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Divergence and physical significance

Short answer

Divergence of D equals volume charge density ρ_v. Positive divergence indicates a source (positive charge), negative a sink (negative charge). Zero divergence means no net charge, field is solenoidal. It's the point form of Gauss's law.

Define divergence and write down its physical significance as it applies to electric fields.

Definition: The divergence of a vector field A at a point is the net outward flux of A per unit volume as the volume shrinks to zero around that point.

∇ · A = lim (Δv → 0)  [ ∮_S A · dS ] / Δv

In Cartesian coordinates: ∇ · A = ∂A_x/∂x + ∂A_y/∂y + ∂A_z/∂z (a scalar).

Physical significance for electric fields

For the electric flux density D, Gauss's law in point form is

∇ · D = ρ_v
  • Divergence of D at a point equals the volume charge density there. Positive divergence means a source (positive charge): flux lines start there and spread outward. Negative divergence means a sink (negative charge): flux lines end there.
  • ∇ · D = 0 means no charge is present at the point: all the flux entering a tiny volume also leaves it. Such a field is called solenoidal.
  • It converts the integral Gauss law, ∮ D · dS = Q, into a local (point) law, using the divergence theorem ∮ D · dS = ∫ (∇ · D) dv.
  • For a point charge, ∇ · D = 0 everywhere except at the charge itself, where it is infinite (a delta function).

Similarly ∇ · B = 0 expresses that magnetic flux lines have no sources (no magnetic monopoles).

✓ Answer checked by Claude Opus 5.5 (high effort)

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