Maxwell’s equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields, as well as their interactions with matter. They form the cornerstone of classical electromagnetism and have wide-ranging applications in physics and engineering. Here are Maxwell’s equations in differential form, commonly written using vector calculus:

1. Gauss’s Law for Electricity (Gauss’s Law for Electric Fields):

This equation states that the divergence of the electric field (E) at any point in space is equal to the electric charge density (ρ) at that point divided by the permittivity of free space (ε0​).

2. Gauss’s Law for Magnetism (Gauss’s Law for Magnetic Fields):

This equation states that the divergence of the magnetic field (B) at any point in space is zero, indicating that there are no magnetic monopoles (isolated magnetic charges).

3. Faraday’s Law of Electromagnetic Induction:

This equation describes how a changing magnetic field induces an electric field (E) in the surrounding space. The curl of the electric field (∇×E) is equal to the negative rate of change of the magnetic field with respect to time.

4. Ampère’s Law with Maxwell’s Addition:
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This equation relates the curl of the magnetic field (∇×B) to the electric current density (J) and the rate of change of the electric field . It includes Maxwell’s addition, which accounts for the displacement current , where μ0​ is the permeability of free space.

These four equations describe the fundamental behavior of electric and magnetic fields in vacuum and in the presence of electric charges and currents. They are used to analyze and predict the behavior of electromagnetic phenomena, including the propagation of electromagnetic waves, the behavior of antennas, the operation of electrical circuits, and many other applications in physics and engineering.

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