Question

a thin wire in which current I is flowing in shaped as shown in figure the value of magnetic field at the centre O of the circular portion will be-​

Answer 1

Given:

A thin wire in which current I is flowing in shaped as shown in figure.

To find:

Magnetic Field Intensity at the centre O.

Diagram:

$\boxed{\setlength{\unitlength}{1cm}\begin{picture}(6,4)\put(1,1){\vector(1,0){1}}\qbezier(2,1)(3,3)(4,1)\put(4,1){\vector(1,0){1}}\put(3,1){\vector(1,1){0.70}}\put(3.5,1.2){r}\put(1.5,1.25){I}\put(3,0.75){O}\end{picture}}$

Calculation:

Field Intensity due to the straight part of conductor is zero , because the axis of the conductor falls along the centre point O.

So, net magnetic field intensity will be contributed only by the semicircular portion of the wire:

$\therefore \: B = \dfrac{ \mu_{0} }{4\pi } \bigg \{ \dfrac{\pi I}{r} \bigg \}$

$= > \: B = \dfrac{ \mu_{0} }{4 \cancel\pi} \bigg \{ \dfrac{ \cancel\pi \: I}{r} \bigg \}$

$= > \: B = \dfrac{ \mu_{0}I }{4r}$

So, final answer is:

$\boxed{ \sf{ \: B = \dfrac{ \mu_{0}I }{4r} }}$