Question

The magnetic field at point $O$ is

• A. $\frac{{\mu }_{0}i}{a}(\frac{1}{\pi }+\frac{1}{4})$
• B. $\frac{{\mu }_{0}i}{2a}(\frac{1}{\pi }+\frac{1}{4})$
  
• C. $\frac{{\mu }_{0}i}{a}(\frac{2}{\pi }+\frac{1}{4})$
• D. $\frac{{\mu }_{0}i}{2a}(\frac{1}{\pi }-\frac{1}{4})$

Answer 1

The correct option is B $\frac{{\mu }_{0}i}{2a}(\frac{1}{\pi }+\frac{1}{4})$


Let the magnetic field at point O because of semi-infinite rod, circular arc and semi-infinite rod be $B_1,B_2\text{and}{B}_3{B}_1=\frac{{\mu }_{0}i}{4\pi a}\otimes \to \text{into the plane}{B}_2=\frac{{\mu }_{0}idl}{4\pi a^2}{B}_2=\frac{{\mu }_{0}i\left(\frac{\pi a}{2}\right)}{4\pi R^2}=\frac{{\mu }_{0}i\pi R}{8\pi a^2}{B}_2=\frac{{\mu }_{0}i}{8a}\otimes \to \text{into the plane}{B}_3=\frac{{\mu }_{0}i}{4\pi a}\otimes \to \text{into the plane}(\text{point O lies close to the one}\text{end of the conductor})B_3=\frac{{\mu }_{0}i}{4\pi a}\otimes \to \text{into the plane}\text{So, net magnetic field at point O}B=B_1+B_2+B_{3}B=\frac{{\mu }_{0}i}{4\pi a}+\frac{{\mu }_{0}i}{8a}+\frac{{\mu }_{0}i}{4\pi a}B=\frac{{\mu }_{0}i}{4\pi a}[\frac{\pi }{2}+2]\otimes \to \text{Into the plane}B=\frac{{\mu }_{0}i}{2a}[\frac{1}{4}+\frac{1}{\pi }]\otimes \to \text{Into the plane}$