Question

The magnetic field at center $O$ of the arc in figure is

• A. $\frac{{\mu }_{0}I}{4\pi \times r}[\sqrt{2}+\pi ]$
• B. $\frac{\mu I}{2\pi r}[\frac{\pi }{4}+(\sqrt{2}-1)]$
• C. $\frac{{\mu }_0}{4\pi }\times \frac{I}{r}\left[(\sqrt{2}-\pi )\right]$
• D. $\frac{{\mu }_0}{4\pi }\times \frac{I}{r}\left[(\sqrt{2}+\frac{\pi }{4})\right]$

Answer 1

The correct option is B $\frac{\mu I}{2\pi r}[\frac{\pi }{4}+(\sqrt{2}-1)]$

Magnetic field due to straight wires

$B_{circular}=\frac{1}{4}=\frac{{\mu }_{0}I}{2r}=\frac{{\mu }_{0}I}{8r}$

$B_{straight}=2\frac{{\mu }_{0}I}{4\pi \left(r\cos {45}^o\right)}[\sin {90}^o-\sin {45}^o]$

$=\frac{{\sqrt{2}\mu }_{0}I}{2\pi r}[1-\frac{1}{\sqrt{2}}]$