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Maxwell's Equations

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Gauss's Law for Electricity

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∇⋅E=ρε0\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}
This equation implies that the electric field divergence at a point is proportional to the electric charge density at that point, which describes how charge can create an electric field.

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Gauss's Law for Magnetism

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∇⋅B=0\nabla \cdot \mathbf{B} = 0
This equation implies that magnetic field lines are continuous with no beginning or end, which indicates that there are no magnetic monopoles.

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Faraday's Law of Induction

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∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
This equation states that a time-varying magnetic field induces an electric field. This is the principle behind electrical generators and transformers.

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Ampere's Law with Maxwell's Addition

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∇×B=μ0J+μ0ε0∂E∂t\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}
This extended version of Ampère's Law shows that a magnetic field is generated by electric currents and also by changing electric fields, explaining the propagation of electromagnetic waves.

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