Two long wires `PQR` and `MNP` carry equal current `I`as shown such that `QR` and `NP` are parallel. Find the magnetic field at origin `O`.

Options

A. `(mu_(0)I)/(2sqrt(3)pia)` into the page
B. `(mu_(0)I)/(2pia)` out of the page
C. `(mu_(0)I)/(2sqrt(3)pia)` out of the page
D. `(mu_(0)I)/(2pia)` into the page

Answer

C

Explanation

Due to `PQ` and `MN`, magnetic field at `O` will be zero.

`r=asin60^(@)=(sqrt(3))/2a`
`B_(QR)=(mu_(0))/(4pi) (I)/(r)[sin90^(@)-sin30^(@)] ox`
`=(mu_(0))/(4pi) I/((sqrt(3)a//2))xx1/2ox=(mu_(0)I)/(4pisqrt(3)a) ox`
`B_(NP)=(mu_(0))/(4pi) (I)/(r)[sin90^(@)+sin30^(@)] o.=(3mu_(0)I)/(4pisqrt(3)a) o.`
Net field at `O`:
`B=B_(NP)-B_(QR)=(mu_(0)I)/(2sqrt(3)pi a) o.`