A long straight wire along the `z`- axis carries a current ` I` in the negative ` z- direction` . The magnetic vector field ` vec(B)` at a point having coordinates (x,y) in the ` Z = 0` plane is

Options

A. `(mu_0I(yhati-xhatj))/(2pi(x^2+y^2))`
B. `(mu_0I(xhati+yhatj))/(2pi(x^2+y^2))`
C. `(mu_0I(xhatj-yhati))/(2pi(x^2+y^2))`
D. `(mu_0I(xhatj+yhati))/(2pi(x^2+y^2))`

Answer

A

Explanation

`vecr=xhati+yhatj`, and `hatr=vecr/r=(xhati+yhatj)/(sqrt(x^2+y^2))`
Magnetic field induction at P,
`|vecB|=(mu_0)/(4pi)(2I)/(r)=(mu_0)/(2pi)(I)/(sqrt(x^2+y^2)`
It acts perpendicular to position vector.

Unit vector perpendicular to the position vector
`hatn=hatrxxhatk=((xhati+yahtj)xxhatk)/(sqrt(x^2+y^2))=(yhati-xhatj)/(sqrt(x^2+y^2)`
`:. vecB=|vecB|hatn=(mu_0I(yhati-xhatj))/(2pi(x^2+y^2))`