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Physics/Maxwell's Equations

Question
Hi Sir,

Maxwell's equations are presented in both differential and integral forms..why is this so and when do we use one form over the other.

One of his equations relate to Ampere's law..do we have an explanation as to why there is a B field surrounding a wire that is carrying a DC current?

Thanks

> Maxwell's equations are presented in both differential and integral forms..
> why is this so

Both forms describe our Universe equally well. We can say, "There are no magnetic monopoles in our Universe," or we can say
∇B = 0
or we can say
∮B dA = 0
All of them are equally good description of the Universe we live in.

> when do we use one form over the other.

Depending on what situation we want to find a solution, we might use one or the other. It all depends on the initial conditions we find.

> do we have an explanation as to why there is a B field
> surrounding a wire that is carrying a DC current?

Depends what you mean by "an explanation."
You could say that we happen to live in a Universe where Ampere's Law is an accurate mathematical description of a phenomena we call a "B-field." When you apply this law to a wire with current going through it, you end up with a circular B-field around the wire.
Or you could say that we happen to live in a Universe where Coulomb's Law and relativity are accurate mathematical descriptions of electrical force, and then apply them rigorously to the case of electrons moving through a wire while the protons stand still. When you do that, you get a net electrical force on any charge moving in relation to that wire, a force that "happens" to match perfectly Ampere's Law and the law of B-field forces.

If you want an explanation for WHY our Universe operates that way, you're beginning to move into the realm of philosophy. Physics is good at revealing the laws that allow us to make useful predictions about our Universe, but it tries NOT to answer the question, "Why does our Universe have these laws and not some other ones?"

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I can help with understanding physics that does not involve eggs. I will NOT help with academic or professional questions, which are NOT limited only to homework. Please do not waste your time by asking a question that comes out of ANY kind of academic, professional, or business matters.

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Have been fascinated by physical laws ever since I learned, at age seven, that magnets work under water. My study continued through college and has not ceased even after I retired.

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B.A. in Physics (with honors) from University of California at Berkeley.M.A. in Physics (with honors) from University of Texas Austin.