Question

A wire carrying current $I$ has the configuration shown in figure. What is the value of $\theta$ for the magnetic induction $B$ to be zero at the centre of the circle

• A. $1\text{rad}$
• B. $2\text{rad}$
• C. $\pi \text{rad}$
• D. $2\pi \text{rad}$

Answer 1

The correct option is B $2\text{rad}$



Magnetic field due to a circular arc at its centre is,

$B_{=}\frac{{\mu }_{0}I}{2R}\left(\frac{\theta }{2\pi }\right)$

Where, $\theta$ =Angle subtended by the arc at the centre.

Magnetic field at distance $R$ due to a straight semi-infinite conductor is,

$B=\frac{{\mu }_{0}I}{4\pi R}$

From the diagram, we can see that wires $1$ and $2$ is a semi-infinite wire and wire $3$ is circular arc.

$\Rightarrow B_1=B_2=\frac{{\mu }_{0}I}{4\pi R}⨀$

$\Rightarrow B_3=\frac{{\mu }_{0}I}{2R}\left(\frac{\theta }{2\pi }\right)⨂$

For, the net field to be zero at the centre of arc,

$B_1+B_2=B_3$

$\Rightarrow \frac{2{\mu }_{0}I}{4\pi R}=\frac{{\mu }_{0}I}{2R}\left(\frac{\theta }{2\pi }\right)$

$\Rightarrow \theta =2\text{rad}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(B\right)$ is the correct answer.

Why this question? To understand the concept of magnetic field due to the combination of straight wire and circular arc.