Parabolic Wire Magnetics

Electricity and Magnetism Level pending

Two parabolas carry current of I = 2 π A I=2\sqrt{π}A of equation y 2 = 4 ( x − 1 ) y^{2}=4(x-1) y 2 = − 4 ( x + 1 ) y^{2}=-4(x+1) Find the magnitude of magnetic force between the part A B AB and C D CD . Details and assumptions Take μ 0 = 1 \mu_{0}=1 (for simplicity)


The answer is 1.541.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Steven Chase
Apr 13, 2020

Nice one. Simulation code is attached. 

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import math

# Scan resolution

N = 5000

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

# Constants

u0 = 1.0
I1 = 2.0*math.sqrt(math.pi)
I2 = 2.0*math.sqrt(math.pi)

dy1 = 4.0/N
dy2 = 4.0/N

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

# Wire 1 is on the right (+x)
# Wire 2 is on the left (-x)

# For every point on wire 1, integrate over wire 2 to find the B field
# Then calculate the infinitesimal force contribution to wire 1
# Add up the infinitesimal force contributions

Fx = 0.0
Fy = 0.0
Fz = 0.0

y1 = -2.0

while y1 <= 2.0:

    x1 = 1.0 + (y1**2.0)/4.0
    z1 = 0.0

    dx1 = (y1/2.0)*dy1
    dz1 = 0.0

    Bx = 0.0
    By = 0.0
    Bz = 0.0

    y2 = -2.0

    while y2 <= 2.0:     # Determine B field at point on wire 1....
                                    # by integrating over wire 2
                                    # Biot Savart
        x2 = -1.0 - (y2**2.0)/4.0
        z2 = 0.0

        dx2 = -(y2/2.0)*dy2
        dz2 = 0.0

        rx = x1 - x2                 
        ry = y1 - y2
        rz = z1 - z2

        r = math.sqrt(rx**2.0 + ry**2.0 + rz**2.0)  

        Bcrossx = dy2*rz - dz2*ry      
        Bcrossy = -(dx2*rz - dz2*rx)
        Bcrossz = dx2*ry - dy2*rx

        dBx = (u0*I2/(4.0*math.pi))*Bcrossx/(r**3.0)  
        dBy = (u0*I2/(4.0*math.pi))*Bcrossy/(r**3.0)
        dBz = (u0*I2/(4.0*math.pi))*Bcrossz/(r**3.0)

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

        y2 = y2 + dy2

    Fcrossx = dy1*Bz - dz1*By        # Add up infinitesimal forces on wire 1
    Fcrossy = -(dx1*Bz - dz1*Bx)     # dF = I*(dL x B)
    Fcrossz = dx1*By - dy1*Bx

    dFx = I1 * Fcrossx
    dFy = I1 * Fcrossy
    dFz = I1 * Fcrossz

    Fx = Fx + dFx            
    Fy = Fy + dFy
    Fz = Fz + dFz

    y1 = y1 + dy1

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

# Print results

print N
print ""
print Fx
print Fy
print Fz
print ""
F = math.sqrt(Fx**2.0 + Fy**2.0 + Fz**2.0)

print F

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

#>>> 
#1000

#-1.54151408185
#-0.00109164846106
#0.0

#1.54151446838
#>>> ================================ RESTART ================================
#>>> 
#2000

#-1.54151434668
#-0.000545824315127
#0.0

#1.54151444331
#>>> ================================ RESTART ================================
#>>> 
#5000

#-1.54177828841
#-4.95827066471e-14
#0.0

#1.54177828841
#>>>

1 Helpful 1 Interesting 1 Brilliant 0 Confused

@Steven Chase Accurate Sir.

A Former Brilliant Member - 1 year, 1 month ago

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