Two infinitely long straight parallel wire are 5 m apart, perpendicular to the plane of paper. One of the wires, as it passes perpendicular to the plane of paper, intersects it at A and carries current I in the downward direction. The other wire intersects the plane of paper at point B and carries current k int the outward direction O in the plane of paper as shown in Fig. With x and y axis shown, magnetic induction at O in the component form can be expressed as

Options

A. `vec(B)=(mu_(0)i)/(2 pi)[-hat(i)+(3hat(j))/(5)]`
B. `vec(B)=(mu_(0)i)/(5 pi)[-hat(i)+(7hat(j))/(24)]`
C. `vec(B)=(mu_(0)i)/(4 pi)[-3hat(i)+(7hat(j))/(18)]`
D. `vec(B)=(mu_(0)i)/(7 pi)[-2hat(i)+(3hat(j))/(5)]`

Answer

B

Explanation

From trogonometry,
`AP=16//5 and PB=9//5`.Thus
`Sin AOP =4//5 and cos AOP =3//5`
`sin BOP=3//5 and cos BOP =4//5`
Magnetic field due to A is `B_(A)=(mu_(0)I)/(8 pi)` which makes angle `theta`
with negative x-axis and lies in third quadrant.

Magnetic field at O due to wire B is `(B_B)=(mu_(0)i)/(6 pi)`which makes angle `theta` with y-axis and lies in second quadrant. In components form,
`vec(B_A)=(mu_(0)i)/(8 pi) (-(4)/(5) i -(3)/(5)j)`
`vec(B_B)=(mu_(0)i)/(6 pi) (-(3)/(5) i -(4)/(5)j)`
Thus, `vec(B_A)+vec(B_B)=(mu_(0)i)/(5 pi) (-i +(7)/(24)j)`.