Vector potential and constant magnetic flux density

[skyboarder2]

Hi,

I would like to verify analytically that a vector potential of the form A=1/2(-yB₀,xB₀,0) produces a constant magnetic flux density of magnitude B₀ in the z direction.
(I guess I'd have to use the relation B=$$\forall$$$$\wedge$$A...)

 

[marcusl]

That is correct (assuming you meant to write the symbol $$\nabla$$). Do you have a question?

 

[skyboarder2]

Nope, I just look for a method to prove it mathematically

 

[marcusl]

Write out the components of $$\vec{\nabla} \times \vec{A}$$ in Cartesian coordinates.

 

[sardar]

Hello,

Thank you for your question. It is indeed possible to verify analytically that the vector potential A=1/2(-yB0,xB0,0) produces a constant magnetic flux density of magnitude B0 in the z direction. To do so, we can use the relation B=\mu_0\wedgeA, where \mu_0 is the permeability of free space.

Substituting the given vector potential into this equation, we get:

B=\mu_0\wedge(1/2(-yB0,xB0,0))

Simplifying this expression, we get:

B=1/2\mu_0(-xB0,yB0,0)

We can see that the z component of this vector is always zero, which means that the magnetic flux density is constant in the z direction. Furthermore, the magnitude of B is given by:

|B|=\sqrt{(1/2\mu_0)^2(-xB0)^2+(1/2\mu_0)^2(yB0)^2}

Simplifying this expression, we get:

|B|=B0

Thus, we have verified that the given vector potential produces a constant magnetic flux density of magnitude B0 in the z direction. I hope this helps to answer your question. Let me know if you have any further inquiries. Thank you.