question_answer

A current loop, having two circular arcs joined by two radial lines shown in the figure. It carries a current of 10 A. The magnetic field at point O will be close to:

A) \[1.0\times {{10}^{-5}}\text{T}\]                      

B) \[1.5\times {{10}^{-5}}\text{T}\]

C) \[1.0\times {{10}^{-7}}\text{T}\]

D) \[1.5\times {{10}^{-7}}\text{T}\]

Correct Answer: A

Solution :

\[\vec{B}=\frac{{{\mu }_{0}}i}{4\pi }\theta \left[ \frac{1}{{{r}_{1}}}-\frac{1}{{{r}_{2}}} \right]\hat{k}\]

\[{{r}_{1}}=3\]\[\text{cm}=3\times {{10}^{-2}}\text{m},\]

\[{{r}_{2}}=5\]\[\text{cm}=5\times {{10}^{-2}}\text{m}\]

\[\theta =\frac{\pi }{4},\] \[i=10A\]

\[\text{\vec{B}}=\frac{4\pi \times {{10}^{-7}}}{16}\times 10\left[ \frac{1}{3\times {{10}^{-2}}}-\frac{1}{5\times {{10}^{-2}}} \right]\hat{k}\]

\[\left| {\text{\vec{B}}} \right|=\frac{\pi }{3}\times {{10}^{-5}}\text{T}\approx 1\times {{10}^{-5}}\text{T}\]