Subject :- Applied Physics 2.

Topic :- Laser.

Difficulty :- Low.

Consider a small volume element ∆v as shown in figure below located inside a conducting medium. The current density has the direction of current flow. If there is no source or sink of charge inside the volume ∆v (=∆x∆y∆z ), the current is steady and continuous and so is current density.\

We know $\vec{\nabla}.\vec{J}=0$ or $\oint_S\vec{J}.\vec{ds}=0$

All three components and their variation are shown in the diagram.

Here $\frac{\partial{J_x}}{\partial{x}},\frac{\partial{J_y}}{\partial{y}},\frac{\partial{J_z}}{\partial{z}}$ are the rate of change of $J_x,\ J_y$ and $J_z$ in x, y, and z directions respectively.

If the current is not steady, the difference between the current flowing into the volume and that flowing out of the volume must equal the rate of change of electric charge inside the volume.

A net flow of current out of the volume (positive current flow) must be equal to the negative rate of change of charge with time (rate of decrease of charge) within the volume.

This is expressed by the continuity equation,

$\vec{\nabla}.\vec{J}=-\frac{\partial{\rho_v}}{\partial{t}}.\ \rho_v$ is the volume charge density

This is the general relation between current density $\vec{J}$and the volume charge density $\rho_v$ at a point.