A wire carrying current \[I\] has the shape as shown in adjoining figure. Linear parts of the wire are very long and parallel to x-axis while semicircular portion of radius B. is lying in YZ plane. Magnetic field at point O is |
[NEET 2015] |
A) \[\mathbf{B}=\frac{{{\mu }_{0}}}{4\pi }\frac{l}{R}(\pi \hat{i}+2\hat{k})\]
B) \[\mathbf{B}=-\frac{{{\mu }_{0}}}{4\pi }\frac{l}{R}(\pi \hat{i}-2\hat{k})\]
C) \[\mathbf{B}=-\frac{{{\mu }_{0}}}{4\pi }\frac{l}{R}(\pi \hat{i}+2\hat{k})\]
D) \[\mathbf{B}=\frac{{{\mu }_{0}}}{4\pi }\frac{l}{R}(\pi \hat{i}-2\hat{k})\]
Correct Answer: C
Solution :
[c] The magnetic field in the different regions is given by |
\[{{\mathbf{B}}_{1}}=\frac{{{\mu }_{0}}}{4\pi }\times \frac{I}{R}(-\mathbf{\hat{k}})\] |
\[{{\mathbf{B}}_{3}}=\frac{{{\mu }_{0}}I}{4\pi R}(-\mathbf{\hat{k}})\] |
\[{{\mathbf{B}}_{2}}=\frac{{{\mu }_{0}}I}{4\pi R}(-\mathbf{\hat{i}})\] |
The net magnetic field at the centre O is |
\[\mathbf{B}={{\mathbf{B}}_{\mathbf{1}}}+{{\mathbf{B}}_{\mathbf{2}}}+{{\mathbf{B}}_{\mathbf{3}}}\] |
\[=\frac{{{\mu }_{0}}I}{4\pi R}(-2\hat{k})+\frac{{{\mu }_{0}}I}{4R}(-\hat{i})\] |
\[=-\frac{{{\mu }_{o}}I}{4\pi R}(2\hat{k}+\pi \hat{i})\] |
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