A current I is flowing in a conductor shaped as shown in figure. The radius of the curved part is r and length of straight portion is very large. Find the magnetic field induction at the centre O.

Options

A. `(mu_(0) I)/(4pi r) ((pi)/(2) + 1)`
B. `(mu_(0) I)/(4pi r) ((pi)/(2) - 1)`
C. `(mu_(0) I)/(4 pi r) ((3pi)/(2) + 1)`
D. `(mu_(0) I)/(4pi r) ((3pi)/(2) - 1)`

Answer

C

Explanation

Magnetic field due to circular arc –
`B_(1) = (mu_0)/(4pi ) * (1 theta)/(R ) = (mu_(0) )/(4pi ) * (I .3 pi //2)/(r )`
`= (3mu_(0) I)/(8 r)` (inside to the paper)
Magnetic field due AD wire - `B_(2) = (mu_(0) I)/(2pi r) xx (1)/(2) = (mu_(0) I)/(4pi r)` (inside)
Total B = `B_(1) + B_(2) = (mu_(0) I)/(4pi r) [ (3pi)/(2) + 1]`