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 $x y$ plane [2022]

Question

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 $x y$ plane? [2022]

Smart Achievers · Accepted Answer

(3)

$\vec{B} = \frac{μ_{0} I}{4 π L} \sin 45 ° (- \hat{k}) + \frac{μ_{0} I π}{4 π (\frac{L}{2})} (- \hat{k}) + \frac{μ_{0} I}{4 π (\frac{L}{4})} \times \frac{π}{2} (- \hat{k})$

$= \frac{- μ_{0} I}{L} [\frac{1}{4 π \sqrt{2}} + 1] \hat{k}$

Solution

Q.
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 $x y$ plane [2022]

1 $\vec{B} = \frac{- μ_{0} I}{L} (\frac{3}{2} + \frac{1}{\sqrt[4]{2 π}}) \hat{k}$

2 $\vec{B} = \frac{- μ_{0} I}{L} (\frac{3}{2} + \frac{1}{2 \sqrt{2 π}}) \hat{k}$

3 $\vec{B} = \frac{- μ_{0} I}{L} (1 + \frac{1}{4 \sqrt{2 π}}) \hat{k}$

4 $\vec{B} = \frac{- μ_{0} I}{L} (1 + \frac{1}{4 π}) \hat{k}$

Ans.
(3)

$\vec{B} = \frac{μ_{0} I}{4 π L} \sin 45 ° (- \hat{k}) + \frac{μ_{0} I π}{4 π (\frac{L}{2})} (- \hat{k}) + \frac{μ_{0} I}{4 π (\frac{L}{4})} \times \frac{π}{2} (- \hat{k})$

$= \frac{- μ_{0} I}{L} [\frac{1}{4 π \sqrt{2}} + 1] \hat{k}$

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Solution

Q. 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 xy plane [2022]

1 B→=-μ0IL(32+12π4)k^

2 B→=-μ0IL(32+122π)k^

3 B→=-μ0IL(1+142π)k^

4 B→=-μ0IL(1+14π)k^

Ans. (3) B→=μ0I4πLsin45°(-k^)+μ0Iπ4π(L2) (-k^)+μ0I4π(L4)×π2 (-k^) =-μ0IL[14π2+1]k^