GraphingCalculator 4;
Window 45 4 743 1176;
PaneDivider 377;
FontSizes 14;
BackgroundType 0;
BackgroundColor 255 255 255;
StackPanes 1;
UseAntialiasing 0;
T -1 1;
U -1 1;
V -1 1;
3D.Axes 0;
3D.Depth 1.1330909743;
3D.View -0.3198340149104613 0.9409665498789046 0.1108519510409181 0.4005883170743735 0.02826999750968362 0.9158219300709054 0.8586240174045219 0.337317001326849 -0.3859819134184331;
3D.Speed 0;
Text "Torque on a loop in a magnetic field.

Plots a square ""frame"" with unit normal N at spherical angles (theta,phi)=(a,b) and passing through a point X0 a distance d from the origin.  Also plots a vector field W in the x-direction.

Unit normal:";
Color 2;
Expr N=vector(sin(a)*cos(b),sin(a)*sin(b),cos(a));
Text "Frame passes through this point:";
Color 3;
Opacity 0.7;
Expr X_0=d*N;
Text "A point along the normal direction a distance f from the origin:";
MathPaneSlider 1;
Expr f=slider([-1,10,40]);
Color 7;
Expr R=f*N;
Color 5;
MathPaneSlider 68;
Expr s=slider([0,0.1]);
Color 6;
Expr O=vector(0,0,0),I=vector(1,0,0),J=vector(0,1,0),K=vector(0,0,1);
Text "T and P are the spherical theta-hat and phi-hat unit vectors:";
Color 4;
Expr T=vector(cos(a)*cos(b),cos(a)*sin(b),-sin(a));
Color 5;
Expr P=vector(-sin(b),cos(b),0);
Text "U and V are unit vectors normal to N that define the plane of the frame, rotated by an angle g about N if we wish to adjust the orientation of the frame.   C and D are the corners of the frame.";
Color 3;
Expr U=T*cos(g)+P*sin(g);
Color 5;
Expr V=-(T*sin(g))+P*cos(g);
Color 2;
Expr C=U+V;
Color 3;
Expr D=U-V;
Text "A square frame X0+uU+vV passing through X0 and oriented at an angle g about N:";
Color 17;
Expr vector(x,y,z)=X_0+U*u+V*v;
Color 17;
Expr vector(x,y,z)=vector(0,v,-u)+X_0;
Text "Sides of frame:";
Color 8;
Expr U+V*t;
Color 8;
Expr -U+V*t;
Color 8;
Expr V+U*t;
Color 8;
Expr -V+U*t;
Text "Magnetic field along the x-direction [GC doesn't display changes in the magnitude W, though]:";
Color 7;
MathPaneSlider 25;
Expr W=slider([-3,3,40]);
Color 3;
Grain 0;
Expr vector(d*x,d*y,d*z)=W*vector(1,0,0);
Text "Magnetic moment u=NiA of loop (up to factor of current):";
Color 17;
Grain 0.1166666666666667;
Expr X_0,X_0+R,'radius'=s;
Color 17;
Expr X_0,X_0+I,'radius'=s;
Text "A ""flux tube"" illustrating the portion of the vector flow that passes through the frame.  In its current form, only works properly when  a and b are varied separately and d=0.  [Tube is 0x+(y/cosb)^m+(z/sina)^m=1; shift center and orientation to generalize; slows down plotting significantly so retained in a separate file.]";
Opacity 0.7;
Expr m=10;
Text "(a,b) are (theta,phi) of unit normal to frame N; d is the distance from the origin to the place; g is the angle of rotation of the frame about N.";
Color 4;
Expr G=slider([0,1,40]);
Color 2;
Expr d=slider([0,10]);
Color 8;
Expr a=pi*A,b=2*pi*B,g=2*pi*G;
Color 6;
MathPaneSlider 20;
Expr A=slider([0,1,40]);
MathPaneSlider 20;
Expr B=slider([0,1,40]);
Text "Edges of loop representing direction of current flow.";
Grain 1;
Expr D,D-(2*U),'radius'=s;
Grain 1;
Expr -C,-C+2*V,'radius'=s;
Grain 1;
Expr -D,-D+2*U,'radius'=s;
Grain 1;
Expr C,C-(2*V),'radius'=s;
Text "Force on each side of loop due to current flowing through loop through the magnetic field B (=WI):";
Color 17;
Expr -V,-V-[cross(W*U,I)],'radius'=s;
Color 17;
Expr V,V+W*[cross(U,I)],'radius'=s;
Color 17;
Expr -U,-U+cross(W*V,I),'radius'=s;
Color 17;
Expr U,U-[cross(W*V,I)],'radius'=s;
Text "Now turn it around.  Switch the current off, but move the loop through the field.   Show the forces on each wire.
Z = velocity of wire, s = speed:";
Color 4;
MathPaneSlider 111;
Expr s=slider([-10,10]);
Expr Z=vector(0,0,1)*s;
Color 17;
Expr O,Z,'radius'=s;
Text "Force on each side of loop due to motion through field B (=WI):";
Color 17;
Expr -V,-V+cross(W*Z,I),'radius'=s;
Color 17;
Expr V,V+W*[cross(Z,I)],'radius'=s;
Color 17;
Expr -U,-U+cross(W*Z,I),'radius'=s;
Color 17;
Expr U,U+W*[cross(Z,I)],'radius'=s;
Text "

Author: David A. Craig <<http://web.lemoyne.edu/~craigda/>";