Maxwell's Equations:
Maxwell's equations describe the fundamental principles of electricity and magnetism. They were formulated by James Clerk Maxwell in the 19th century and have become one of the cornerstones of classical physics. Maxwell's equations are a set of four differential equations that relate electric and magnetic fields to their sources, namely electric charges and currents.

1. Gauss's Law for Electric Fields:
This equation relates the electric flux through a closed surface to the total charge enclosed by that surface. It states that the electric flux is proportional to the total charge enclosed by the surface divided by the permittivity of the medium. Mathematically, it can be written as:

∮E · dA = (1/ε₀) ∫ρ dV

where E is the electric field, dA is an infinitesimal area element, ρ is the charge density, ε₀ is the permittivity of free space, and the integrals represent surface and volume integrals respectively.

2. Gauss's Law for Magnetic Fields:
This equation relates the magnetic flux through a closed surface to the total magnetic charge enclosed by that surface. It states that the magnetic flux is always zero, implying the absence of magnetic monopoles. Mathematically, it can be written as:

∮B · dA = 0

where B is the magnetic field and dA is an infinitesimal area element.

3. Faraday's Law of Electromagnetic Induction:
This equation describes how a changing magnetic field induces an electric field. It states that the induced electromotive force (emf) along a closed loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, it can be written as:

∮E · dl = -dΦ/dt

where E is the electric field, dl is an infinitesimal path element, Φ is the magnetic flux, and dt is the time interval.

4. Ampere's Law with Maxwell's Addition:
This equation describes the relationship between magnetic fields and electric currents. It states that the circulation of the magnetic field around a closed loop is equal to the sum of the electric current passing through the loop and the displacement current. Mathematically, it can be written as:

∮B · dl = μ₀(I + ε₀(dΦE/dt))

where B is the magnetic field, dl is an infinitesimal path element, I is the electric current, μ₀ is the permeability of free space, ε₀ is the permittivity of free space, dΦE/dt is the rate of change of electric flux, and the integrals represent line integrals.

Relation to Photoelectric Effect and Wave Theory of Light:
The photoelectric effect is the phenomenon where light incident on a material surface causes the emission of electrons. It cannot be explained solely by the wave theory of light but requires the concept of photons and the dual nature of radiation and matter.

Maxwell's equations, in conjunction with the wave theory of light, provide a comprehensive framework to understand the propagation and interaction of electromagnetic waves. The equations describe how electric and magnetic fields are generated, transmitted, and interact with matter.

The photoelectric effect can be explained by considering light as a stream of particles (photons), each carrying energy proportional to its frequency. When a photon interacts