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The situation with forces due to magnetism is more complicated. First, we
find that, despite numerous searches, no evidence of magnetic charges
exist. In other words, there are no particles which create a radial
magnetic field (usually termed
Again, the nature of the force is different in the magnetic field-electric current case. The force takes on the following characteristic:
where 
Normally, we make life easy for the student by choosing situations to
be set up in such a way that
The magnetic force is understood to be due to the presence of a magnetic field produced by charges in motion. You can see a 3-D visualization of the production of the field from a wire carrying positive charges (or equivalently and more realistically, a wire carrying negative charges in the opposite direction) in the following video. Note that the scene shows a wire in which the charges are initially at rest. As the camera pans, the charges start to move and the magnetic field expands outward from the wire at the speed of light. The direction of the magnetic field is shown by the white arrows whose rotation depicts the fact that the field is circular around the wire. Enjoy by clicking here or on the image at the beginning of this document.
The aim of this demonstration is to produce plots which illuminate the concept of flux. This is inherently 3-dimensional, so we will make a lot of use of 3-d plots and some animation.
We begin with the concept of describing a flow. We have already introduced the concept of vectors as a way to describe the path of something in motion. We can simulate that by an animation in Maple by using drawing a line in 3-space. The line represents the path of a particle as it moves. How would we draw a line in 3 dimensions? We can make use of the concept of a parametric specification of the line, i.e. instead of writing down an equation for a line, we write down 3 specifications for a line. Each specification is a description of the path of the line along the x, y, or z axis (we call this a parametrization of the line). To parametrize a path along an axis, we simply ask ourselves how the path position changes. For example, let's say we want to draw a path which is parallel to the x-axis. That means that the position of the particle never changes as far the y and z axes are concerned. If the particle started at y=1.5, and z=1.5 (as an example), then, its y and z position will remain (1.5,1.5) no matter what the x position is. In Maple, parametrizations are specified by placing the changes in square brackets: [xp, yp, zp] where xp is the x-parametrization, yp is the y-parametrization, and so on. For a path along the x-axis, we want y and z to stay constant, so let's go to Maple and try plotting:
#
plot3d([x, 1.5, 1.5], x=1..5, y=1..4);
#
#We see a line in space! Set the axes to BOXED to get a good view of
#what we drew. The line is parallel to x and every point on the line
#has a y,z value of (1.5, 1.5). Now let's try to draw lines parallel
#to the y axis and z axis. We draw simulaneous plots.
#
plot3d({[x, 1.5, 1.5], [1.5, y, 1.5], [1.5, 1.5, y]}, x=0..5, y=0..5,
scaling=CONSTRAINED);
#
#Not bad. Note that we couldn't specify z in the last plot since plot3d
#only accepts 2 independent variables. But the variable doesn't matter
#since all we want to do is to represent something that changes linearly.
#Now let's add a little life by animating things. Suppose we want to
#show the path of a particle that starts at the origin and zooms outward
#along the x-axis. We just add a third variable that multiplies the x
#term in our parametric specification. Make it so ...
#
animate3d([x*u, 0.0, 0.0], x=0..5, y=0..5, u=0..1,scaling=CONSTRAINED);
#
#To specify three particles zooming out from the origin along three different
#directions, we say
#
animate3d({[x*u, 0.0, 0.0], [0.0, y*u, 0.0], [0.0, 0.0, y*u]}, x=0..5,
y=0..5, u=0..1,scaling=CONSTRAINED);
#
Alright, enough with setup. What has all this to do with flow? Well, we
can now make it appear as though particles are zooming out from some source
at the origin. For example, maybe we are looking at drops spraying out from
a garden hose sitting at the origin. Let's consider motion along one
dimension for the moment. How would we characterize how much of something
is flowing away from the origin? What we are trying to specify is the
amount of water (the mass) that flows away in a given amount of time.
One way to be quantitative about it is to set up a gate near the source and
have the gate measure how much water flows through it in, say, one second.
The gate can be represented simply as a plane, but how do we position the
plane? Common sense says that to measure all the water leaving, we have
to have the gate closed, i.e. it must be perpendicular to the flow. If the
gate is open, then the water goes past it, but not through it, and hence
is not measured. Let's look at an animated picture of the gate closed and
the gate open... back to Maple
#
animate3d({[x*u, 0.0, 0.0], [2.5, x-2, y-2]}, x=0..5, y=0..5, u=0..1,
title=`Gate_closed_Measure_flow`, labels=[x, y, z], scaling=CONSTRAINED);
#
animate3d({[x*u, 0.0, 0.0], [x, 3.5, y-1]}, x=0..5, y=0..5, u=0..1,
title=`Gate_closed__Flow_is_0`, labels=[x, y, z],scaling=CONSTRAINED);
#
#Notice that the second parametrization is the one specifying the plane.
#Let's make it a little more realistic, by giving the flow a little
#thickness. That is, we stretch it out in y and z a little.
#
animate3d({[x*u, 0.0, 0.0], [x*u, -0.2, 0.0], [x*u, 0.2, 0.0],
[x*u, 0.0, -0.2], [x*u, 0.0, 0.2], [2.5, x-2, y-2]}, x=0..5, y=0..5, u=0..1,
title=`Gate_closed__Measure_flow`, labels=[x, y, z], scaling=CONSTRAINED);
#
animate3d({[x*u, 0.0, 0.0], [x*u, -0.2, 0.0], [x*u, 0.2, 0.0],
[x*u, 0.0, -0.2], [x*u, 0.0, 0.2], [x, 3.5, y-1]}, x=0..5, y=0..5, u=0..1,
title=`Gate_closed__Flow_is_0`, labels=[x, y, z],scaling=CONSTRAINED);
#
What we notice is that to make an accurate measure of the flow, our gate
must be perpendicular to the flow. Thus, the orientation of the gate is
set by the direction of the flow. Now let's extend this reasoning to 3-D.
To explain why everything falls off as
where 
where 
#
animate3d({[sin(t)*cos(p), sin(t)*sin(p), cos(t)], [u*sin(p), 0., 0.],
[0., u*sin(p), 0.], [0., 0., u*sin(p)],[u*sin(p), 0, u*sin(p)],
[0, u*sin(p), u*sin(p)]}, p=0..2*Pi, t=0..Pi, u=0..2, labels=[x, y, z],
title=`Source_at_origin`, style=WIREFRAME, scaling=CONSTRAINED, axes=BOXED);
#
Quite a picture isn't it? But it's more than just a picture. If we
imagine anything flowing out from a point, then the measurement of
its flow is the amount that flows past our spherical surface in a given
amount of time. For that same amount of time, if we look at a surface
further away from the source, we see that the amount emitted by the source
is the same, but now the area of our gate is larger. What we see is that
our measurement of the strength of the source or amount of flow from it
seems to go down as 1/(Area of a sphere)! Thus, long before Newton wrote
down the theory of gravity or Coulomb measured the electric force,
Renaissance scholars expected that all forces from a central source must
go down as (the
part is usually absorbed into the force
constant, G for gravity and
) for electric force),
i.e. the further you are from the source, the weaker the sources flow of water,
force, or whatever else it emits appears to look to you. The source is
emitting in all directions and dissipating its flow over wider and wider
spheres as you get further away from it. We can picture two spheres
with flow going outward as shown below.
#
animate3d({[sin(t)*cos(p), sin(t)*sin(p), cos(t)], [u*sin(p), 0., 0.],
[0., 0., u*sin(p)],[0., u*sin(p), 0.], [u*sin(p), 0, u*sin(p)],
[0, u*sin(p), u*sin(p)], [2*sin(t)*cos(p), 2*sin(t)*sin(p), 2*cos(t)]},
p=0..2*Pi,t=0..Pi, u=0..4, labels=[x, y, z], title=`Source_at_origin`,
style=WIREFRAME, scaling=CONSTRAINED, axes=BOXED);