Question

A long wire is bent into the shape shown in figure without cross contact at $\text{P}$. The radius of the circular section is $R.$ If current $I$ flows in the wire, then the magnetic field at the center $\text{C}$ is-

• A. $\frac{{\mu }_{0}I}{2R}(1+\frac{1}{\pi })⨀$
• B. $\frac{{\mu }_{0}I}{2R}(1+\frac{1}{\pi })⨂$
• C. $\frac{{\mu }_{0}I}{2R}(1-\frac{1}{\pi })⨀$
• D. $\frac{{\mu }_{0}I}{2R}(1-\frac{1}{\pi })⨂$

Answer 1

The correct option is D $\frac{{\mu }_{0}I}{2R}(1-\frac{1}{\pi })⨂$


Since straight part of the long wire has very small gap, so we can assume infinite straight wire directly.

The magnetic field due to a straight current carrying conductor is,

$B=\frac{{\mu }_{0}I}{4\pi d}[\sin {\theta }_1+\sin {\theta }_2]$

$B_1=\frac{{\mu }_{0}I}{4\pi R}[\sin {90}^{\circ }+\sin {90}^{\circ }]=\frac{{\mu }_{0}I}{2\pi R}⨀$

The magnetic field due to a circular coil at the center is,

$B=\frac{{\mu }_{0}I}{2R}$

$\therefore B_2=\frac{{\mu }_{0}I}{2R}⨂$

The resultant magnetic field at point $\text{O}$ is,

$\therefore B_{net}=B_2-B_1=\frac{{\mu }_{0}I}{2R}(1-\frac{1}{\pi })⨂$

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

Why this question ? To understand the concept of magnetic field at a point due to combination of straight conductor and circular core.