Word Count : 478

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In classical electrodynamics, the behavior of electromagnetic fields can be described in terms of scalar and vector potentials. The scalar potential (\phi(\mathbf{r}, t)) and vector potential (\mathbf{A}(\mathbf{r}, t)) are related to the electric field (\mathbf{E}) and the magnetic field (\mathbf{B}) by the relations: [ \mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}, \quad \mathbf{B} = \nabla \times \mathbf{A}. ] For a localized source such as a small oscillating electric dipole, the potentials at a point (\mathbf{r}) in space are expressed as retarded potentials to account for the finite speed of light (c): [ \phi(\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \int \frac{\rho(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} d^3\mathbf{r}', \quad \mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \frac{\mathbf{J}(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} d^3\mathbf{r}', ] where ____ ___ ______ _____ _______ ___ ________ ___ _______ __________.
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