Three long, straight wires are carrying currents that have the same magnitude. In C the current is opposite to the current in A and B. The wires are equally spaced. Each wire experiences a net force due to the other two wires. Which wire experiences a net force with the greatest magnitude?

The magnetic force felt by wire B is the highest.

Explanation:

With the three wires arranged in the order A, B and C equally spaced a distance d from one another.

Using the right hand rule (if the thumb is the direction of current, the direction of the magnetic field is in the direction of the fingers), we can obtain the magnetic field on each of the wires and subsequently obtain the net magnetic force on each wire.

Magnetic field = [μI/2πr]

Since all μ, I and 2π are all constants for this setup under consideration, we can refer to all of them as k.

r = distance between the point where the magnetic field is required from the wire causing the magnetic field.

Magnetic field = (k/r)

Let's take a convention of pointing out of the page of the book to be positive and pointing into the book to be negative.

On wire A,

The magnetic field on wire A due to wire B is in the direction that is coming out of the page of the book and the magnetic field due to wire C is going into the page, hence, the net magnetic field is calculated thus

Distance between wire A and B = d

Magnetic field on wire A due to wire B = (k/d)

Distance between wire A and C = 2d

Magnetic field on wire A due to wire C = -(k/2d)

Total magnetic field on wire A

= (k/d) - (k/2d) = (k/2d)

On wire B,

The magnetic field on wire B due to wire A is in the direction that is pointing into the page of the book and the magnetic field due to wire C is also going into the page, hence the net magnetic field is calculated thus,

Distance between wire A and B = d

Magnetic field on wire B due to wire A = -(k/d)

Distance between wire B and C = d

Magnetic field on wire B due to wire C = -(k/d)

Total magnetic field on wire B

= -(k/d) - (k/d) = -(2k/d)

Magnitude of B = (2k/d)

On wire C,

The magnetic field on wire C due to wire A is in the direction that is pointing into the page of the book and the magnetic field due to wire B is also going into the page, hence the net magnetic field is an addition of these two magnetic fields.

Distance between wire A and C = 2d

Magnetic field on wire C due to wire A = -(k/2d)

Distance between wire B and C = d

Magnetic field on wire C due to wire B = -(k/d)

Total magnetic field on wire C

= -(k/2d) - (k/d) = -(3k/2d)

Magnitude of the magnetic field = (3k/2d) = (1.5k/d)

From these calculations, it is evident that the magnitude of the magnetic field felt by wire B is the highest [(2k/d) > (1.5k/d) > (k/2d)].

Since the magnetic force felt by each wire is directly proportional to the strength of the magnetic field felt at the point where the wire is, it is evident that the magnetic force felt by wire B is the highest then. (Since the magnetic field at B is highest).

Hope this Helps!!!

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