Write Maxwell's equations for electric and magnetic fields in a charge free dielectric medium. Using these equations, obtain the wave equations for the electric and magnetic fields in the dielectric medium. Will the speed of electromagnetic waves in the dielectric medium be same as that in the free space? Justify your answer.

Maxwell’s equations govern the behavior of electric and magnetic fields and form the foundation of classical electromagnetism and optics. In a charge-free, non-conducting dielectric medium, where free charges and currents are absent, Maxwell’s equations are expressed in their differential forms as follows:

1. Gauss's Law for Electricity:
   $\nabla \cdot \mathbf{E} = 0$

2. Gauss's Law for Magnetism:
   $\nabla \cdot \mathbf{B} = 0$

3. Faraday’s Law of Electromagnetic Induction:
   $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

4. Ampère-Maxwell Law:
   $\nabla \times \mathbf{B} = \mu \epsilon \frac{\partial \mathbf{E}}{\partial t}$

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