than whether 4 and 5 are really same or not, any way to see two or more of these mechanisms as related in any way?

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    $\begingroup$ I just realized my answer had a flaw. I will correct it tomorrow (I am too sleepy now), and also check if everything you said is correct. $\endgroup$
    – user65081
    Aug 24, 2021 at 6:06
  • $\begingroup$ @Wolphramjonny Ok thanks much. Sleep well. $\endgroup$
    – Al Brown
    Aug 24, 2021 at 6:11

2 Answers 2

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A not-so-well-known fact is that it is possible to obtain a complete solution for the Maxwell equations provided you assume the charge and current distributions fall sufficiently fast as you go to spatial infinity. These solutions are generalizations of the Coulomb and Biot--Savart laws for time-dependent cases and are known as Jefimenko equations. They are presented in usual Electromagnetism textbooks and I quote here their expressions from Griffiths' book (tags according to 4th edition): $$\begin{align} \mathbf{E}(\mathbf{r},t) &= \frac{1}{4\pi \epsilon_0} \int \left[\frac{\rho(\mathbf{r}', t_r)}{R^2} \hat{\mathbf{R}} + \frac{\dot{\rho}(\mathbf{r}', t_r)}{c R} \hat{\mathbf{R}} - \frac{\dot{\mathbf{J}}(\mathbf{r}',t_r)}{c^2 R}\right] \mathrm{d}\tau', \tag{10.36} \\ \mathbf{B}(\mathbf{r},t) &= \frac{\mu_0}{4\pi} \int \left[\frac{\mathbf{J}(\mathbf{r}',t_r)}{R^2} + \frac{\dot{\mathbf{J}}(\mathbf{r}',t_r)}{c R}\right] \times \hat{\mathbf{R}} \,\mathrm{d}\tau', \tag{10.38} \end{align}$$ where I use SI units and I denote $\mathbf{R} = \mathbf{r} - \mathbf{r}'$ (Griffiths uses a cursive $r$), $t_r = t - \frac{R}{c}$ is the retarded time.

These equations are not particularly convenient for direct computation (the integrals might get quite cumbersome), but they make it some phenomenological considerations much clearer. For example, notice the dependence of the fields on the time derivative of the current: it confirms that a changing electric field isn't really the cause of induction of a magnetic field. In fact, the change in current that you need to change the electric field happens to be the very same you would need to induce the corresponding magnetic field. It's sort of a coincidence, so to speak, not a cause-consequence relation. Maxwell's equations can't really make this distinction, but once they are solved you can see it directly from the expressions.

Given this, let me try to address each one of your questions.

Q1

  1. At the level of Classical Electrodynamics, this is correct, but I should mention magnetic monopoles are an active line of research in both Particle Physics and Condensed Matter Physics.
  2. Sort of. We do observe this, but it is by pure coincidence. Fundamentally, electromagnetic fields are generated by charges and currents, sometimes by means of their time derivatives.
  3. Correct.

Q2

  1. In the absence of magnetic monopoles, magnetic fields are caused by electric currents, such as the one of a moving charge. The electric field caused by a current is due to the charge density, its time derivative and the time derivative of the current. The same current that generates an electric field generates a magnetic field, and they often can be mistaken as the cause of one another.

Q3

I'd rather see Maxwell's equations in a different way. As you mentioned, there exist electric charges and they are subject to forces related to electric and magnetic fields. This is my view of the Lorentz force law. Then the Maxwell equations say

  1. Wherever there is charge, the electric field diverges or converges, making close charges be attracted or repelled according o their signs
  2. Magnetic field lines always are closed. There are no sources or drains of magnetic field.
  3. The electric field always curls around a variation of magnetic field. From Jefimenko's equations we know this isn't a cause-consequence relation, but rather some sort of coincidence.
  4. The magnetic field curls around currents and changes in the electric field. While the former is a cause-consequence effect (the current generates the field that is curling around it), the latter isn't: it is a coincidence due to the changes in current that generates both the change in electric field and the magnetic field (we know this due to Jefimenko's equations).
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    $\begingroup$ @AlBrown I've updated my answer accordingly. Please take another look at it $\endgroup$ Aug 24, 2021 at 6:55
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    $\begingroup$ Wow you wrote that magnetic and electric fields are caused by charges and currents and their derivatives. Could we make the equations with no fields as dependent variables? Oh thats what jeffemenk is. Thanks a lot $\endgroup$
    – Al Brown
    Aug 24, 2021 at 8:20
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    $\begingroup$ @AlBrown Yes. When making computations, there is no problem in assuming this point of view. It yields the correct results and makes things quite simpler to compute, I'd say. $\endgroup$ Aug 24, 2021 at 14:41
  • $\begingroup$ Question after all? 1. Hoping to use jefimenko. We’re getting E and B at point $r$, time t, by integrating over every other point $r’ \neq r$ in $\mathbb{R}^3$ (as if $r$ was (0,0,0) is one way to say it). So the integral is just a volume integral? On wikipeda they have $\int [...]~ d^3r’$ and you wrote from griffiths $d \tau ‘$. Nevet seen $d^3$ or $d\tau$. All three of those are same (Incl “a volume integral”)? So in reality we might have $\int~ [\int ~( \int ... dx)~ dy]~dz$ depending on functional form?Thanks 2. The time delay means it’d be what the config WAS not is. Why does that work? $\endgroup$
    – Al Brown
    Aug 24, 2021 at 19:26
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    $\begingroup$ @AlBrown Both $\textrm{d}\tau$ and $\textrm{d}^3r$ mean volume integral. The delay time does mean you should integrate on past configuration, not present ones. That is due to the finite speed of light, which is already a feature of Classical Electrodynamics. The assumption of falling off fast enough is there in order for the integrals to converge. If you pick $\rho(\mathbf{r},t) = \text{constant}$, the integrals will be divergent. Luckily, situations of physical interest will often (if not essentially always) have charges extending over only a finite portion of space. $\endgroup$ Aug 25, 2021 at 2:14
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Pulling some highlights from Alves’ detailed answer:

Q3: Maxwell Equations

The four Maxwell Equations (five relationships) are best understood as:


1.Electric charges cause electric fields that converge/diverge to/from the charge: $$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0} $$

(In other words, charges attract/repel: $F_{1,2}= k_e \frac{q_1q_2 }{r^2}$)


2.Currents cause magnetic fields that curl around the current: $$\underbrace{\nabla {\times} \vec{B} = \mu_0 \vec{J}}~\text{ } ~(+ \mu_0\varepsilon_0 \frac{\partial \vec{E}}{\partial t})$$

(In other words, currents attract/repel: $F_{1,2}= \mu_0 \frac{\vec{I_1}\cdot\vec{I_2}L}{2\pi r}$)


3.Magnetic field lines are always closed, with no sources or drains of field: $$\nabla \cdot \vec{B} =0 $$

(Quite straightforward until here.)


4.The electric field curls around changes in the magnetic field: $$\nabla \times \vec{E} = \frac{-\partial \vec{B}}{\partial t}$$

This is a consequence, but during Maxwell is considered an additional relationship.


5.The magnetic field curls around changes in the electric field:

$$\underbrace{\nabla {\times} \vec{B} = \mu_0\varepsilon_0 \frac{\partial \vec{E}}{\partial t} }~\text{ } ~(+ \mu_0 \vec{J})$$

This is a consequence, but during Maxwell is considered an additional relationship.


From Jefimenko's equations we know that 4., 5. are not causal - not from the curl to the derivative nor vice versa. The terms are due to current variations affecting each field individually. But without fields, the analysis is more difficult. If fields (Maxwell) are used, the terms in 2 and 5 must both be included.



Q1

A. Electric charges exist, and attract/repel. We can call the would-be force per unit-charge from them, the “electric field”.

True, and furthermore currents also attract/repel. We can call the would-be force from a current on another current if it was present, the “magnetic field”.

B. No monopole “magnetic charges” exist?

Partly True. Normally yes, and until recently was true everywhere. Now magnetic monopoles are an active line of research in both Particle Physics and Condensed Matter Physics.

C. In addition to charges, electric fields can also be made from the magnetic field changing at that point?

Not Technically True See Q3

D. Magnetic fields put force on moving charges

True, and as a direct cause. Currents put forces on other currents



Q2

False, from all perspectives

Under Maxwell, the effect on $B$ from current and $\frac{\partial \vec{E}}{\partial t}$ are separate terms. Both aspects must be considered.

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