Magnetic field of a moving charged particle

Write a VPython program to calculate and display the magnetic field due to a moving proton in region of space surrounding the path of the proton. Initial conditions: start the proton at location <-2e-10, 0, 0>, with velocity 3e5 m/s in the +x direction. A reasonable value for the radius of the sphere representing the proton is about 3e-12 m (reasonable in the sense that the proton will be visible in your display - physically the radius of a proton is much smaller!) At 64 observation locations located on a cylindrical surface surrounding the path of the proton, a distance of 3e-11 m radially outward from the proton's path, create arrows to visualize the magnetic field at each location. You will update these arrows each time you move the proton, so you see the magnetic field in space changing as the proton moves. You will need to scale the arrows, as usual, to make an understandable display.

1. Calculating a vector cross-product in VPython

VPython has a built-in cross product function to make it easy to find vector cross products.

If < 2, 0, 0> and <0, 3, 0>, what is ? (Do the calculation on paper).

= ___________________________

Note that the cross product produces a vector which is perpendicular to the plane containing the original vectors.

Start a VPython program in the usual way:

from visual import *
from __future__ import division

Now ask VPython to do the same computation. Define two vectors A and B with the values given above, and print the result of the following statement:

C = cross(A,B)

2. Calculating a magnetic field in VPython

Creating objects

· Create a sphere representing the proton at the initial location specified in the problem statement above.
· Create a vector quantity velocity representing the velocity of the proton specified above.
· Create an arrow at the observation location < 0, 3e-11, 0> m, with axis=vector(0,0,0), and color=color.cyan. Name the arrow ba so you can refer to it later.
· Run the program

Why don't you see the arrow? _____________________

Calculating magnetic field

To calculate the magnetic field at the observation location due to the moving proton at its current location, you need a vector which, as usual, extends from the source particle to the observation location.

· Write a VPython statement to calculate the vector .
· Write VPython statements to calculate and .
· Write VPython statements to calculate the magnetic field at the observation location, due to the moving proton. Print the value of the magnetic field (which should be a vector).

Is the direction of the magnetic field you calculated correct? Check with the right-hand rule.

· Now set the axis of the arrow ba to be the magnetic field you calculated, multiplied by an appropriate scalar factor to make the size of the arrow appropriate for this display. Run the program.

Do you in fact see a cyan arrow at the observation location, pointing in the appropriate direction?

3. Using velocity to model the motion of a particle

A computer animation of the motion of an object uses the same principle as a movie or a flip book does. You display the object at successive locations, each at a slightly later time; to your eye, the motion can look continuous. You calculate the position of the object, and it is displayed in that position. You calculate where the object will be a very short time deltat later, and change its position to the new location, so it is displayed in the new location.

You will use the relation between velocity (a vector) and position (a vector) to calculate the position of a moving proton at successive times, separated by a short time step Dt. Remember that, in vector terms:

, or

In VPython, this translates to a statement like:

proton.pos = proton.pos + velocity*deltat

where velocity is a vector quantity which you have initialized earlier in the program.

This statement may look odd to you if you have not had much programming experience. The equals sign here has the meaning of ``assignment" rather than equality. This instruction tells VPython to read up the old position of the proton, add to it the quantity , and store the new position as the current position of the proton. This will automatically move the proton to the new position.

· Now write a loop in which the particle starts at the specified initial position, and moves from left to right across the screen. The position of the particle should be advanced by each time through the loop. Use deltat = 1e-20 s. Stop the loop when the particle has gone off the right side of the screen (i.e. when the x-component of the particle's position becomes greater than 2e-10 m).
· To keep the camera from zooming in and out again, put this line just before the loop:

scene.autoscale = 0

4. Magnetic field changing with time

As the proton moves, the magnetic field that it makes at the observation location changes. The program we are writing is different from the others we've written to calculate electric fields, because we want to observe the magnetic field at a particular location changing with time. We want to create only one arrow at the location, but change its magnitude and direction as necessary.

· Leave the code that creates the arrow outside of the loop, but move all of the code you wrote to calculate B and change the axis of the arrow into the loop, underneath the statement:

proton.pos = proton.pos + velocity*deltat

· Make everything is indented to the same level. Run the program.

Did the magnetic field at the observation location increase in magnitude as the proton approached, and decrease as the proton moved away? ________________

Did you find yourself inside the magnetic field vector? You will need a different scale factor now.

5. More observation locations

We want to observe the magnetic field at more than one location as the proton moves.

Before the loop:

Put the following code before the loop to create a list of observation arrows (note - this is a list of arrows, whose positions are at the observation locations).

obsarrows = []

for theta in arange(0,2*pi,pi/4):

for x in arange(-1.8e-10, 2.0e-10, 0.2e-10):

obsarrows.append(arrow(pos=(x,3e-11*sin(theta),3e-11*cos(theta)),

axis=vector(0,0,0), color=color.cyan))

Run the program and look for error messages. If you don't find any, go on.

Inside the loop:

You will need to modify your code inside the loop to calculate B at each location and update every arrow each time you move the proton one step. This will involve a nested loop (a loop inside a loop). Pay attention to indenting. The basic organization is:

while proton.pos.x < 1e-10:

## your code to move the proton here

for ba in obsarrows:

## put your code to calculate magnetic fields here. get indentation correct.

You may wish to increase deltat slightly if your program runs too slowly.