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To derive the wave equation for the x-component of the electric field vector in free space, we start with Maxwell’s equations in vacuum, where there are no charges ((\rho = 0)) and no currents ((\mathbf{J} = 0)): 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 induction: [ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} ] 4. Ampère-Maxwell law: [ \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} ] Here, (\mathbf{E}) and (\mathbf{B}) are the electric and magnetic fields, (\epsilon_0) is the permittivity of free space, ______ __________ ____ ___ ____ ______ ___ _____.
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