0) In your 2nd paragraph, I presume you meant to write "a changing magnetic field can induce an electric field" (Faraday's law.)
1) Your first question concerns, I think, a static charge distribution plus a set of changing magnetic fields (produced, perhaps, by a set of dynamic currents somewhere off-stage). In this case,
- the static charges will produce a conservative (curl-less) electric field.
- the changing magnetic fields will induce a divergence-less (non-conservative) electric field
The total electric field will be the sum of these two. (They obviously cannot be equal, since one has a divergence but no curl, and the other has a curl but no divergence.)
2) While a current is by definition charges in motion, magnetostatics studies cases where the currents do not change with time (and net charge density = 0). In those cases the current density $\boldsymbol{j}$ is constant in time, the resultant fields are magnetic fields that are also unchanging in time, and no changing electric fields are induced. So they're not being neglected, they just don't exist in these cases.
3) (your first bullet) The displacement current (aka Maxwell correction term) is required to maintain the consistency of the equation. The classic demonstration is of a capacitor being charged via wires. Here the current is constant but the charge on the capacitor plates is increasing linearly with time.
Imagine trying to calculate the line integral of the magnetic field around a path surrounding the wire, without the displacement current term.
- If one constructs a surface, bounded by the path, that intersects the wire, you can apply Stoke's theorem and Ampere's law and get a non-zero result.
- However, a second surface, also bounded by the path, that passes between the plates of the capacitor, has no current crossing it and so gives a zero result for the line integral.
The displacement current term removes this inconsistency, since the charging capacitor has a changing electric field between its plates.
4) (your 2nd bullet)
- In the static charge case, $V$ is just the normal Coulomb potential.
- For the time-varying case, the potentials are not unique: various possibilities, all related by gauge transformations, give the same fields. For example, the Coulomb gauge (in which $ \boldsymbol{\nabla \cdot A} = 0$) gives the same formula for the potential as for the static case. Other gauge choices will give different results. I'm sure your text discusses the gauge conditions.
I assume what is meant by Faraday's law of induction is what Griffiths refers to as the "universal flux rule", the statement of which can be found in this question. This covers both cases 1) and 2), even though in 1) it is justified by the third Maxwell equation1 and in 2) by the Lorentz force law.
The universal flux rule is a consequence of the third Maxwell equation, the Lorentz force law, and Gauss's law for magnetism (the second Maxwell equation). To the extent that those three laws are fundamental, the universal flux rule is not.
I won't comment on whether the universal flux rule is intuitively true. But the real relationship is given by the derivation of the universal flux rule from the Maxwell equations and the Lorentz force law. You can derive it yourself, but it requires you to either:
- know the form of the Leibniz integral rule for integration over an oriented surface in three dimensions
- be able to derive #1 from the more general statement using differential geometry
- be able to come up with an intuitive sort of argument involving infinitesimal deformations of the loop, like what is shown here.
If you look at the formula for (1), and set $\mathbf{F} = \mathbf{B}$, you see that
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{a} &= \iint_{\Sigma} \dot{\mathbf{B}} \cdot \mathrm{d}\mathbf{a} + \iint_{\Sigma} \mathbf{v}(\nabla \cdot \mathbf{B}) \cdot \mathrm{d}\mathbf{a} - \int_{\partial \Sigma} \mathbf{v} \times \mathbf{B} \cdot \mathrm{d}\boldsymbol\ell \\
&= - \iint_{\Sigma} \nabla \times \mathbf{E} \cdot \mathrm{d}\mathbf{a} - \int_{\partial\Sigma} \mathbf{v} \times \mathbf{B} \cdot \mathrm{d}\boldsymbol\ell \\
&= -\int_{\partial \Sigma} \mathbf{E} + \mathbf{v} \times \mathbf{B} \cdot \mathrm{d}\boldsymbol\ell
\end{align*}
where we have used the third Maxwell equation, Gauss's law for magnetism, and the Kelvin--Stokes theorem. The final expression on the right hand side is of course the negative emf in the loop, and we recover the universal flux rule.
Observe that the first term, $\iint_\Sigma \dot{\mathbf{B}} \cdot \mathrm{d}\mathbf{a}$, becomes the electric part of the emf, so if the loop is stationary and the magnetic field changes, then the resulting emf is entirely due to the induced electric field. In contrast, the third term, $-\int_{\partial\Sigma} \mathbf{v} \times \mathbf{B} \cdot \mathrm{d}\boldsymbol\ell$, becomes the magnetic part of the emf, so if the magnetic field is constant and the loop moves, then the resulting emf is entirely due to the Lorentz force. In general, when the magnetic field may change and the loop may also move simultaneously, the total emf is the sum of these two contributions.
If you are an undergrad taking a first course in electromagnetism, you should know the statement of the universal flux rule, and you should be able to justify it by working out specific cases using the third Maxwell equation, the Lorentz force law, or some combination thereof, but I can't imagine you would be asked for the proof of the general case from scratch, as given above.
The universal flux rule only applies to the case of an idealized wire, modelled as a continuous one-dimensional closed curve in which current is constrained to flow, that possibly undergoes a continuous deformation. It cannot be used for cases like the Faraday disc. In such cases you will need to go back to the first principles, that is, the third Maxwell equation and the Lorentz force law. There is no shortcut or generalization of the flux rule that you can apply. You should be able to do this on an exam.
1 This equation is also often referred to as "Faraday's law" (which I try to avoid) or the "Maxwell--Faraday equation/law" (which I will also avoid here because of the potential to cause confusion).
Best Answer
A long straight wire is also an inductor, with the direction of the magnetic field outside of the wire circulating in a direction determined by the right hand rule.
Such a wire doesn't have very much inductance (like a solenoid for instance), so it is not unreasonable to expect that the current through it could be decreased linearly.
When the current passes zero, the magnetic field surrounding the wire collapses, and due to Lenz's law, will induce an electric field in the wire in the direction opposite that of the original current flow (the answer).
In solenoids that are part of an LRC circuit, it is not unusual for this arrangement to produce "ringing", which is a continuation of this process in which there can be several cycles of damped alternating current produced, magnetic field generation and collapse.