I think I have tried both. I have no easy access to an academic library, but that's possible if someone provides me a book to look at.

Ion sources are not always used in accelerators: they are often used to obtain special plating on metals, and also to irradiate biological tissues. Definitely not vacuum.

Isn't ${\partial j \over \partial t} = 0$ the definition of a steady current?

Sorry, never heard about gyrosurfing.

@Jon Custer. Do you mean that the term "vibrate" is inadequate or something else?

@ZeroTheHero. I think this expression is said to be valid even for finite solenoid (according to the Wikipedia article).

I'm always surprised that there are questions that receive no upvote, but at the same time answers to that question that receive 4 upvotes or more: if an answer is so good that it deserve 4 upvotes, then the corresponding question that HAS LEAD to that answer should receive at least 1 upvote. That's my humble opinion.

@Ján Lalinský. Indeed, you are completely right. I have rewritten my answer. Thank you again for helping me and pointing out my mistakes.

@Ján Lalinský. I tried to answer to your objection in an edit of my answer.