Deriving W_{int} from (1.57) to (1.58) in Jackson Classical Electrodynamics helps to understand the relationship between the electric and magnetic fields in an electromagnetic field and how they contribute to the energy density of the system. It also allows for the calculation of the total energy stored in the field.
The energy density of an electromagnetic field is directly related to W_{int}, as it is a measure of the amount of energy stored in the field per unit volume. W_{int} is the integration of the energy density over the entire volume of the field.
The mathematical expression for W_{int} is given by the integral of 1/2(E^2 + B^2) over the volume of the electromagnetic field. This represents the sum of the electric and magnetic energy densities.
W_{int} is related to the Poynting vector, as it is the energy flow per unit area of the field. The Poynting vector is defined as the cross product of the electric and magnetic fields, and it represents the direction and magnitude of energy flow in the field. W_{int} is the integration of the Poynting vector over the entire volume of the field.
Understanding W_{int} can have various real-world applications, such as in the design and optimization of electromagnetic devices, calculating the energy efficiency of electrical systems, and studying the behavior of electromagnetic fields in different materials. It can also aid in the development of new technologies, such as wireless power transfer and electromagnetic shielding.