Figure a shows, in cross section, two wires that are straight, parallel, and ver

Question

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 uB set up by the two currents at the origin. Figure b gives uB as a function of the position x of wire 2. The curve has an asymptote of uB = 2.12 nJ/m3 as x , and the horizontal axis scale is set by xs = 90.0 cm.

(a) What is the value of i1?

(b) What is the value of i2?

Explanation / Answer

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 uB = 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

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