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Question
A current carrying loop consists of 3 identical quarter circles of radius R, lying in the positive quadrants of the x-y, y-z and z-x planes with their centres at the origin, joined together. Find the direction and magnitude of B at the origin.
Solution
From Biot-Savart law we find the relation of magnetic field at centre of the current carrying coil which subtends an angle θ, B = `mu_0/(4pi) (Iθ)/R`.
Magnetic field at origin due to the quarter circle lying in x-y plane:
`vecB_1 = (mu_0)/(4pi) (I(pi/2))/R hatk = (mu_0)/4 I/(2R) hatk`
Similarly, magnetic field at origin due to the quarter circle lying in the y-z plane:
`vecB_2 = (mu_0/4) I/(2R) hati`
Similarly, magnetic field at origin due to the quarter circle lying in the z-x plane:
`vecB_3 = (mu_0/4) I/(2R) hatj`
Now, vector sum of magnetic field at origin due to each quarter is given by
`vecB_("net") = 1/4 ((mu_0I)/(2R)) (hati + hatj + hatk)`.
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