Figure a shows in cross section two wires that are straight

Figure a shows, in cross section, two wires that are straight, parallel, and very long. The ratio i₁/i₂ of the current carried by wire 1 to that carried by wire 2 is 1/3. Wire 1 is fixed in place. Wire 2 can be moved along the positive side of the x axis so as to change the magnetic energy density u_B set up by the two currents at the origin. Figure b gives u_B as a function of the position x of wire 2. The curve has an asymptote of u_B = 2.12 nJ/m³ as x , and the horizontal axis scale is set by x_s = 90.0 cm.

(a) What is the value of i₁?

(b) What is the value of i₂?

Solution

Considering the direction of two currents are same.

a) The equation for intensity of magnetic field for a straight wire carrying current I₁ is,

    B₁ = mu(0) * I₁ / 2 * pi * d

mu (0) = 4 * pi * 10^-7

pi = 3.14

d = 90 cm = 0.90m

Provided energy density of magnetic field u_B = B² / 2 * mu(0) = 2.12 nJ/m³ = 2.12 * 10^-9 J/m³

B² = 2.12 * 10^-9 * 2 * 4 * 3.14 * 10^-7 = 6.689 * 10^-21 T²

B = 8.178 * 10^-11 T

Here B₁ = 8.178 * 10^-11 T

therefore I₁ = B₁ * 2 * 3.14 * 0.90 / mu(0) = 3.680 * 10^-4 A = 0.368 mA

b) I₁ / I₂ = 1 / 3   (Provided)

     I₂ = 3 I₁ = 0.368 * 3 = 1.104 mA