Question

Find the magnetic induction of the field at the point $O$ of a loop with current $I$, whose shape is illustrated. A current $I$ flows along a thin wire shaped as shown in figure. The radius of a curved part of the wire is equal to $R$ the angle is $2\varphi$. Find the magnetic induction of the field at the point $O$.

• A. $B=(\pi -\varphi +\sin \varphi ){\mu }_0\frac{I}{2\pi R}$
• B. $B=(\pi -\varphi +\tan \varphi ){\mu }_0\frac{I}{2R}$
• C. $B=(\pi -\varphi +\tan \varphi ){\mu }_0\frac{I}{2\pi R}$
• D. None of these

Answer 1

The correct option is C $B=(\pi -\varphi +\tan \varphi ){\mu }_0\frac{I}{2\pi R}$


We know that, Magnetic field due to circular arc at centre is $B_{arc}=\frac{{\mu }_{0}i}{4\pi R}\theta \otimes$

Magnetic field due to given circular arc is
$B_1=\frac{{\mu }_{0}I}{4\pi R}(2\pi -2\varphi )$
$B_1=\frac{{\mu }_{0}I}{2\pi R}(\pi -\varphi )\otimes$

We know that, Magnetic field due to a line wire is given by,
$B=\frac{{\mu }_{0}I(\sin {\theta }_1+\sin {\theta }_2)}{4\pi \left(d\right)}$
Here ${\theta }_1={\theta }_2=\varphi ,d=R\cos \varphi$

Magnetic field due to given straight wire
$B_2=\frac{{\mu }_{0}I}{4\pi R\cos \varphi }2\sin \varphi$
$B_2=\frac{{\mu }_{0}I}{2\pi R}\tan \varphi \otimes$

Net Magnetic field is
$B_{net}=B_1+B_2$
$B_{net}=\frac{{\mu }_{0}I}{2\pi R}[\pi -\varphi +\tan \varphi ]$