Significance of div D and potential

Physical significance of div D

∇·D = ρ_v is Gauss's law in point form. The divergence of D at a point is the net outward electric flux per unit volume leaving an infinitesimal volume around that point:

∇·D = lim(Δv→0) (∮ D·dS)/Δv = ρ_v

• ∇·D > 0 at a point: a source (positive charge) is present, and flux lines diverge from it.
• ∇·D < 0: a sink (negative charge), flux lines converge.
• ∇·D = 0: no free charge there, so flux lines pass through without beginning or ending.

So div D tells how much free volume charge is located at each point. In Cartesian form ∇·D = ∂D_x/∂x + ∂D_y/∂y + ∂D_z/∂z.

Importance of potential in electrostatics

1. Scalar quantity: V is a scalar, so it is far easier to find than the vector E. Add scalars from many charges, then find E = −∇V.
2. Energy meaning: V is the work per unit charge in bringing a charge from a reference point (infinity or ground); the potential difference gives the work done on a charge, W = QΔV.
3. Conservative field: ∮E·dl = 0, so V depends only on position, and equipotential surfaces are always perpendicular to E.
4. Boundary value problems: Poisson's equation ∇²V = −ρ_v/ε and Laplace's equation ∇²V = 0 let us find fields, charge and capacitance from boundary potentials.
5. Practical: Circuit quantities such as voltage, capacitance (C = Q/V) and stored energy (W = ½CV²) are defined through potential.