In Fig. Two long wires `W_1 and W_2`, each carrying current I, are placed parallel to each other and parallel to z-axis. The direction of current in `W_1`is outward and in `W_2` it is inward. Find `vecB` at p and Q.

Explanation

Let `vecB` due to `W_(1)` be `vecB_(1)` and due to `W_(2)` be `vecB_(2)`
By symmetry `|vecB_(1)|=|vecB_(2)|=B`
`B_(p)=2B cos 60^(@)=B=(mu_(0)l)/(2pi2a)=(mu_(0)l)/(4pia)`
`:. vecB_(p)=(mu_(0)l)/(4pia)hatj`
For `theta B_(1)=(mu_(0)I)/(2pisqrt5a) , rArr B_(2)=(mu_(0)I)/(2pia)`
`tan theta=a/(2a)=1/2 rArr barB=(B_(1) cos theta hatj)+(B_(2)-B_(1) sin theta)hati`
`sin theta=1/sqrt5 rArr barB=(mu_(0)I)/(5paia)hatj+((mu_(0)I)/(2pia)+(mu_(0)I)/(10pia))hati`
`cos theta=2/sqrt5 rArr barB=(2mu_(0)I)/(5paia)hatj+((mu_(0)I)/(5pia)+(mu_(0)I)/(5pia))hati`