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Maxwell's Equations

\[\vec{\nabla}\cdot \vec{E} = \frac{\rho }{\epsilon_{0}}     ........(1)\]

\[\vec{\nabla}\cdot \vec{B} = 0........(2)\]

\[\vec{\nabla}\times \vec{E} = -\frac{\partial \vec{B}}{\partial t}..........(3)\]

\[\vec{\nabla}\times \vec{B} = \mu_{0}\vec{J}+\mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}..........(4)\]

This is a traditional way of writing Maxwell's equation, however a bit misleading. Because they reinforce the notion that electric fields can be produced either by charges(ρ) or by changing magnetic fields(∂B/∂t), and magnetic fields can be produced either by currents (J) or by changing electric fields (∂E/∂t). Actually, this is misleading, because ∂B/∂t and ∂E/∂t are themselves due to charges and currents. It will better if we write,

 \[\vec{\nabla}\cdot \vec{E} = \frac{\rho }{\epsilon_{0}}     ........(1)\]

\[\vec{\nabla}\cdot \vec{B} = 0........(2)\]

\[\vec{\nabla}\times \vec{E} +\frac{\partial \vec{B}}{\partial t} = 0..........(3)\]


\[\vec{\nabla}\times \vec{B} -\mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}= \mu_{0}\vec{J}..........(4)\]

Continuity Equation:

Taking divergence of equation (4) ,
\[\vec{\nabla} \cdot (\vec{\nabla}\times \vec{B}) = \mu_{0}(\vec{\nabla} \cdot \vec{J})+\mu_{0}\epsilon_{0}\frac{\partial(\vec{\nabla} \cdot \vec{E})}{\partial t}\]

\[\Rightarrow 0 = \mu_{0}(\vec{\nabla} \cdot \vec{J})+\mu_{0}\frac{\partial\rho}{\partial t}      .............[div(curl) = 0]\]

\[\Rightarrow  \mu_{0}(\vec{\nabla} \cdot \vec{J}) =-\mu_{0}\frac{\partial\rho}{\partial t}\]

\[\Rightarrow  \vec{\nabla} \cdot \vec{J} =-\frac{\partial\rho}{\partial t}\]


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