An IE IRODOV Electrodynamics problem A ring (radius R) is given a negative charge -q and at the centre a positive charge +q is kept. We have to calculate the electric field on the axis of the ring at a distance X from the centre. Take X>>R. 
I calculated the field. At the first attempt I  got 0, which I  had got by considering the ring as a point charge. So +q and -q should give a field 0 as  the charges are of opposite nature. But the answer is something different, the author has first calculated the field due to the ring (not taking it as a point charge) and subtracted the field of point charge from it. In the final expression he has neglected the radius R.
I need a logical explanation as to what is different in both the solutions and why couldn't we take the ring as a point charge at the start?
 A: Consider infinitesimal charges distributed around the ring. At a point on the axis the radial (perpendicular to the axis) components of their fields cancel by symmetry. The field in the axial direction is the sum of the axial components of their fields. Thus, the field (due to the ring) on the axis is equal to the axial component of the field of a  single point charge located on the ring.
So you need to calculate the field strength at distance $(X^2+R^2)^\frac{1}{2}$ and multiply by $X/(X^2+R^2)^\frac{1}{2}$ to get the axial component (and then add the field of the center charge).
A: You've run into a classic approximation problem!
Usually, you can approximate $X + \epsilon$ as just $X$ if $\epsilon$ is very small compared to $X$. And usually, you can do this to every variable in a problem, and the answer will come out fine.
It only doesn't work when the answer itself is also very small; for example, let's say here the field from the ring is $X + \epsilon$ and the field from the charge is $-X$. Then the answer is $X + \epsilon - X = \epsilon$, and if you make the approximation $X + \epsilon \approx X$ then you've thrown out the whole answer!
To fix this, you can do two things. You can avoid doing any approximations at all until you hit your final answer. Or, you can approximate as long as you keep track of the order of the terms you're throwing out. Then if you arrive at a trivial answer, you need to back up and keep terms of higher orders.
