Eight identical spherical drops of mercury charged to \(12 \mathrm{~V}\) above carth potential are made to coalesce into a single spherical drop. What is the new potential and how has the internal electric energy of the system changed?

Electrostatics is the branch of physics that deals with the study of stationary or slow-moving electric charges. It encompasses the forces, fields, and potentials arising from static charges. The basics of electrostatics are critical in understanding everyday phenomena and are foundational for technological applications such as capacitors and electrostatic precipitators.

In the context of a charged sphere, a key principle of electrostatics is at play: the electric charge on the surface of a conductor will redistribute itself so as to produce a constant potential over the entire surface. When we speak about the electric potential of a sphere, we're referring to the work needed to bring a positive charge from infinity to the surface of the sphere against the electrostatic field created by the sphere's charge.

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative. Like charges repel each other, while opposite charges attract. The electric charge of an object can be quantified and is measured in coulombs (C).

When it comes to multiple objects, such as charged drops of a liquid, the total charge is the sum of the individual charges. This property is used in electrostatic calculations to determine how the charge affects the electric potential and energy of the system. Conservation of charge is a key principle, meaning that in any electrical process, charge is neither created nor destroyed but may be transferred from one form to another.

Electric energy, in electrostatic systems, refers to the potential energy stored due to the positions of charged objects in an electric field. This energy can be calculated for a system of charges and is especially relevant in configurations like a collection of charged spheres. The energy of an isolated charged sphere is given by the formula \( U = \frac{Q^2}{8\pi \epsilon_0 r} \), where \( Q \) is the charge, \( r \) is the radius of the sphere, and \( \epsilon_0 \) is the vacuum permittivity.

Understanding how the electric energy changes when multiple charged objects interact, such as in the coalescence of charged drops, helps to reveal insights into the work required to bring the charges together and the influence of electric fields on the energy of the system.

The coalescence of charged drops is a process where smaller droplets merge to form a larger droplet. This phenomenon is not only fascinating in fluid dynamics but also plays a significant role in understanding electrostatic principles. When charged drops coalesce, the volume of the resulting drop is equal to the sum of the individual volumes.

However, the relationship between the radius and charge influences the electric potential of the coalesced drop. If several small charged droplets merge together, due to the conservation of charge, the total charge remains the same but is now distributed over a larger surface. The potential of a sphere being related to its radius means that when smaller charged drops coalesce into a larger one, the electric potential of the resulting drop increases, assuming the total charge remains conserved.

The principle of conservation of charge is a cornerstone in the study of electrostatics. It states that the total electric charge in an isolated system remains constant irrespective of any changes occurring within the system. This fundamental principle ensures that during the coalescence of charged spheres or drops, the total charge before the process is equal to the total charge after the process.

However, while the charge remains the same, other properties, such as potential and electric energy, may change according to the configuration of the system. In the example of merging charged drops, the overall volume and surface area change, thereby affecting the potential and electric energy, but the total charge — as dictated by conservation of charge — stays constant. This is a crucial concept in analyzing the consequences of combining charged objects.