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In classical electrodynamics, the behavior of electromagnetic fields in vacuum is governed by Maxwell's equations. In free space, where there are no charges or currents, these equations reduce to: [ \nabla \cdot \mathbf{E} = 0, \quad \nabla \cdot \mathbf{B} = 0, ] [ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}. ] To derive the electromagnetic wave equation for the electric field (\mathbf{E}), we take the curl of Faraday’s law: [ \nabla \times (\nabla \times \mathbf{E}) = \nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t}\right) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}). ] Substituting Ampère’s law in vacuum (\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}), we obtain: [ \nabla \times (\nabla \times \mathbf{E}) = _______ ____ ________ ________ ______ _________ __________.
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