Gauss's Law in differential form is a mathematical equation that relates the electric field at a point to the charge density at that point. It is a more general version of Gauss's Law in integral form, which relates the total electric flux through a closed surface to the enclosed charge.
Gauss's Law in differential form is derived from Maxwell's equations, specifically the divergence theorem. It states that the divergence of the electric field is equal to the charge density divided by the permittivity of free space.
Gauss's Law in differential form is significant because it allows us to calculate the electric field at any point in space, given the charge distribution. This is extremely useful in solving many electrostatic problems in physics and engineering.
The main difference between the two forms of Gauss's Law is that the differential form is used to calculate the electric field at a particular point, while the integral form is used to calculate the total electric flux through a closed surface. The integral form is also more general, as it applies to both electrostatic and time-varying electric fields.
Gauss's Law in differential form is most useful in situations where the electric field needs to be calculated at a specific point, such as at the location of a charged particle or in the vicinity of a complex charge distribution. It is also useful in solving problems with symmetrical charge distributions, as it allows for easier calculation of the electric field.