electromagneticelectromagnetism
The vector potential in magnetic field (represented by A), what does that physically mean? I have serious problem in understanding it intuitively.
$$\nabla^2\text{A} = -\mu_0 \text{j}$$
I read that the above equation can be derived using continuity equation. Can it be derived? if possible how? Thank you.
The reason for the introduction of the 'displacement current' was exactly to solve cases like that of a capacitor. A magnetic field cannot have discontinuities, unlike the electric field (there are electric charges, but there are not magnetic monopoles, at least as far as we know in the Universe in its current state). There cannot be a magnetic field outside the capacitor and nothing inside. en.wikipedia.org/wiki/Displacement_current
Probably the best way to understand what's going on is to use an electrical circuit analogy again.
The MMF is like a voltage source, and is the number of turns in your coil times the current in the coil.
The permeability of your core is like a resistor connected to your MMF source. It's high permeability so just like a low value resistor. You want a high magnetic flux, so your voltage source wants to cause a high current to flow in the resistor.
Now, if your magnetic core were a closed path like a toroid you would be all set.
However, you have a break in your magnetic circuit path. This is like an additional resistance in series with the small resistor you have from the ferrite core.
This "air gap" resistance analog is very much larger than your ferrite resistance analog. So no matter how high a permeability your ferrite has your MMF voltage source will not be able to drive a large current in it. (You won't get a large magnetic flux.) See the schematic below. The resistor values are arbitrary, just to illustrate the point.
So with a large air gap the only way to increase the magnetic flux is to increase the number of turns, N and/or the current in the coil.
simulate this circuit – Schematic created using CircuitLab
Best Answer
The short answer is the vector potential is such that the curl of it gives the B field. (Taking a curl is often easier than integrating.) There is also a fascinating quantum mechanical effect, such that the vector potential leads to a phase change of particles. I went looking on line and it's hard to beat the Feynman lectures. Enjoy!