Question

A part of a long wire carrying a current $i$ is bent into a circle of radius $r$ as shown in the figure. The net magnetic at the centre $O$ is the circular loop is

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

Answer 1

The correct option is C $\frac{{\mu }_{0}i}{2\pi r}(\pi +1)$

The given shape is equivalent to the following diagram containing two wires : one circular loop and one infinitely long straight wire

Field at $O$ due wire $AB$,

$B_{AB}=\frac{{\mu }_{0}i}{2\pi r}⨀$

Direction of field can be checked by ampere’s right hand thumb rule.

Field at $O$ due to circular loop ,
$B_{\text{loop}}=\frac{{\mu }_{0}i}{2r}⨀$

Direction of field can be checked by ampere’s right hand thumb rule.

Since both fields are in same direction so resresultant field at $O$

$\Rightarrow B_{net}=B_{AB}+B_{\text{loop}}=\frac{{\mu }_{0}i}{2\pi r}+\frac{{\mu }_{0}i}{2r}$

$\Rightarrow B_{net}=\frac{{\mu }_{0}i}{2\pi r}(\pi +1)⨀$