What happens to the electric field lines when an electron and positron collide? A month back I began my course in classical electrodynamics. I know about the field lines of moving charges. But I have this question that if an electron and a positron are situated some distance apart and assuming that the electric force is 'somehow' great enough to accelerate them towards each other, then what happens to the electric field lines after the 2 have collided? The net charge becomes 0 and taking into the fact that photons are released, my question is what happens to the electric field? 
Edit - So is it that the Electric Field exists outside the radius ct and is zero within the circle of radius ct which increases in diameter with time and spreads in space ?
 A: The electron and positron are two point charges with opposite sign, and classically , as the field lines are an iconal representation of the  charge, when the charge  becomes zero there will be no electric field lines from the spot where the two point particles overlap. 
BUT electrons and positrons are quantum mechanical particles and when close enough classical electrodynamics has to be replaced by quantum mechanical equations .
For the low energies you are discussing, when they get close enough they may get caught in each other's potential forming a positronium, similar to the hydrogen atom. . This has energy levels and the annihilation will happen from one of these energy levels.
The postron and electron will be in probability locuses, called orbitals and these, depending on the quantum numbers of the energy levels, will have asymmetries which will allow for dipole and multipole electric fields as long as the positronium survives intact. It will fast annihilate into two photons, which will carry electric and magnetic field information  in their complex wave functions, and balance the quantum numbers, energy and momentum and angular momentum. 
The calculations for the probability of annihilation can only be done accurately using quantum electrodynamics, QED.
EDIT after comment.

But what happens if they do collide or suppose we make them collide somehow ?

There are experiments with  electron positron colliders, where the particles have higher energy ( not starting at zero kinetic energy as in your question). Again the collision dynamics is successfully modeled using QED. When electrons scatter on positrons, with enough energy other channels open and the scattering produces  particles and resonances, i.e. the electron and positron"disappear" and other elementary particles come out.
Here is what the crossection looks like for this scattering.

the top figure, go to the link for the caption. 
It needs serious study of QED to understand these scatterings.
For the simple case of annihilation to two photons, the Feynman diagram which defines the probability of this to happen is 

The e- radiates a photon and becomes virtual, and meets the e+ and annihilates into another photon, conserving momentum and energy in the center of mass system. It has no meaning to be talking of fields within this context of the quantum mechanical framework. Again the two photons will have in their complex wavefunction information on electric and magnetic fields, but the classical picture does not work at this level.
