Two Magnetic Loops (Part 3)

Electricity and Magnetism Level 4

There are two circular loops of wire, each carrying a current of magnitude $1$ .

Loop $1$ has a radius of $1$ and center $(x,y,z) = (0,0,0)$ . The vector $(1,-2,3)$ is normal to the plane of loop 1.
Loop $2$ has a radius of $2$ and center $(x,y,z) = (1,2,3)$ . The vector $(-5,4,1)$ is normal to the plane of loop 2.

What is the magnitude of the magnetic force exerted by one loop on the other?

Details and Assumptions:
1) Magnetic permeability $\mu_0 = 1$ , for simplicity

The answer is 0.0728.

1 solution

Steven Chase
Oct 28, 2019

Simulation code below. This one can actually solve any of the three from this set of problems, depending on the "opt" setting.

import math

N = 5000

I = 1.0
u0 = 1.0

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

dtheta1 = 2.0*math.pi/N
dtheta2 = dtheta1

opt = 3

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

if opt == 1:

    R1 = 1.0
    R2 = 2.0

if opt == 2:

    R1 = 1.0
    R2 = 2.0

if opt == 3:

    R1 = 1.0
    R2 = 2.0

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

if opt == 1:

    C1x = 0.0
    C1y = 0.0
    C1z = 0.0

    C2x = 0.0
    C2y = 0.0
    C2z = 1.0

if opt == 2:

    C1x = 0.0
    C1y = 0.0
    C1z = 0.0

    C2x = 1.0
    C2y = 0.0
    C2z = 0.0

if opt == 3:

    C1x = 0.0
    C1y = 0.0
    C1z = 0.0

    C2x = 1.0
    C2y = 2.0
    C2z = 3.0

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

if opt == 1:

    N1x = 0.0
    N1y = 0.0
    N1z = 1.0

    N2x = 0.0
    N2y = 0.0
    N2z = 1.0

if opt == 2:

    N1x = 0.0
    N1y = 0.0
    N1z = 1.0

    N2x = 1.0
    N2y = 0.0
    N2z = 10.0**(-6.0)

if opt == 3:

    N1x = 1.0
    N1y = -2.0
    N1z = 3.0

    N2x = -5.0
    N2y = 4.0
    N2z = 1.0


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

# Nx*ux + Ny*uy + Nz*uz = 0

u1x = 1.0
u1y = 1.0
u1z = -(N1x*u1x + N1y*u1y)/N1z

u2x = 1.0
u2y = 1.0
u2z = -(N2x*u2x + N2y*u2y)/N2z

u1 = math.sqrt(u1x**2.0 + u1y**2.0 + u1z**2.0)
u2 = math.sqrt(u2x**2.0 + u2y**2.0 + u2z**2.0)

u1x = u1x/u1
u1y = u1y/u1
u1z = u1z/u1

u2x = u2x/u2
u2y = u2y/u2
u2z = u2z/u2

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

v1x = N1y*u1z - N1z*u1y
v1y = -(N1x*u1z - N1z*u1x)
v1z = N1x*u1y - N1y*u1x

v2x = N2y*u2z - N2z*u2y
v2y = -(N2x*u2z - N2z*u2x)
v2z = N2x*u2y - N2y*u2x

v1 = math.sqrt(v1x**2.0 + v1y**2.0 + v1z**2.0)
v2 = math.sqrt(v2x**2.0 + v2y**2.0 + v2z**2.0)

v1x = v1x/v1
v1y = v1y/v1
v1z = v1z/v1

v2x = v2x/v2
v2y = v2y/v2
v2z = v2z/v2

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

print (u1x**2.0 + u1y**2.0 + u1z**2.0)
print (u2x**2.0 + u2y**2.0 + u2z**2.0)

print ""

print (v1x**2.0 + v1y**2.0 + v1z**2.0)
print (v2x**2.0 + v2y**2.0 + v2z**2.0)

print ""

print (N1x*u1x + N1y*u1y + N1z*u1z)
print (N1x*v1x + N1y*v1y + N1z*v1z)
print (v1x*u1x + v1y*u1y + v1z*u1z)

print ""

print (N2x*u2x + N2y*u2y + N2z*u2z)
print (N2x*v2x + N2y*v2y + N2z*v2z)
print (v2x*u2x + v2y*u2y + v2z*u2z)

print ""


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

count = 0

Fx = 0.0
Fy = 0.0
Fz = 0.0

theta1 = 0.0

while theta1 <= 2.0*math.pi:

    x1 = C1x + R1*math.cos(theta1)*u1x + R1*math.sin(theta1)*v1x
    y1 = C1y + R1*math.cos(theta1)*u1y + R1*math.sin(theta1)*v1y
    z1 = C1z + R1*math.cos(theta1)*u1z + R1*math.sin(theta1)*v1z

    dx1 = (-R1*math.sin(theta1)*u1x + R1*math.cos(theta1)*v1x)*dtheta1
    dy1 = (-R1*math.sin(theta1)*u1y + R1*math.cos(theta1)*v1y)*dtheta1
    dz1 = (-R1*math.sin(theta1)*u1z + R1*math.cos(theta1)*v1z)*dtheta1

    Bx = 0.0
    By = 0.0
    Bz = 0.0

    theta2 = 0.0

    while theta2 <= 2.0*math.pi:

        x2 = C2x + R2*math.cos(theta2)*u2x + R2*math.sin(theta2)*v2x
        y2 = C2y + R2*math.cos(theta2)*u2y + R2*math.sin(theta2)*v2y
        z2 = C2z + R2*math.cos(theta2)*u2z + R2*math.sin(theta2)*v2z

        dx2 = (-R2*math.sin(theta2)*u2x + R2*math.cos(theta2)*v2x)*dtheta2
        dy2 = (-R2*math.sin(theta2)*u2y + R2*math.cos(theta2)*v2y)*dtheta2
        dz2 = (-R2*math.sin(theta2)*u2z + R2*math.cos(theta2)*v2z)*dtheta2

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

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

        crossBx = dy2*rz - dz2*ry
        crossBy = -(dx2*rz - dz2*rx)
        crossBz = dx2*ry - dy2*rx

        dBx = Bmult*crossBx/rcube
        dBy = Bmult*crossBy/rcube
        dBz = Bmult*crossBz/rcube

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

        theta2 = theta2 + dtheta2

        count = count + 1

        if count%10**6 == 0:
            print count/10**6

    crossFx = dy1*Bz - dz1*By
    crossFy = -(dx1*Bz - dz1*Bx)
    crossFz = dx1*By - dy1*Bx

    dFx = I * crossFx
    dFy = I * crossFy
    dFz = I * crossFz

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

    theta1 = theta1 + dtheta1

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

print ""
print ""

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

@Steven Chase sir i want to know how do you make this type of solution. What's this import math???? Can you bit explain Please

A Former Brilliant Member - 1 year, 7 months ago

This is the Python programming language. The "math" library allows you to do things like trig functions and square roots