A thin straight rod of length `2a` carrying a uniformly distributed charge `q` is located in vacumm. Find the magnitude of the electric field strength as a function of the distatance `r` from the rod's centre along the streaight line
(a) perpendicular to the rod and passing through its centre ,
(b) coinciding with the rod's direction (at the point lying outside the rod).

Explanation

(a) It is clear from symmetry condiderations that vector `vec(E)` must be directed as shwon in the figure. This shown the way of solving theis problem , we must find the component `dE`, of the field created by the element `dl` of the rod, having the charge `dq` and then intergation the result over all teh elements of the rod. In this case
`dE_(r) = d E cos alpha = (1)/(4pi epsilon_(0)) (lambda dl)/(r_(0)^(2)) cos alpha`,
whew `lambda = (q)/(2a)` is the linear charge density. Let us reduice this equaction this equaction of the from conventlent for intergation. Figure shows that `dl cos alpha = r_(0) alpha` and `r_(0) = (r )/(cos alpha)`,
Consequently,
`dE_(r) = (1)/(4pi epsilon_(0)) (lambda r_(0) d alpha)/(r_(0)^(2)) = (lambda)/(4pi epsilon_(0) r) cos alpha d alpha`
This expression can be easily intergated :
`E = (lambda)/(4pi epsilon_(0) r) 2 int_(0)^(alpha_(0)) cos alpha d alpha = (lambda)/(4pi epsilon_(0) r) 2 sin alpha_(0)`
where `alpha_(0)` is the maximum value of the angle `alpha, sin alpha_(0) = a// sqrt(a^(2) + r^(2))`
Thus, `E = (q//2a)/(4pi epsilon_(0)r)2 (a)/(sqrt(a^(2) + r^(2))) = (q)/(4pi epsilon_(0) r sqrt(a^(2) + r^(2)))`
Note that in this case also `E = (q)/(4pi epsilon_(0) r^(2))` for `r gt gt a` as of the fiedld of a point charge.
(b) Let, us consider the element of length `dl` at a distanace `l` from the centre of the rod, as shown in fig.
Then field at `P`, due to this element.
`dE = (lambda dl)/(4pi epsilon_(0) (r-l)^(2))`,
If the element lies on teh side, shown in the diagram, and `dE = (lambda dl)/(4pi epsilon_(0) (r + l)^(2))`, if it lies on other side.
Hence `E = int dE = int_(0)^(a) (lambda)/(4pi epsilon_(0) (r-l)^(2)) + int_(0)^(a) (lambda)/(4pi epsilon_(0) (r + l)^(2))`
On intergating and putting `lambda = (q)/(2a)`, we get, `E = (q)/(4pi epsilon_(0)) (1)/((r^(2) - a^(2)))`
For `r gt gt a, E = (q)/(4pi epsilon_(0) r^(2))`