Conservation Laws in Electro Magnetism
Overview
Conservation of Charge
{% Q = \int_v \rho(\vec{r}) dV %}
The current that exits the volume is given by {% \int_v \vec{J} \cdot \vec{dA} %},
so
{% \frac{dQ}{dt} = \int_v \vec{J} \cdot \vec{dA} %}
{% \int_v \frac{\partial{\rho}}{\partial{t}} dV = -\int_v \nabla \cdot \vec{J} dV %}
which leads to the continuity equation
{% \frac{\partial{\rho}}{\partial{t}} = - \nabla \cdot \vec{J} %}
