The figure is an idealized schematic drawing of a rail gun. Projectile P sits be

Question

The figure is an idealized schematic drawing of a rail gun. Projectile P sits between two wide rails of circular cross section; a source of current sends current through the rails and through the (conducting) projectile (a fuse is not used). (a) Let w be the distance between the rails, R the radius of each rail, and i the current. Find the force acting on the projectile directed to the right along the rails. (b) If the projectile starts from the left end of the rails at rest, find the speed v at which it is expelled at the right. Assume that i = 451 kA, w = 34 mm, R = 9.4 cm, L = 4.2 m, and the projectile mass is 10 g.

Explanation / Answer

Magnetic field due to a current carrying conductor = 2kI/a
for the given situation, consider a small part of projectile, of length dx, and at a distance x from the rail

dB = 2kI/x
dF = IdxdB = 2kI^2dx/x
F = 2kI^2ln(x)
applying limits, from R tp R+w
F = 2kI^2*ln[(R+w)/R]

same force is applied by the other rail, so net F = 4kI^2 ln((R+w)/R)
a) F = 4*10^-7 * (451000)^2 ln((0.094+0.034)/0.094) = 25118.8422 N
b) a = F/m = 2511884.228 m/s/s
2*a**4.2 = v^2
v = 4593.454 m/s

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