Deduce the relation for the magnetic induction at a point due to an infinitely long straight conductor carrying current.

Explanation

Magnetic field due to long straight conductor carrying current : Consider a long straight wire NM with current I flowing from N to M . Let P be the point at a distance a from point O. wire
Consider an element of length dl of the wire at a distance l from point O and `vec(r)` be the vector
joining the element dl with the point P. Let `theta` be the angle between d`vec(l) and vec(r)`. then, the magnetic field at P due to the element is
`d vec(B) = (mu_(0) l vec(dl))/(4 pi r^(2)) sin theta` (unit vector perpendicular to d`vec(l) and vec(r)`
the direction of the field is perpendicular to the plane of the paper and going into it. this can be determined by taking the cross product between two vectors d`vec(l) and vec(r)` (let it be `hat(n)`). The net magnetic field can be determined by integrating equation with proper limits.
`vec(B) int d vec(B)`
From the figure, in a right angle trinalge PAO,
l = - `(a)/(tan theta) ("since tan "(pi - theta)= - tan theta) rArr (1)/(tan theta) = cot theta`
l = - a cot `theta` and r = a cosec `theta`
Differentiating ,
dl = a `cosec^(2) theta d theta`
`d vec(B) = (mu_(0) I)/(4 pi ) (("a cosec"^(2) theta d theta))/(("a cosec" theta)^(2)) sin theta d theta hat(n)`
= `(mu_(0) I)/(4 pi ) (("a cosec"^(2) theta d theta))/("a cosec"^(2)theta ) sin theta d theta hat(n)`
` d vec(B) = (mu_(0)I)/(4 pi a ) sin theta d theta hat(n) `
this is the magnetic field at a piont P due to the current is small elemental length, Note tht we have expressed the magnetic field OP in terms of angular coordinate i.e. `theta`. therefore, the net magnetic field at the point P which can be obtained by integrating d`vec(B)` be varying the angle from `theta = phi_(1) "to" theta = phi_(2) ` is
`vec(B) = (mu_(0)I)/(4 pi a) int_(phi_(1))^(phi 2) sin theta d theta hat(n) = (mu_(0)I)/(4 pi a ) (cos phi_(1) - cos theta_(2))hat(n)`
For a an infinitely long straight wire, I = 0 and 2 = , the magnetic field is
`vec(B) = (mu_(0)I)/(2 pi a)hat(n)`
Note that here `hat(n) ` represents the unit vector from the point O to P .