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?
Best Answer
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
Q2
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