Final answer:
The continuity of electric field lines and the condition that ∇⋅E=0 in free space relate to the principles of Gauss's law, which states that the net electric flux through a closed surface is proportional to the charge enclosed. In free space without charges, this implies continuity in the electric field, as any field line entering must also leave the enclosed space.
Step-by-step explanation:
The continuity of electric field lines generated by a point charge and the condition that ∇⋅E=0 in free space are consequences of the physical laws described by Gauss's law. The law states that the electric flux through a closed surface is proportional to the charge enclosed by the surface. In free space, where there are no charges, the net flux through any closed surface is zero. This implies that the electric field is continuous because any field line entering the surface must also leave, ensuring there are no discontinuities. The premise of the electric field being a continuous, immaterial substance that fills all of space around a charge-bearing object, to infinite in all directions, is fundamental. It is by this medium that electric forces are transmitted. If the electric field had discontinuities, charges in the vicinity would be affected in an abrupt and unphysical manner, which contradicts observations and the continuous nature of physical laws. In essence, the absence of charges in a given volume of free space leads to ∇⋅E equating to zero, thereby ensuring the continuity of the electric field generated by charges outside that volume. When charges do exist, they alter the space around them and the field lines emanate or terminate at these charges, but still, do so in a continuous fashion.