Find the magnetic field at O. | vedclass

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Question

(A) The magnetic field at the center $O$ due to straight segments $1, 3, 5,$ and $7$ is zero because the point $O$ lies on the axis of these segments.

Vedclass · Accepted Answer

$\frac{5{\mu _0}i	heta }{24\pi r}$

Vedclass · Answer

(A) The magnetic field at the center $O$ due to straight segments $1, 3, 5,$ and $7$ is zero because the point $O$ lies on the axis of these segments.For the circular arcs,the magnetic field is given by $B = \frac{{\mu _0}i	heta }{4\pi R}$.For arc $2$ (radius $3r$,current $i$ clockwise): $B_2 = \frac{{\mu _0}i	heta }{4\pi (3r)} = \frac{{\mu _0}i	heta }{12\pi r}$ (directed into the page,$\otimes$).For arc $4$ (radius $2r$,current $i$ counter-clockwise): $B_4 = \frac{{\mu _0}i	heta }{4\pi (2r)} = \frac{{\mu _0}i	heta }{8\pi r}$ (directed out of the page,$\odot$).For arc $6$ (radius $r$,current $i$ clockwise): $B_6 = \frac{{\mu _0}i	heta }{4\pi r}$ (directed into the page,$\otimes$).The net magnetic field at $O$ is $B_{net} = B_2 - B_4 + B_6 = \frac{{\mu _0}i	heta }{4\pi r} (\frac{1}{3} - \frac{1}{2} + 1) = \frac{{\mu _0}i	heta }{4\pi r} (\frac{2 - 3 + 6}{6}) = \frac{5{\mu _0}i	heta }{24\pi r}$ (directed into the page).

Find the magnetic field at $O$.

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