Flux Through a Circle - 2

Electricity and Magnetism Level 4

Consider a circle of unit radius centered at the origin of the x y xy -plane. There exists a current-carrying wire along the straight line x = 2 x = 2 . The wire is of finite length which extends from y = − 5 y = -5 to y = 5 y = 5 . 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.

Bonus: Compare your answer to the case when the length of the wire is infinite. The solvers are encouraged to present their observations and comments about this comparison, along with a detailed solution.

Also Try: Flux Through a Circle

Note: The problem is based on a suggestion by Steven Chase. There will be more variants of this coming up soon.


The answer is 1.57047967.

Karan Chatrath by Karan Chatrath , 27, Netherlands

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1 solution

Steven Chase
Aug 10, 2019

Here is a computational Biot-Savart approach, based on a triple integral. The result comes out to be about 94 % 94\% as large as when the wire is infinite. So obviously, the contributions from the wire become much less important as the distance from the disk increases. 

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import math

# Constants

N = 500   # Number of divisions for r (disk),theta (disk) ,y (wire)
          # Triple integral - number of loop iterations is N^3

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

xw = 2.0  # wire x coordinate
zw = 0.0  # wire z coordinate

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

# Infinitesimals

dr = R/N
dtheta = 2.0*math.pi/N
dyw = 10.0/N

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

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

        x = r*math.cos(theta)    # Coordinates of patch within unit disk
        y = r*math.sin(theta)
        z = 0.0

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

        yw = -5.0

        while yw <= 5.0:   # Add up all contributions from wire segment to each disk patch

            rx = x - xw    # Displacement vector from wire to disk
            ry = y - yw
            rz = z - zw

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

            dx = 0.0       # Infinitesimal path vector along wire ("dl")
            dy = dyw
            dz = 0.0

            crossx = dy*rz - dz*ry    # cross product of dl and r
            crossy = -(dx*rz - dz*rx)
            crossz = dx*ry - dy*rx

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

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

            yw = yw + dyw

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

        dAx = 0.0  # Area-weighted disk normal vector
        dAy = 0.0
        dAz = dA

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

        lam = lam + dlam # Update flux

        theta = theta + dtheta

    r = r + dr

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

print N
print lam

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

# Results

# N = 100 (1 million loops in triple integral)
# lam = 1.57868129706

# N = 200 (8 million loops in triple integral)
# lam = 1.57465955855

# N = 500 (125 million loops in triple integral)
# lam = 1.56748645025

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