Two long parallel wires (shown in the figure below) are attracted to each other

Question

Two long parallel wires (shown in the figure below) are attracted to each other by a force per unit length k. One wire carries a current of I_1 to the right and is located along the line y = a. The second wire lies along the x-axis and it carries current I_2. Answer the following questions based on this information. What is the direction of I_2? What is the value of h in terms of k, I_2 and a? Find the value of y for the line in the plane of the two wires along which the total magnetic field is zero.

Explanation / Answer

(A) magnetic force is attractive and that is when currents are in same direction.

so I2 is to the right.

(B) force per unit length = u0 I1 I2 / (2 pi r )

n = u0 I1 I2 / (2 pi a)

I2 = (2 pi a n) / (u0 I1)

(C) that point will be in between wire( y < a)

B = u0 I / (2 pi r)

Bnet = [ u0 I2 / (2 pi y)] - [ u0 I1 / 2 pi (a-y)] = 0

u0 (2 pi a n ) / (2 pi y u0 I1) = u0 I1 / 2pi (a -y)

a n / y I1 = u0 I1 / 2 pi (a-y)

2 pi n a^2 - 2pi a n y = u0 y I1^2

y = 2 pi a^2 n / (2pi a n + u0 I1^2 )

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