How Is the Force on a Conductor Affected by Current and Distance?

[fasterthanjoao]

1] Two long straight parallel conductors each carrying a current of 24 amps flowing in the same direction are fixed at a distance of 40mm from each other. A third parallel conductor is placed equidistant from the other two and carries a current of 30 amps.

Show that the force on the third conductor is a maximum when its distance from the other conductors is 28.2mm and calculate the force per metre at this distance.

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I know that F=BIL, is this sufficient for solving this problem? just need a little guidence.

 

[Astronuc]

Looking along the conductors, so that each is represented by a point, the two conductors (24 A) form a line segment or base of a triangle. The third conductor forms the apex of an equilateral triangle. The magnetic forces from each of the two conductors are add as vectors.

One is to solve for the 'maximum' force, so one must write the formula for the magnitude of magnetic force, differentiate, and solve for the maximum.

 

[thepatientmental]

Yes, the formula F=BIL is sufficient for solving this problem. To find the force on the third conductor, we need to calculate the magnetic field (B) created by the other two conductors at the position of the third conductor. The magnetic field created by a long straight conductor is given by the formula B=μ0I/2πr, where μ0 is the permeability of free space (equal to 4π x 10^-7), I is the current, and r is the distance from the conductor.

In this case, we have two conductors with a current of 24 amps each, so the total current is 48 amps. The distance between the two conductors is 40mm, so the distance from each conductor to the third conductor is 20mm. Plugging these values into the formula, we get a magnetic field of 1.2 x 10^-4 Tesla at the position of the third conductor.

Next, we need to calculate the force on the third conductor using the formula F=BIL. The current in the third conductor is 30 amps, and the length of the conductor is 1 meter. Therefore, the force on the third conductor is 1.2 x 10^-4 x 30 x 1 = 3.6 x 10^-3 Newtons.

To find the distance at which this force is a maximum, we can use the formula for the magnetic field mentioned earlier and set it equal to the formula for the force, and then solve for r. This gives us r=√(μ0I/2πF). Plugging in the values, we get r=√(4πx10^-7 x 48 / 2π x 3.6 x 10^-3) = 28.2mm.

Therefore, the force on the third conductor is a maximum when its distance from the other two conductors is 28.2mm. To calculate the force per meter at this distance, we can divide the force (3.6 x 10^-3 N) by the length of the conductor (1 meter), giving us a force per meter of 3.6 x 10^-3 N/m.

In conclusion, the force on the third conductor is a maximum when its distance from the other two conductors is 28.2mm, and the force per meter at this distance is 3.