The figure at the top shows, in cross section, two wires that are straight, para

Question

The figure at the top 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. The lower figure gives uB as a function of the position x of wire 2. The curve has an asymptote of uB = 1.62 nJ/m3 as x goes to infinity, and the horizontal axis scale is set by xs = 48.0 cm. What is the value of i1? (**.0187 A is incorrect for i1?)

Explanation / Answer

We have, xs = 48 cm

and the total magnetic energy density is zero at location

           x= 2*(48/6).= 16 cm = 0.16 m

At this point magnitudes of magnetic fields are same.  

     uoi1/2d = uoi2/2*0.16

Hence , d = 0.16*(1/3) = 0.05333 m

the field,

B1 = sqrt (2uBuo)

The magnetic field due to first wire is

B1 = uoi1/2d

so, the current i1 is,

   i1 = 2d*sqrt (2uB/uo)

     = 2*0.05333*sqrt (2*1.62*10^-9 / 4*10^-7)

      = 0.017 A = 17.01*10^-3 A = 17.01 mA

the current i2 is,

         i2 = 3i1 = 51.04 mA

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