Very Involved Biot-Savart Problem

Electricity and Magnetism Level 5
Here is a more demanding follow-up to this problem
We begin with a regular hexagon of radius $1$ centered on the origin in the $xy$ plane. Then we extend the sides of the hexagon to create a six-pointed star, as shown in the diagram. The distance $D$ between the top and bottom tips of the star is equal to $2 \, \sqrt{3}$ .
A uniform current $I$ circulates around the star, which can be thought of as a closed loop of wire. The magnetic flux density magnitude $B$ at test point $(x,y,z) = (1,2,3)$ can be expressed as:
$\large{B = \alpha \, \frac{\mu_0 \, I}{\pi \, D}}$
Determine the parameter $\alpha$ , and give your answer as $\lfloor 1000 \, \alpha \rfloor$ , where $\lfloor \cdot \rfloor$ denotes the floor function.
The answer is 137.

1 solution

Steven Chase
Oct 25, 2019
Simulation code below
import math

###########################################

# Simulation parameters

N = 10**5

u0 = 1.0
I = 1.0
D = 2.0*math.sqrt(3.0)

Bmult = u0*I/(4.0*math.pi)

Tx = 1.0
Ty = 2.0
Tz = 3.0

###########################################

# Point coordinates

shift = math.pi/6.0

Rbig = D/2.0
Rsmall = 1.0

R1 = Rbig
R2 = Rsmall
R3 = Rbig
R4 = Rsmall
R5 = Rbig
R6 = Rsmall
R7 = Rbig
R8 = Rsmall
R9 = Rbig
R10 = Rsmall
R11 = Rbig
R12 = Rsmall

theta1 = math.pi/2.0 - 0.0 * (math.pi/6.0)
theta2 = math.pi/2.0 - 1.0 * (math.pi/6.0)
theta3 = math.pi/2.0 - 2.0 * (math.pi/6.0)
theta4 = math.pi/2.0 - 3.0 * (math.pi/6.0)
theta5 = math.pi/2.0 - 4.0 * (math.pi/6.0)
theta6 = math.pi/2.0 - 5.0 * (math.pi/6.0)
theta7 = math.pi/2.0 - 6.0 * (math.pi/6.0)
theta8 = math.pi/2.0 - 7.0 * (math.pi/6.0)
theta9 = math.pi/2.0 - 8.0 * (math.pi/6.0)
theta10 = math.pi/2.0 - 9.0 * (math.pi/6.0)
theta11 = math.pi/2.0 - 10.0 * (math.pi/6.0)
theta12 = math.pi/2.0 - 11.0 * (math.pi/6.0)

x1 = R1*math.cos(theta1)
x2 = R2*math.cos(theta2)
x3 = R3*math.cos(theta3)
x4 = R4*math.cos(theta4)
x5 = R5*math.cos(theta5)
x6 = R6*math.cos(theta6)
x7 = R7*math.cos(theta7)
x8 = R8*math.cos(theta8)
x9 = R9*math.cos(theta9)
x10 = R10*math.cos(theta10)
x11 = R11*math.cos(theta11)
x12 = R12*math.cos(theta12)

y1 = R1*math.sin(theta1)
y2 = R2*math.sin(theta2)
y3 = R3*math.sin(theta3)
y4 = R4*math.sin(theta4)
y5 = R5*math.sin(theta5)
y6 = R6*math.sin(theta6)
y7 = R7*math.sin(theta7)
y8 = R8*math.sin(theta8)
y9 = R9*math.sin(theta9)
y10 = R10*math.sin(theta10)
y11 = R11*math.sin(theta11)
y12 = R12*math.sin(theta12)

###############################################################

# Field components
Bx = 0.0
By = 0.0
Bz = 0.0

###############################################################

# Numerically integrate the Biot-Savart expression over all line segments

###############################################################

# Segment 1-2

xs = x1
ys = y1

xe = x2
ye = y2

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 2-3

xs = x2
ys = y2

xe = x3
ye = y3

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 3-4

xs = x3
ys = y3

xe = x4
ye = y4

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz
###############################################################


# Segment 4-5

xs = x4
ys = y4

xe = x5
ye = y5

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz
###############################################################

# Segment 5-6

xs = x5
ys = y5

xe = x6
ye = y6

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 6-7

xs = x6
ys = y6

xe = x7
ye = y7

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 7-8

xs = x7
ys = y7

xe = x8
ye = y8

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 8-9

xs = x8
ys = y8

xe = x9
ye = y9

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 9-10

xs = x9
ys = y9

xe = x10
ye = y10

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 10-11

xs = x10
ys = y10

xe = x11
ye = y11

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 11-12

xs = x11
ys = y11

xe = x12
ye = y12

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Segment 12-1

xs = x12
ys = y12

xe = x1
ye = y1

dx = (xe - xs)/N
dy = (ye - ys)/N
dz = 0.0

for j in range(0,N):

    x = xs + j*dx
    y = ys + j*dy
    z = 0.0

    rx = Tx - x
    ry = Ty - y
    rz = Tz - z

    rcube = (rx**2.0 + ry**2.0 + rz**2.0)**(3.0/2.0)

    crossx = dy*rz - dz*ry
    crossy = -(dx*rz - dz*rx)
    crossz = dx*ry - dy*rx

    dBx = Bmult*crossx/rcube
    dBy = Bmult*crossy/rcube
    dBz = Bmult*crossz/rcube

    Bx = Bx + dBx
    By = By + dBy
    Bz = Bz + dBz

###############################################################

# Print results

ansmult = u0*I/(math.pi*D)

Bmag = math.sqrt(Bx**2.0 + By**2.0 + Bz**2.0)

alpha = Bmag/ansmult

print alpha
print math.floor(1000.0*alpha)

#0.137415895853
#137.0
Very Involved Biot-Savart Problem

The answer is 137.

1 solution

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