Attractive Couple

Electricity and Magnetism Level pending

There are two identival curves of s i n x sinx .Both of them carry 100 A 100A of current. Both are parallel to X − Y X-Y plane. First one has the equation y = s i n x y=sinx , z = 0 z=0 from x = 0 x=0 to x = 2 π x=2π means from position A A to B B . While the other is at z = 1 z=1 , y = s i n x y=sinx from x = 0 x=0 to x = 2 π x=2π . I have provided the figure but u can't see both because they are overlapping each other.

What is the magnitude of the magnetic force exerted by one curve on the other? Details and Assumptions 1) Magnetic permeability μ 0 = 4 π × 1 0 − 7 \mu_{0}=4π \times 10^{-7}


The answer is 0.012.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Steven Chase
Mar 17, 2020

For every point on Wire 1 1 , integrate over the entire Wire 2 2 using Biot-Savart to find the B field at that point on Wire 1 1 . Then calculate the infinitesimal force on the tiny current-carrying segment of Wire 1 1 . Add up all of these infinitesimal forces. 

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import math

# Scan resolution

N = 2000

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

# Constants

u0 = 4.0*math.pi * (10.0**-7.0)
I1 = 100.0
I2 = 100.0

dx1 = 2.0*math.pi/N
dx2 = 2.0*math.pi/N

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

# 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

x1 = 0.0

while x1 <= 2.0*math.pi:

    y1 = math.sin(x1)
    z1 = 0.0

    dy1 = math.cos(x1) * dx1
    dz1 = 0.0

    Bx = 0.0
    By = 0.0
    Bz = 0.0

    x2 = 0.0
    z2 = 1.0

    while x2 <= 2.0*math.pi:     # Determine B field at point on wire 1....
                                # by integrating over wire 2
                                # Biot Savart
        y2 = math.sin(x2)             

        dy2 = math.cos(x2) * dx2        
        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

        x2 = x2 + dx2

    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

    x1 = x1 + dx1

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

# 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

#-1.73618460781e-17
#-3.46065898513e-17
#0.0122130707706

#0.0122130707706
#>>> ================================ RESTART ================================
#>>> 
#2000

#-3.97581383573e-17
#5.83027600672e-17
#0.0122048920311

#0.0122048920311
#>>> ================================ RESTART ================================
#>>> 
#5000

#-2.06119414016e-07
#7.64037550909e-07
#0.0121967215849

#0.0121967216105
#>>>

1 Helpful 1 Interesting 1 Brilliant 0 Confused

@Legend of Physics Could you double check your result on the Sinusoid Magnetics Part 3 problem?

Steven Chase - 1 year, 2 months ago

Log in to reply

Yes let me check??

A Former Brilliant Member - 1 year, 2 months ago

@Steven Chase I founded a mistake. Finally reposted and corrected the answer. Thank you

A Former Brilliant Member - 1 year, 2 months ago

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