How to picture the magnetic vector potental A

In summary, the vector potential A can be thought of as circulating around points where the magnetic field B is nonzero, with its net circulation giving the B field according to the right-hand rule. However, different A's can produce the same B field, making it a mathematical convenience in classical E&M. In quantum physics, particles can directly interact with A, as seen in the Aharanov-Bohm effect. A gauge invariant quantity, A can be pictured in terms of the geometric interpretation of the curl and each component of A acts as an independent scalar potential for the corresponding current component in the Lorenz gauge.
  • #1
si22
4
0
whats a good way to picture the vector potental A in terms of B & like what exactly is A & how does it even exist outside a torus where B & etc =0

for example its easy to see the electric potential uses the electric field E like E*ds & its quite obvious,
wheras how does A not even contain the B field

also why is A sometimes said to not even exist or is just a paper shortcut when it actualy seems to work or exist in some way. thanks
 
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  • #2
Since the curl of the vector potential A is equal to the magnetic field B, a good way to think of it is that A circulates around any point where B is nonzero--its net circulation around a point gives the B field at that point, according to the right-hand rule. It is important to remember though that you can always write down different A's to produce the same B field--this is called choosing a gauge. For example, a uniform B field in the z direction could be represented by any of the following:
A = -By i
A = Bx j
A = -By/2 i + Bx/2 j
where i is the unit vector in the x direction, and j is the unit vector in the y direction, and B is the magnitude of B.
If you plot these, you will see that they all look quite different, but they all circulate around in a similar fashion.

In classical E&M, the B field is the measurable quantity, so A is said to just be a mathematical convenience. However, in quantum physics, particles can be affected by magnetism even if they never pass through a region of nonzero B--instead they directly interact with A. A good example is the Aharanov-Bohm effect: http://en.wikipedia.org/wiki/Aharanov-Bohm_effect
 
  • #3
What do you mean the vector potential ##A## isn't given in terms of the magnetic field ##B##? ##\nabla \times A = B## so you can picture it in terms of the usual geometric interpretation of the curl (think of the vorticity of velocity fields of fluids). The reason classically that ##A## is said to simply be a purely mathematical field (and not a physical field) is because it is not a gauge invariant quantity. I can take ##A \rightarrow A + \nabla \varphi## and I will still get the same physical magnetic field ##B## i.e. ##\nabla \times (A + \nabla \varphi) =\nabla \times A##.
 
  • #4
I've found it helpful to look at the vector potential in the Lorenz gauge -- where each component of the vector potential acts like an independent scalar potential for the corresponding current component...so you can imagine each infinitesimal current-element in the <x, y, z> direction as a source for a corresponding 1/r A field whose vector points in the same <x, y, z> direction. What you lose, though, is the ability to see the direction of the Lorentz force by just comparing the directions of two vectors at a single point.
 
  • #5


The magnetic vector potential A is a mathematical construct that is used to describe the magnetic field in a given space. It is related to the magnetic field B through the equation B = ∇ x A, where ∇ is the gradient operator.

One way to picture the magnetic vector potential A is to imagine it as a series of arrows pointing in different directions, with the direction and length of each arrow representing the strength and direction of the magnetic field at that point. This visualization can help to understand how A is related to B and how it exists outside a torus where B = 0.

It is important to note that A is not a physical quantity like the magnetic field B. It is a mathematical tool that is used to simplify calculations and describe the behavior of the magnetic field. A does not contain the B field, but rather is a way to mathematically represent it.

The reason why A is sometimes said to not exist or be just a paper shortcut is because it is not a measurable quantity and cannot be directly observed. However, it is still a useful concept in the study of electromagnetism and is an important part of the mathematical framework used to understand and describe the behavior of magnetic fields.
 

1. How can I visualize the magnetic vector potential A?

The magnetic vector potential A cannot be directly visualized as it is a vector field in three-dimensional space. However, it can be represented graphically using vector plots or by drawing magnetic field lines based on the curl of A.

2. What is the physical significance of the magnetic vector potential A?

The magnetic vector potential A is a mathematical tool used to simplify calculations and understand the behavior of magnetic fields. It is related to the magnetic field B through the relationship B = ∇ x A, and is particularly useful in situations involving symmetrical or steady-state magnetic fields.

3. How is the magnetic vector potential A calculated?

The magnetic vector potential A is calculated using the vector potential formula, which takes into account the current density and the distance between the point of interest and the current element. It is an integral of the current density over the volume in which the magnetic field is being calculated.

4. Can the magnetic vector potential A be negative?

Yes, the magnetic vector potential A can have both positive and negative values. The sign of A depends on the direction of the current flow and the convention used in the calculation. The magnitude of A is more important than its sign in understanding the behavior of magnetic fields.

5. How does the magnetic vector potential A relate to the magnetic scalar potential?

The magnetic vector potential A and the magnetic scalar potential Φ are related through the equation A = -∇Φ. This means that the magnetic scalar potential is the negative gradient of the magnetic vector potential. Together, they are used to describe the magnetic field in a given space.

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