1. 1
Magnetic Field Report
Leah Klein Borden
Graduate student mentor: Indu Venugopal
PI: Prof. Andreas Linninger
Linninger Research Group
Submission date: 5/10/16

2. 2
INTRODUCTION
Generation of a magnetic vector potential field (A) and magnetic flux density (B) from a single
point source of current density in vacuum
Outline
The aim of this report is to write a MATLAB program to understand, program and visualize the
generation of a magnetic field from a single point source of current density in vacuum in a three
dimensional space. Several such current density sources (from unpaired electron spins of
ferromagnetic atoms) when reinforced can generate the magnetic field of a permanent magnet
Relevant Background and Methodology
Maxwell’s equations
The classical problem of magnetostatic field theory starts with the Maxwell's equations:
∇× H = J (1 a)
B = µH (1 b)
∇ ⋅ B = 0 (1 c)
where
H is the magnetic field strength
B is the magnetic flux density
J is the current density that generates the magnetic field
µ is the magnetic permeability which is a function of the magnetizing material under
consideration
Thus the gradient operator ∇ is defined as
∇=
d
dx
,
d
dy
,
d
dz
Relationship of the magnetic flux density (B) to the magnetic vector potential (A)
In the classical vector potential approach, the system of the three equations (1a-c) can be further
simplified by the introduction of an auxiliary variable vector A (representing the magnetic vector
potential) which is related to B (magnetic flux density) by the equation:
∇× A = B

3. 3
This equation indicates that the curl of the magnetic vector potential field gives us the magnetic
flux density field (B). The curl of the vector A is the vector B that points along the axis of the
rotation of vector A and whose length corresponds to the speed of the rotation of A. Suppose
the A field has three components in x, y and z directions given by Ax, Ay and Az respectively, then
∇× A = B =
dAz
dy
−
dAy
dz
i +
dAx
dz
+
dAz
dx
j +
dAy
dx
+
dAx
dy
k
Equation (1c) is now automatically satisfied by the identity
∇. ∇× A = 0
Relationship between the source current (J) and the magnetic vector potential (A)
Now, on eliminating B and H from equations (la) and (1b) we have the governing vector potential
equation in terms of only the magnetic vector potential:
∇ ×
7
8
(∇ × A) = J (1d)
Proof that 𝛁 ⋅ 𝐁 = 𝟎 for any 𝐁 = (𝛁 × 𝐀)
Let B = (b1 b2 b3) ; ∇ ⋅ B =
FG7
FH
+
FGI
FH
+
FGJ
FH
b1 b2 b3 = (bI7z − bJ7y bJ7x − b77z b77y − bI7x)
∇ ⋅ B = bI7z − bJ7y + bJ7x − b77z + b77y − bI7x = 0

4. 4
PART A: Relationship between source current ( J) in one direction (z) and the magnetic field ( B)
In three dimensions any numerical formulation must solve for all three components of B. In two-
dimensional plane problems it is, however, assumed that the x and y components of the current
density vector J can be assumed to be zero. Therefore, JX = JY = O.
The current density vector J can be thought of as an infinitely long wire in the z-direction (JK).
This implies that the resulting magnetic field B must be symmetric along the z-direction.
Let B = (b1 b2 b3)
Therefore,
FG7
FK
=
FGI
FK
=
FGJ
FK
= 0 (1g)
Also, J = ∇ × B = (0 0 JK)
This implies that
FGJ
FH
=
FGJ
FL
= 0 (1h)
Equations 1g and 1h imply that b3 = 0
B = b1 b2 0 à B lies in a planar x-y field
∇ × B = (0 0 JK)
This can be reduced to
b11x + b21y = 0 (1i)
b11y + b21x = Jz (1j)
Introduction of scalar field Φ
Let Φ be a scalar field with ∇ Φ = (ΦX ΦY) such that
b1 = ΦY and b2 = - Φx
Hence,
b1x = Φyx b1y = Φyy b2y = - Φxy b2x = - Φxx
Substituting in equations 1i and 1j, we get
Φyx - Φxy = 0 (1k)
Φyy + Φxx = Jz (1l)
Equation (1l) is a simple diffusion equation that we have attempted to solve via MATLAB

5. 5
Simulation Parameters and Method
A 2D Cartesian mesh of 21 elements was used to generate the vacuum space in which the
computations were performed. A constant current density source (JK) of 10000 A/m2
was applied
at the center of this mesh in the ‘z’ direction (no x and y components). This acted as a source
term. This generates the scalar field Φ , as determined in equation 1j.
Initialization conditions
-Internal vacuum area:
The initial value of the Φ field in all the elements was set to 0.
The initial current density term was applied at a single element at the center of the mesh. This
acted as a source term.
-Boundary conditions: The boundaries value of the Φ field was set to 0. This represented the Φ
field value at infinity (which is assumed to be zero).
Assumptions
1. The boundaries region of the mesh represents an infinite distance at which Φ field value is
equal to zero.
2. The magnetic permeability of vacuum (µ0) is 1.25 x 10-6
N·A−2
.
3. Each element has a uniform dimension of 1mm x 1mm.

6. 6
Results
The code given in the appendix section generates the scalar field Φ.
Φ-field
As can be seen from figure 2, the magnitude of the Φ field value is the highest at the center of
the mesh closest to the current density source and gradually decreases and reaches a very small
value at the elements closest to the boundaries.
Figure 2: The magnitude of the Φ field (highest at center and decreases towards the border)
The Φ field produced as a result of the current density vector in the z direction is not a dirac delta
function. On reducing the mesh size by half, 1/10th and 1/100th times, the Φ field does not tend
to become a dirac function, as shown in the images below –
This is because the Φ field is a obtained from a diffusion equation from the current source at
the center of the mesh.
Figure 3: Plot showing the magnitude of the Φ field with mesh sizes of 0.5 mm, 0.1 mm and 0.01 mm (left to right) respectively

7. 7
Gradient of scalar field (𝛁 Φ)
Figure 4 : Vector plot of the 𝛁 Φ field and contour plot of the magnitude of the 𝛁 Φ field
Magnetic flux density field (𝐁 field)
Figure 5 : Vector plot of the 𝐁 field and contour plot of the magnitude of the 𝐁 field

8. 8
Obtaining the 𝐀 field
We know that ∇ × A = 𝐵
Therefore,
(dAz/dy) – (dAy/dz) = Bx = dΦ/dy
(dAx/dz) – (dAz/dx) = By = -dΦ/dx
We know that our 𝐵 field lies in the x-y plane. Therefore, by definition, the A field needs to lie
in the z plane.
This implies that the terms (dAy/dz) and (dAx/dz) are zero. Therefore, we obtain the result -
A = (0 0 Az ) = (0 0 Φ)
Therefore, plot of A field would be the same as plot of gradient of the Φ field.
Magnetic vector potential field (𝐀 field)
Figure 6 : Vector plot of the 𝐀 field and contour plot of the magnitude of the 𝐀 field
ANN⃑. 𝐵N⃗ = 0
AxBx + AyBy +AzBz = 0
We know that Bz = 0 and Bx & By ≠ 0
AxBx + AyBy = 0
One solution for this is Ax = Ay = 0, which corroborates our understanding of the ANN⃑ field

9. 9
MATLAB CODE FOR GENERATING THE A and B fields
%Generating Phi field
clear
close all
clc
mSize = 22;
MPI= zeros(mSize,mSize); %(magnetic pole 1 for north and 0 for south)
MPO= zeros(mSize,mSize);
dx =1; % Value of dx was changed to (.5mm, .1mm, .01mm) to verify this was not a dirac
function
dy =1; % Value of dy was changed to (.5mm, .1mm, .01mm) to verify this was not a dirac
function
%(e is the Coulomb's constant (ke) which is a proportionality const in equations relating electric
variables and is exactly equal to ke = 8.9875517873681764×10^9 N·m2/C2 (i.e. m/F).
%thr- refers to threshold)
e = 9e10;
thr = 1000; % thresh-hold
while (e>thr)
for i= 2:mSize-1
for j= 2:mSize-1
if(i==11 && j==11); MPO(i,j)=1e4;
else
[a1, a2, a3, a4, a5, a6]=computecoefficients(i,j);
MPO (i,j) = ((a4*(dx/dy)*MPI(i,j-1))+(a2*(dx/dy)*MPI(i,j+1))+(a1*(dy/dx)*MPI(i-
1,j))+(a3*(dy/dx)*MPI(i+1,j)))/a5 ;
end
end
end
e=max(max((MPI-MPO).^2));
MPI = MPO;
end
%Plotting Phi field
W = MPO;
U = zeros(mSize,mSize);

10. 10
V = zeros(mSize,mSize);
[X,Y] = meshgrid(0:21,0:21);
Z = zeros(mSize,mSize);
% 2D
quiver(X,Y,W,Z);
hold on
surf(X,W); % can graph only a gradient field (quiver) or a surface (surf) at a time. Mute either
command based on desired result.
hold on
%3D, can graph either a 2D or a 3D at a time. Mute either the 2D or the 3D depending on
preference.
quiver3(X,Y,U,V,W,Z);
hold on
surf(X,Y,W); % can graph only a gradient field (quiver) or a surface (surf) at a time. Mute either
command based on desired result.
hold on
% Plotting the gradient of Phi field
[FX,FY] = gradient(W);
figure()
quiver(FX,FY)
contour (FX,FY) % can graph only a gradient field (quiver) or contour at a time. Mute either
command based on desired result.
% Plotting Magnetic flux density filed (B Field)
[FX,FY] = gradient(W);
figure()
quiver(FY, -FX)
%contour(FY,-FX) % can graph only a gradient field (quiver) or contour at a time. Mute either
command based on desired result.
% Function used for computing coefficients
function [a1,a2,a3,a4,a5,a6]= computecoefficients(i,j)

11. 11
a1 = 1; %N
a2 = 1; %E
a3 = 1; %S
a4 = 1; %W
if i == 2; a1= a1*2; end
if i == 20; a3= a3*2; end
if j == 2; a4= a4*2; end
if j == 20; a2= a2*2; end
a5 = a1+a2+a3+a4;
a6=0;
% if(i==50 && j==50); a6=1000; end

12. 12
Table describing the comparison between electrostatics and magnetostatics
Equation MAGNETOSTATICS ELECTROSTATICS
Potential Magnetic Scalar
potential φm Magnetic Vector
potential A
Electric
Scalar potential
φ or V
Poisson’s Equation ∇I
ϕ = −
RS
T
, where ρVis the free
charge density
∇I
A = −µoȷ
H = −∇ϕ (outside magnet) E = −∇ϕ
∇
7
8
∇ Az = Jz (inside magnet)
(at all points)
Theorem (differential form) Ampere’s law: B. dl = ∇×
B. ds = µoI
Gauss theorem: ϕ=
E. dA =
]^
T_
Theorem (integral form) ∇ × A = B, ∇. B = 0,
∇. A = 0
∇ × E = 0,
∇. E =
R
T_
Gauge Transformation gauge for the
scalar potential
transformation , leaves the
electric field invariant
transformation ,
leaves the magnetic field
invariant
Potential Energy U = −pE U = −µB
Force Law Biot and Savart’s Law B =
8_
bc
⨜
eFf×g
gh
Coulomb’s Law E =
7
bcT_
⃒j⃒
gh
Main three vectors B,M,H E,D,P
Their relationship B = µ(H + M) D = εoE + P
Analogy in vectors B associated with all
currents, H associated with true
currents, M associated with
magnetizing currents
E associated with all
charges, D associated
with free charges, P
associated with induced
charges
Boundary conditions at the
interface between 2 media
The tangential component of E & the
normal component of
D are continuous across the
boundary.
Tangential component of H &
the normal component of B
are continuous
across the boundary