Question

The radius of the curved part of the wire is $R=100mm$, the linear parts of the wire are very long. Find the magnetic induction at the point $O$ if the wire carrying a current $I=8.0A$ has the shape shown in figure.

• A. $0.44\mu T$
• B. $0.34\mu T$
• C. $0.56\mu T$
• D. $0.78\mu T$

Answer 1

The correct option is B $0.34\mu T$

Given,

$I=8A$

$R=100mm=0.1m$

$\frac{{\mu }_0}{4\pi }={10}^{-7}$

Magnetic field due to semi-circular wire=$\frac{{\mu }_{0}i}{4R}(-\hat{i})$

Magnetic field due to semi-infinite wire along x axis=$\frac{{\mu }_{0}i}{4\pi R}(-\hat{k})$

Magnetic field due to semi-infinite wire along z axis=$\frac{{\mu }_{0}i}{4\pi R}(-\hat{i})$

Thus net magnetic field at O is $\frac{{\mu }_{0}i}{4R}(1+\frac{1}{\pi })(-\hat{i})+\frac{{\mu }_{0}i}{4\pi R}(-\hat{k})$

$B_0=-\frac{{\mu }_{0}I}{4\pi R}(\hat{k}+(\pi +1\left)\hat{i}\right)$

It's magnitude is,

$|B_0|=\frac{{\mu }_{0}I}{4\pi R}.\sqrt{(1{)}^2+(\pi +1{)}^2}$

$|B_0|={10}^{-7}\times \frac{8}{0.1}\sqrt{1+(4.14{)}^2}$

$|B_0|=0.34\mu T$

The correct option is B.