Difference between Math and Physics

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Many of us just think physics as subfield of mathematics. But it is totally wrong. I will explain why.

1. Original Ampère's Circuital Law

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In physics, Maxwell's Euqations are the four fundamental equations in electromagnetism. Among those, the Ampère's law, is stated as follows.

Ampère's Law - [3]
If is a closed loop enclosing a current , then up to a constant, the current through the loop is euqal to the circulation of the magnetic field around . To be precise,

Here we assume that is oriented so that and are related by a right-hand rule.

When Maxwell examined the original form of Ampère's Law above, he found that it was invalid for time varying electric and magnetic fields. If we restate the above original Ampère's Law in differential form, we get

where is the current density, and using Stoke's Theorem,

we finally have

2. What is Wrong?

2-1. Equation of Continuity [4]

First, recall what we've done in Fluid mechanics. Suppose a fluid of density flows with velocity vector field in a solid region in space enclosed by a smooth surface . Then if there are no sources or sinks in the region , the rate of fluid flowing into should equal to the rate of fluid flowing out of . The rate of fluid flowing into is

and the rate of fluid flowing out of is

Using divergence theorem, we get

so that

because the region was arbitrary.

2-2. Direct application to Current Flow

Current itself also can be considered as flow (of electrons), so that , is the charge density. And what we get is

2-3. So what is Wrong here?

If we assume is of class , we have . But by original Ampère's Law and continuity equation this is equivalent to saying that

so that it forces charge density to be constant over time (even if is time varying!) ; which is a serious flaw.

3. How do we Overcome this?

3-1. Mathematical Point of View [5]

First let's look at the differential form

If is time invariant (i.e steady flow), by original Amère's Law , it is equal to zero. If varies over time, it should be non-zero. Hence, introducing an aribtrary vector field by setting

will solve the issue. Observe that when is time invariant.

Now, take the divergence. Using continuity equation,

Also using Gauss's Law,

so that

Then most general form would be

where is any divergence free vector field, . So in purely mathematical sense, the modified Ampère's Law would be

3-2. Physical Point of View [6]

So the question reduces to

Can we specify the vector field ?

In complete mathematical sense, as long as it satisfies divergenceless, it can be nonzero. But

Ampère's Law is a law in physics not in mathematics!

So the arbitrariness of is not the ultimate solution. Specification should be done in purely physical sense - by the experiment. Since no one has observed any physical evidence for (and that's what Maxwell did in the past), it is assumed to be , which then reduces to modified Ampère's Law in Physics,