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Electricity and Magnetism Problems

Problem A):

Consider the railgun shown below:
See attached file for full problem description.

Consider a 1 kilogram projectile that is accelerated using a 10 meter long, 3 cm wide, railgun
with a 1 million amp constant current pulse.

1) Derive an analytic expression for B everywhere in the plane between two infinitely long parallel wires.

For parts 2-5, assume the magnetic field from part 1:

2) Calculate the resulting Lorentz force on the projectile.

3) Calculate the final velocity of the projectile.

4) Is this motion relativistic? For what length rail does the motion first become relativistic (v/c=0.3).

5) Calculate the position-dependence of the back EMF induced in the projectile. What is the minimum voltage between the rails required to sustain the current pulse?

6) Of course, the magnetic field in a real railgun is not the same as the field between two infinite parallel wires because the current returns through the projectile. Assuming that the current is maintained at 1 million amps during the entire launch, calculate the position-dependent magnetic field that acts on the projectile.

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Solution Preview

An element needed for specific numerical evaluations, the radius of the rail (wire) is omitted in the statement of the problem.

I therefore shall answer the question in terms of formulae, and if you manage to obtain the missing numerical data on the radius of the rail, you can substitute the numbers.

The calculations are written in the attached file 109291.pdf
I also attach RailgunTheory.pdf taken from web page http://www.matthewmassey.com/RailgunTheory.pdf
in which much of the same is calculated as in your assignment

Various formulae can be found in your textbook(s). In case you have the Introduction to Electrodynamics by David J. Griffiths, it has all the formulae needed.
However, just in case, here are some web pages with various formulae and data:
http://www.netdenizen.com/emagnet/solenoids/frommaxwellonly.htmhttp://www.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_feb_28_2003.shtml
http://en.wikipedia.org/wiki/Rail_gun
http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/MagneticField/MFOnWire.html
http://www.matthewmassey.com/RailgunTheory.pdf
http://www.nasa.gov/worldbook/mercury_worldbook.html
http://library.thinkquest.org/16600/advanced/inducedemf.shtml
http://scitec.uwichill.edu.bb/cmp/online/p10d/sodha/lecture14/lect14.htm

Please see the attached file.

=======
Here is the plain TEX source

magnification=magstep1
baselineskip=12pt
parindent = 0pt
parskip = 12pt

defl{left}
defr{right}
defla{langle}
defra{rangle}
defp{partial}

centerline{bf Rail Gun}

A crucially important element which is the radius of the rail (wire) is omitted in the

statement of the problem.
I therefore shall answer the question in terms of formulae, and if you manage to obtain the

missing data on the radius of the rail, you can ...

Solution Summary

This solution considers the railgun provided. The expert explains how to derive an analytic expression for B.

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