A rectangular loop of wire with sides H = 30 cm and W = 50 cm carries current I

Question

A rectangular loop of wire with sides H = 30 cm and W = 50 cm carries current I2 = 0.381 A. An infinite straight wire, located a distance L = 24 cm from segment ad of the loop as shown, carries current I1 = 0.597 A in the positive y-direction.

Fad,x, the x-component of the force exerted by the infinite wire on segment ad of the loop is equal to 5.69*10^-8 N

Fbc,x, the x-component of the force exerted by the infinite wire on segment bc of the loop is equal to -1.84*10^-8 N

Another infinite straight wire, aligned with the y-axis is now added at a distance 2L = 48 cm from segment bc of the loop as shown. A current, I3, flows in this wire. The loop now experiences a net force of zero.

5) What is the magnitude of I3?

Explanation / Answer

Given that,

Fad = 5.69*10^(-8) N

Fbc = -1.84*10^(-8) N

Force on loop due to i1,

F1 = 5.69*10^(-8) - 1.84*10^(-8) = 3.85*10^(-8) N

Force on loop due to i3,

Fad = u0*i3*i2*H / 2*pi*(2L + W)

Fad = 4*pi*i3*0.381*0.30 / 2*pi*(2*0.24 + 0.50)

Fad = i3*2.33*10^(-8) (i) N  

Fbc = u0*i3*i2*H / 2*pi*2L

Fbc = 4*pi*i3*0.381*0.30 / (2*pi*2*0.24)

Fbc = i3*4.76*10^(-8) (-i) N

Net force, F3 = i3*2.43*10^(-8) N

Net force on loop = 0

F1 - F3 = 0

F1 = F3

3.85*10^(-8) =  i3*2.43*10^(-8)

i3 = 1.58 A

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