Magnetic Vector Potential

Back in electrostatics, since ∇×E=0, we could write E=-∇V, and likewise,

in magnetostatics, since ∇·B=0, we can apparently write B=∇×A…..

Ha… what on earth does this mean TTTT

Today I’m just going to think only about this…

First, the fact that the curl of the vector field representing E is 0 means that anywhere in the vector field representing E,

a hella tiny pinwheel (can I call this an “electric-field wind(?)”) has no ‘rotation’ caused by the electric-field wind.

That’s about how I thought of it.

(Curl is the thing that shows how much something rotates, and by “electric-field wind,” I just imagined the electric force-line as a sort of wind-blast lol)

I’m not sure if this will make sense, but let me just keep going with my take.

Okay okay okay, the pinwheel has no rotation~ because the pinwheel isn’t spinning.

The pinwheel doesn’t ‘spin like a propeller’ under the electric-field wind, but the pinwheel itself goes shoom~~~` flying around

(The reason we can be sure it flies is that ∇·E = ρ/ε0, so there’s a value.)

Then why, because ∇·B = 0, can we write ∇xA = B?

This time, let’s stick that ridiculously tiny pinwheel on a skewer and shove it into the vector field B,

hmm~ since the divergence of B is 0, the pinwheel itself won’t go shoom~ flying off!

And since ∇xB=μ0J (because there’s a value), I think it will just stay in place spinning byong-geul byong-geul.

(At what speed is it spinning byong-geul byong-geul? It’s spinning as fast as μ0J, I guess.)

Now for what we were really curious about

∇xA = B, what on earth does this mean~~~~~~~

If you stick that tiny pinwheel from before into some A vector field, it’ll spin, sure it will, right? Because that means it spins with the value B!

You’ve stuck a pinwheel like this in, but

apparently it rotates due to the A vector (if you think of the A vector just like an F vector, as something like a kind of force)

For example, if it spins like this, we express ‘it’s spinning like that’ as the vector B!

So the field that shows how the pinwheel spins at every point of the A vector field, and displays that as the B vector — that is the B vector field,

in other words, we can think of it as being the magnetic field, right?!

Oho

Okay, and one more thing

Now let me go over the meaning of this expression one more time.

When J flows along a wire like this, way~~~ over there A is~~~~

((Since J is a constant current, the magnitude of the red J vectors should be the same, right????

So I drew the magnitude (the length of the vectors) like that — hard to see, isn’t it T_T T.T

The sum of vectors like those is A!!!!!!

Now

why why why why why whyyyy why on earth!

let me interpret in my own way why we divide the J vectors by distance to the 1st power and sum them to get the A(r) vector.

Let’s say current flows in a freakin’ tiny segment like that (that’s why I wrote J instead of i)

Then over there at a distance η, B will form kinda like that, right????

But what did we say B was a moment ago? It was the degree of rotation of the pinwheel due to vector A.

If you imagine bringing that pinwheel from before to a spot η away over there, the source of the torque (turning force) that spins the pinwheel is A.

But! So very naturally, the direction of J and the direction of A are the same.

The meaning of that A is a vector that is 1/η times smaller than J relative to η,

(That’s why I drew the A vector on the left bigger than the A vector on the right.)

Clearly, what B means is “at a distance η away from a constant current, there arises a magnetic field B of that amount by the Biot–Savart law.”

What the A vector (vector potential) we learned today means is: at a spot η away from J(r’),

there’s some effect of J decreased in proportion to the 1st power of the distance,

and let’s call it A if you add up all those effects of J and multiply by μ0 over 4π.

(Multiplying by μ0 over 4π means…

it’s no longer ‘current’, right? Something… in the same direction as the current? sort of?? I dunno I dunno T.T potential? lol)

In short, I interpreted A as something like the effect(?) of J at a spot some distance away from J.

P.S. Just in case — you all know, right, that among the ways of notating vectors, besides drawing an arrow over the letter, there’s also using boldface (thick lettering)?????

J → magnitude of the vector J

J → vector J (has magnitude and direction)

Originally written in Korean on my Naver blog (2014-12). Translated to English for gdpark.blog.