Which one of the following options represents the magnetic field $\vec{B}$ at O due to the current flowing in the given wire segments lying on the $xy$ plane?

(C) $\vec{B}=\frac{-\mu_{0} I}{L}\left(1+\frac{1}{4 \sqrt{2} \pi}\right) \hat{k}$

(A) $\vec{B}=\frac{-\mu_{0} I}{L}\left(\frac{3}{2}+\frac{1}{4 \sqrt{2} \pi}\right) \hat{k}$

(B) $\vec{B}=-\frac{\mu_{0} I}{L}\left(\frac{3}{2}+\frac{1}{2 \sqrt{2} \pi}\right) \hat{k}$

(D) $\vec{B}=\frac{-\mu_{0} I}{L}\left(1+\frac{1}{4 \pi}\right) \hat{k}$

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Magnetic Effects of Current

Which one of the following options represents the magnetic field

B

at O due to the current flowing in the given wire segments lying on the

x y

plane?

Answer: (C)

The magnetic field at O is the sum of contributions from three segments:

Straight segment at 45°: $B_{1} = \frac{μ _{0} I}{4 π L} sin 45° (- \hat{k}) = \frac{μ _{0} I}{4 2 π L} (- \hat{k})$

Semicircular arc of radius L/2: $B_{2} = \frac{μ _{0} I π}{4 π ( L /2 )} (- \hat{k}) = \frac{μ _{0} I}{2 L} (- \hat{k})$

Quarter circular arc of radius L/4: $B_{3} = \frac{μ _{0} I ( π /2 )}{4 π ( L /4 )} (- \hat{k}) = \frac{μ _{0} I}{2 L} (- \hat{k})$

Total:

B = \frac{μ _{0} I}{L} (\frac{1}{2} + \frac{1}{2} + \frac{1}{4 2 π}) (- \hat{k}) = \frac{- μ _{0} I}{L} (1 + \frac{1}{4 2 π}) \hat{k}

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Single correct option