Obtain the magnetic field from this experimental setup?

[pondzo]

Homework Statement

Hi all, I would appreciate some help with the following problem;

I need to obtain and visualize the current flow and magnetic field profile of an elliptic cylinder (made from ferromagnetic material) which has a left section set at 0 volts and a right section set at 5 volts with a current allowed to flow. The following picture is a top view of the situation:

I am using Mathematica to visualize and obtain the information needed. The answer I am looking for does not need to be relevant to mathematica, I just need a way to obtain the magnetic field.

Homework Equations

Major axis (a) = 2 microns
Minor axis (b) = 1 micron
Thickness (L) = 30 nm
Current density magnitude = 1*10^10 A/m^2

Laplaces equation for electric potential.
Laplaces equation for magnetic potential.

The Attempt at a Solution

The following code describes the elliptic cylinder:

R1 = ImplicitRegion[(x/(2*10^(-6)))^2 + (y/(1*10^(-6)))^2 <= 1 \[And]
Abs[z] <= 15*10^(-9), {{x, -2*10^(-6),
2*10^(-6)}, {y, -(1*10^(-6)), 1*10^(-6)}, {z, -15*10^(-9),
15*10^(-9)}}]

The following code sets the Dirchlet Boundary Conditions for the electric potential function on the disk:

DCB1 = {DirichletCondition[u[x, y, z] == 0,
(x/(2*10^(-6)))^2 + (y/(1*10^(-6)))^2 <= 1 \[And]
x <= -1.7*10^(-6) \[And] Abs[z] <= 15*10^-(9)], DirichletCondition[
u[x, y, z] ==
5, (x/(2*10^(-6)))^2 + (y/(1*10^(-6)))^2 <= 1 \[And]
x >= 1.7*10^(-6) \[And] Abs[z] <= 15*10^-(9)]}

The following code numerically solves for the laplaces equation for the electric potential on the disk with the boundary conditions above:

pot = NDSolveValue[{Laplacian[u[x, y, z], {x, y, z}] == 0, DCB1}, u,
{x, y, z} \[Element] R1]

The following code obtains the electric field:

Efield = -Grad[pot[x, y, z],{x, y, z}]

If I am thinking correctly (?) a vector plot of this Electric field will represent the conventional current flow through the disk:

Show[VectorPlot3D[
Evaluate[Efield], {x, -1.7*10^(-6), 1.7*10^(-6)}, {y, -1*10^(-6),
1*10^(-6)}, {z, -15*10^(-15), 15*10^(-15)},
ViewPoint -> {0, 0, 100}],
RegionPlot3D[
ImplicitRegion[(x/(2*10^(-6)))^2 + (y/(1*10^(-6)))^2 <=
1, {{x, -2*10^(-6), 2*10^(-6)}, {y, -(1*10^(-6)), 1*10^(-6)}}],
PlotStyle -> Opacity[0.3], Boxed -> False]]

The numerical solver i am using has to extrapolate the Efield in regions beyond the disk and so aren't quite accurate (hence the oddly placed vector arrows, i think).

Q1 ) Will someone please confirm whether this is actually representative of the conventional current flow?

Q2) How could I obtain the magnetic field from the information I have?
Attempt: I am more interested in visualising the direction and profile of the magnetic field. If the above is truly representative of the current flow then for the sake of visualising the magnetic field (I don't think this is actually correct) I have set the current density vector field equal to the electric vector field and tried to use this as a means of solving the magnetic potential equation:$\nabla^2\vec{A}=-\mu_0\times\vec{j}$

Subscript[\[Mu], 0] = 4 Pi*10^(-7)

Magneticpotential =
NDSolveValue[{Laplacian[
A[x, y, z], {x, y, z}] == -Subscript[\[Mu], 0]*Efield},
A, {x, y, z} \[Element] R1]

Bfield = Curl[Magneticpotential, {x, y, z}]

I then tried to do a vectorplot of Bfield but it yielded nothing desired... so I must be doing something wrong.

I'll iterate again that an answer to this does not need to be relevant to Mathematica, I would just like to know if I have approached the current flow visualisation correctly and a method to obtain the magnetic field of the disk. Thank you for your time. Michael.

 

[mark726]

Hello Michael,

Thank you for reaching out for help with this problem. I am a physicist with experience in electromagnetism and I am happy to assist you.

Firstly, your approach to visualizing the conventional current flow is correct. The vector plot of the electric field will indeed represent the direction and magnitude of the current flow through the disk. However, the accuracy of the plot may be affected by the numerical solver you are using and the extrapolation of the electric field outside of the disk.

To obtain the magnetic field, you can use the equation you have mentioned, $\nabla^2\vec{A}=-\mu_0\times\vec{j}$, where $\vec{A}$ is the magnetic potential and $\vec{j}$ is the current density. However, there are a few things to consider in order to obtain an accurate result.

Firstly, you need to make sure that the current density vector field you are using is correct. In your code, you have set the current density to be equal to the electric field. This is not correct. The current density is a vector quantity that represents the flow of electric charge per unit area. In this case, the current density will be constant throughout the disk since the current is flowing in a uniform manner. Therefore, you can set the current density to be a constant vector with magnitude equal to the current density magnitude you have provided, and direction perpendicular to the surface of the disk (since the current is flowing in a circular path).

Secondly, you need to make sure that the units of your variables are consistent. In your code, you have used units of microns and nanometers for length, and Amperes per square meter for current density. However, for the magnetic field equation, you need to use units of meters for length and Amperes for current density. Therefore, you will need to convert your units before solving for the magnetic potential.

Finally, once you have obtained the magnetic potential, you can use the equation $\vec{B}=\nabla\times\vec{A}$ to calculate the magnetic field. This will give you a vector field that represents the direction and magnitude of the magnetic field at each point in space.

I hope this helps. If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your project!