Far, far away

Charge per unit length is $1 C/m$

For an element $dx$ on the wire, charge is $1 \times dx C/m$

We know,

Field due to charge $dx$ at any distance $r$ is

$dx/r^{2}$

To find out field at $r$ ,

Let us assume the point at distance $d$ lies on the perpendicular from the centre $O$

Therefore by Pythogoras Theorem , $r^{2}=x^{2}+d^{2}$ ,

Where $x$ is the distance of element $dx$ on the wire from mid-point $O$

Therefore the total field at distance $d$ from the wire is

$\int_{-1/2}^{1/2} dx./(x^{2}+d^{2})=1/d \times [tan^{-1} x/d]_{-1/2}^{1/2}$

Since $d>>x$

Therefore, $tan^{-1}x/d$ tends to $x/d$

Substituting $x/d$ for $tan^{-1}x/d$

We get-

$1/d \times [x/d]_{-1/2}^{1/2}$

And that leads to

$1/d^{2}$

Therefore, the answer is $2$

take a slice of the road and make sure you get a circle whose area will be pi (radius)^2 now think about the situation#a rod(a pure rod)#charge#uniform distribution#etc(given problem)!! *remember the closer you look the less you will actually see

Very far away from the rod, we see it is a point charge rather than as a line charge. If the rod were thin infinite then the force would fall of as 1/r but since the rod is finite it will be equivalent to a point charge at very far away distance hence 1/r2.