Magnetic field due to currentMedium
Question
An infinitely long straight wire carrying current I is bent in a planer shape as shown in the diagram. The radius of the circular part is r . The magnetic field at the centre O of the circular loop is :
Options
A.$\frac{\mu_{0}}{2\pi}\frac{I}{r}(\pi + 1)\widehat{i}$
B.$- \frac{\mu_{0}}{2\pi}\frac{I}{r}(\pi - 1)\widehat{i}$
C.$\frac{\mu_{0}}{2\pi}\frac{I}{r}(\pi - 1)\widehat{i}$
D.$- \frac{\mu_{0}}{2\pi}\frac{I}{r}(\pi + 1)\widehat{i}$
Solution
$${{\overrightarrow{B}}_{0} = {\overrightarrow{B}}_{AB} + {\overrightarrow{B}}_{DE} + {\overrightarrow{B}}_{BCD} }{= \frac{\mu_{0}i}{4\pi r}\widehat{i} + \frac{\mu_{0}i}{4\pi r}\widehat{i} - \frac{\mu_{0}i}{2r}\widehat{i} }{= \frac{\mu_{0}i}{2\pi r}\widehat{i} - \frac{\mu_{0}i}{2r}\widehat{i} }{= \frac{\mu_{0}i}{2\pi r}(1 - \pi)\widehat{i} }{= - \frac{\mu_{0}i}{2\pi r}(\pi - 1)\widehat{i}}$$
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