Flux Through a Circle - 4 - Corrected Version

Electricity and Magnetism Level 4

Consider a circle of unit radius centered at the the point ( 1 , 1 , − 11 / 9 ) (1,1,-11/9) . The circle lies on the plane:

5 x + 7 y + 9 z = 1 5x + 7y + 9z = 1

There exists an infinitely long current-carrying wire along the straight line:

x − 5 2 = y − 3 4 = z 9 \frac{x-5}{2} = \frac{y-3}{4} = \frac{z}{9}

The magnitude of the current flowing through this wire is:

I = 2 π μ o I = \frac{2\pi}{\mu_o}

Here μ o \mu_o is the permeability of free space. Compute the magnitude of the magnetic flux through the circle.

Also Try:

Flux Through a Circle

Flux Through a Circle - 2

Flux Through a Circle - 3

Note: Thanks to Steven Chase for his inputs.


The answer is 0.10746.

Karan Chatrath by Karan Chatrath , 27, Netherlands

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1 solution

Steven Chase
Aug 17, 2019

Simulation code is below. Results are at the bottom. 

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import math

# Constants

ND = 200   # Number of divisions for r (disk),theta (disk)
NL = 200  # Number of divisions for line

R = 1.0   # loop radius
u0 = 1.0  # permeability

I = 2.0*math.pi/u0  # current
Bmult = u0*I/(4.0*math.pi) # Biot Savart multiplier

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

# Line parameters

P1x = 5.0   # First point on line
P1y = 3.0
P1z = 0.0

P2x = 7.0   # Second point on line
P2y = 7.0
P2z = 9.0

vx = P2x - P1x  # vector from one to the other
vy = P2y - P1y
vz = P2z - P1z

v = math.sqrt(vx**2.0 + vy**2.0 + vz**2.0)

ux = vx/v  # line unit vector
uy = vy/v  # "alpha" is the distance parameter for the line
uz = vz/v

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

# Disk Parameters

Cx = 1.0         # Disk center
Cy = 1.0
Cz = -11.0/9.0

Nx = 5.0   # Disk normal
Ny = 7.0
Nz = 9.0

N = math.sqrt(Nx**2.0 + Ny**2.0 + Nz**2.0)

uNx = Nx/N  # Unit disk normal
uNy = Ny/N
uNz = Nz/N

# Nx*v1x + Ny*v1y + Nz*v1z = 0

v1x = 2.0
v1y = 3.0
v1z = -(Nx*v1x + Ny*v1y)/Nz

v1 = math.sqrt(v1x**2.0 + v1y**2.0 + v1z**2.0)

u1x = v1x/v1   # First unit basis vector in disk plane
u1y = v1y/v1
u1z = v1z/v1

v2x = Ny*u1z - Nz*u1y    # Get second disk plane vector from cross product
v2y = -(Nx*u1z - Nz*u1x)
v2z = Nx*u1y - Ny*u1x

v2 = math.sqrt(v2x**2.0 + v2y**2.0 + v2z**2.0)

u2x = v2x/v2   # Second unit basis vector in disk plane
u2y = v2y/v2
u2z = v2z/v2

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

# Vector checks

print (uNx**2.0 + uNy**2.0 + uNz**2.0)
print (u1x**2.0 + u1y**2.0 + u1z**2.0)
print (u2x**2.0 + u2y**2.0 + u2z**2.0)
print ""
print (uNx*u1x + uNy*u1y + uNz*u1z)
print (uNx*u2x + uNy*u2y + uNz*u2z)
print (u1x*u2x + u1y*u2y + u1z*u2z)
print ""
print ""

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

# Infinitesimals

dr = R/ND
dtheta = 2.0*math.pi/ND
dalpha = 200.0/NL

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

lam = 0.0   # flux

r = 0.0

while r <= R:           # Integrate over unit disk
                        # Biot Savart
    theta = 0.0

    while theta <= 2.0*math.pi:  # Calculate B for each infinitesimal patch

        # Coordinates of patch within unit disk

        x = 1.0*Cx + r*math.cos(theta)*u1x + r*math.sin(theta)*u2x    
        y = 1.0*Cy + r*math.cos(theta)*u1y + r*math.sin(theta)*u2y 
        z = 1.0*Cz + r*math.cos(theta)*u1z + r*math.sin(theta)*u2z 

        Bx = 0.0  # B-field components
        By = 0.0
        Bz = 0.0

        alpha = -100.0  # line distance parameter

        while alpha <= 100.0:   # Add up all contributions from wire segment to each disk patch

            xw = P1x + alpha*ux
            yw = P1y + alpha*uy
            zw = P1z + alpha*uz

            Dx = x - xw    # Displacement vector from wire to disk
            Dy = y - yw
            Dz = z - zw

            Dcube = (Dx**2.0 + Dy**2.0 + Dz**2.0)**(3.0/2.0)

            dx = dalpha * ux       # Infinitesimal path vector along wire ("dl")
            dy = dalpha * uy
            dz = dalpha * uz

            crossx = dy*Dz - dz*Dy    # cross product of dl and r
            crossy = -(dx*Dz - dz*Dx)
            crossz = dx*Dy - dy*Dx

            dBx = Bmult * crossx / Dcube  # Infinitesimal B-field contributions
            dBy = Bmult * crossy / Dcube
            dBz = Bmult * crossz / Dcube

            Bx = Bx + dBx  # Update B-field
            By = By + dBy
            Bz = Bz + dBz

            alpha = alpha + dalpha


        dA = r*dr*dtheta   # Infinitesimal disk patch area

        dAx = dA * uNx  # Area-weighted disk normal vector
        dAy = dA * uNy
        dAz = dA * uNz

        dlam = Bx*dAx + By*dAy + Bz*dAz  # Infinitesimal flux contribution (dot product)

        lam = lam + dlam # Update flux

        theta = theta + dtheta

    r = r + dr

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

print ND
print NL
print (ND*ND*NL)
print ""
print lam

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

# Results


#200
#200
#8000000  # 8 million loops

#-0.107393257782

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

#400
#400
#64000000 # 64 million loops

#-0.107809493851

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

#1000
#1000
#1000000000  # 1 billion loops

#-0.107432096547

1 Helpful 1 Interesting 1 Brilliant 0 Confused

@Karan Chatrath How about instead of a straight wire, we have a circular loop of wire? Or maybe a parabolic wire?

Steven Chase - 1 year, 9 months ago

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Interesting idea. The number of variants to this situation can become countless. I'll give this a thought.

Karan Chatrath - 1 year, 9 months ago

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@Karan Chatrath Sir, the first problem in this series was doable manually.....but the next questions became tooo tedious to solve by hand.….could you think of a calculative question, pls???

Aaghaz Mahajan - 1 year, 9 months ago

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@Aaghaz Mahajan – I will keep your suggestion in mind.

Karan Chatrath - 1 year, 9 months ago

Also, check my new question ....

Aaghaz Mahajan - 1 year, 9 months ago

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It is a nice problem.

Karan Chatrath - 1 year, 9 months ago

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