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In classical electrodynamics they are strictly speaking not needed but they simplify things since writing E and B fields in terms of the potentials automatically solves the two homogeneous Maxwell equations. The freedom to choose different gauges also allows to simplify calculations.
In quantum mechanics the potentials play a more fundamental role since they directly appear in the Schrödinger equation (also see the Aharanov-Bohm effect).
Just wanted to say I appreciate your name.
Thanks, although I have to admit that I am not actually an expert on Superconducters.
You probably deserve it either way.
Thanks. I am very new to the concept of fields. Can you recommend any book for the same as YouTube and google doesn't really explain these in more depth. I want to get a more indepth feeling of what fields do, how they behave, etc.
What have you read so far, i.e. where did you come across the scalar and vector potentials?
There are treatments of electrodynamics and field theory at all kinds of levels. I am not up to date on the best "Introduction to physics" texts in english, so maybe someone else can jump in. As a complement to a more standard text, volume 2 of the Feynman lectures in physics is a classic.
To add to my previous post: for an approachable introduction to special relativity and relativistic field theory there is a corresponding book in the series "The theoretical minimum" by Leonard Susskind, but you would need his book on classical mechanics as a prerequisite. I think his lectures are also on youtube.
But it could just as easily occur classically, such as with a classical complex scalar field with a coupled Klein-Gordon action. Single-particle bosonic wavefunction equations (with or without coupling to a classical electromagnetic field) can always be reinterpreted in terms of classical fields.
Is there a physical example for this with a classical complex scalar field? Maybe in an effective treatment of some condensate.
You can then put electromagnetic fields in the same footing as matter through the Lagrangian constructed from the potentials. When matter is interacting with EM fields, the momentum of matter and the EM potentials actually need to be considered together. From the potentials you can also find "Gauge symmetry" which will become very important when quantum mechanics enters the picture. Some field calculations are also easier when you start from the potentials first (I believe Lienard-Wiechert potentials for moving charge fall in this category?).
Feel free to add/correct.
To calculate the E and B fields?
Because the potential is a 4-vector. If you fix the time basis and split the spatial components from the temporal components, you get a 3-vector and a scalar.