you are given a wire of length L and that carries a uniform current i through it.(adsbygoogle = window.adsbygoogle || []).push({});

the wire can bent into either a circle or a square

which shape gives the maximum magnetic field at its center?

for the circle

[tex] B = \frac{\mu_{0}}{4 \pi} \int \frac{i ds \cross r}{r^3} [/tex]

im not qutie sure about the s part since ds = L, riught? so ds is constant value... and so is R so we should get

[tex] B = \frac{\mu_{0} i}{4 \pi R^2} [/tex]

where [tex] R = \frac{L}{2 \pi} [/tex]

so [tex] B = \frac{\mu_{0} i \pi}{L^2} [/tex]

now for the squar

the problem is ds is a cosntant value, but r is not because it varies from s/2 to [itex] \frac{s}{\sqrt{2}} [/itex]

so what do i do? Can i use the square as an Amperian loop and solve it like so

[tex] B (s^2) = \mu_{0} i [/tex]

since 4S = L, s = L/4 so

[tex] B = \frac{16 \mu_{0} i}{L^2} [/tex]

thus the magnetic field at the center due to th square loop is greater becasue 4 > pi?

Also i am a bit confused as to when i use the Ampere's law and Biot Savart Law, can the Ampere's law be used for the above described situations or not?

Please help! Your advice is greatly appreciated!

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# Homework Help: Maximize the magnetic field

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