## Equation of continuity for current density

We know the equation of continuity is

$\overrightarrow{\nabla}. \overrightarrow{J}+ \frac{\partial \rho}{\partial t}=0 \qquad(1)$

According to Maxwell's first differential equation

$\overrightarrow{\nabla}. \overrightarrow{D}=\rho \qquad(2)$

From equation $(1)$ and equation$(2)$

$\overrightarrow{\nabla}. \overrightarrow{J}+ \frac{\partial }{\partial t}(\overrightarrow{\nabla}. \overrightarrow{D})=0$

 $\overrightarrow{\nabla}. (\overrightarrow{J}+ \frac{\partial \overrightarrow{D} }{\partial t}) =0$

Where the term →
$(\overrightarrow{J}+ \frac{\partial \overrightarrow{D} }{\partial t})$ → solenoidal vector and it is also regarded as total current density for time varying electric field.
$D$ → The displacement vector
$\frac{\partial \overrightarrow{D} }{\partial t}$ → Displacement current density

The above equation is known as the "Equation of continuity for current density".