Why do the electrostatic field lines not form a closed loop?

The electrostatic field lines do not form closed loops in most situations because they represent the direction of the electric field vector at each point in space. The electric field arises from electric charges, which can be positive or negative.

According to Gauss's law, the electric field lines start at positive charges and end at negative charges. This means that if we have a positive charge, the electric field lines will emanate from it and extend outward indefinitely. Similarly, if we have a negative charge, the field lines will converge onto it. These field lines do not loop back to the charge because they represent the direction of the electric field at each point, and the field points away from positive charges and towards negative charges.

If the electric field lines were to form closed loops, it would imply that the field lines start and end at the same charge. This situation is not possible in the absence of other charges because it would violate the principle of charge conservation. In other words, the total electric flux leaving or entering a closed surface enclosing a charge must be proportional to the charge enclosed. If the field lines formed closed loops, the flux would be zero, which is inconsistent with the presence of a non-zero charge.

However, it's worth noting that there are situations in electromagnetism where the electric field lines can form closed loops. For example, in the presence of a changing magnetic field or in the context of electromagnetic waves, the electric field lines can form closed loops as they interact with magnetic fields. This behavior is described by Maxwell's equations and is beyond the scope of electrostatics.