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First, we need to remember the magnetic field in the center of a coil, which is:
[tex]B=\frac{\mu_0iN}{2R}[/tex]Where R is the radius, and N is the number of turns. So, we have two unknows. However, the total length of the wire also gives us a constraint to the problem. We can equate this as: (L is the total length of 3m)
[tex]L=(2\pi R)N[/tex]As 2piR is the length of a single turn, multiplied by the number of turns. If we replace our values on these equations, we get:
[tex]2.795*10^{-3}=\frac{\mu_0*1.29*N}{2R}[/tex]And
[tex]3=2\pi NR[/tex]So, we have two equations and two unknowns. We can find out their values. As we want to find the diameter, let us first isolate the value of N on the second equation:
[tex]N=\frac{3}{2\pi R}[/tex]By replacing this value of N on the first equation, we get:
[tex]2.795*10^{-3}=\frac{\mu_0*1.29}{2R}*\frac{3}{2\pi R}[/tex]Then, if we isolate R, we get:
[tex]R^2=\frac{\mu_0*1.29*3}{4\pi *2.795*10^{-3}}[/tex]Then our final answer is D=2R=2.35cm
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