# Homework Help: Obtain the magnetic field from this experimental setup?

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1. Jul 23, 2015

### pondzo

1. The problem statement, all variables and given/known data
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.

2. Relevant 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.

3. 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.

Last edited: Jul 23, 2015
2. Jul 28, 2015