Figure 29-89 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. Show that the force on the projectile is directed to the right along the rails and is given approximately by
(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 = 450 kA, w = 12 mm, R = 6.7 cm, L = 4.0 m, and the projectile mass is 10 g.
As the projectile is moving in the presence of magnetic field due to the two rails (1 and 2), it is also carrying current. So small current element of projectile will experience force due to both rails.Net force on current element () will be the vector sum of individual forces due to rail 1 and 2.
Consider a portion of projectile between .We can find magnetic force on this segment due to both rails (1 and 2)
The net force on the segment is , where and are the forces on portion of projectile due to rail 1 and 2 respectively.
We can find field due to rail 1 at distance from the center of rail 1.We can use Ampere law for this, then we get
Similarly, B field due to rail 2 at distance from the center of rail 2, is
And using this we get
By using cross product rule
Total force is obtained by integrating over
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Hence the force on the projectile is, role="math" localid="1664355437114"
To find the speed of projectile we can use work energy theorem
Substitute all the value in the above equation.
Hence the speed of the projectile is,
Fig. 29-63 shows wire 1 in cross section; the wire is long and straight, carries a current of out of the page, and is at distance from a surface. Wire 2, which is parallel to wire 1 and also long, is at horizontal distance from wire 1 and carries a current of into the page. What is the x component of the magnetic force per unit length on wire 2 due to wire 1?
A long wire is known to have a radius greater than and to carry a current that is uniformly distributed over its cross section. The magnitude of the magnetic field due to that current is at a point from the axis of the wire, and at a point 10 mm from the axis of the wire. What is the radius of the wire?
Figure 29-32 shows four circular Amperian loops (a, b, c, d) and, in cross section, four long circular conductors (the shaded regions), all of which are concentric. Three of the conductors are hollow cylinders; the central conductor is a solid cylinder. The currents in the conductors are, from smallest radius to largest radius, 4 A out of the page, 9 A into the page, 5 A out of the page, and 3 A into the page. Rank the Amperian loops according to the magnitude of around each, greatest first.
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