What are some common mistakes students make with Coulomb's law?
Unlike Newton's law of gravitation, Coulomb's force depends between the intervening medium of the charges. It reduces from its vacuum value when some material medium is present (It actually reduces by a factor of the dielectric constant of the medium) .
So, one should pay special attention if a dielectric is present in between the charges.
Another important issue is that, students many a times ignore the concept of point charges. We should always bear in mind that the sizes of the charges should be much smaller than their separation. If we for example encounter a charged rod 1 m long and wish to determine the force it exerts on a small charge 50 cm away on its perpendicular bisector, and we simply apply Coulomb's law, we are in trouble. We then have to consider the force an infinitesimally small charge element exerts on the test charges and resort to the process of integration.