Continuity equation for time varying field?

Answer 1

The Continuity Equation for a time varying field Eris:

dEr/cdt = Del.Ev where Ev is the vector field associated with the real time varying field.

Er + Ev =E, constitute a quaternion field.

Del.Ev is the Divergence of the vector field.

The Continuity Equation is a statement that the time variation of the real field is equal to the Divergence of the vector field. or more succinctly, the quaternion field E=Er + Ev is Real invariant.

The Vector part of the variation is 0= dEv/cdt + Del Er + DelxEv , this is Vector Invariance of E. This is not the Continuioty Equatin but the Induction Equation. Together they represent the Invariance of the quaternion field E=Er + Ev.

Because quaternions are not taught in schools yet, few realize the relationship between Continuity and Induction, they are the Real and Vector parts of Invariance!

Answer 2

The continuity equation for a time-varying field relates the divergence of the field with the rate of change of field strength at any given point. Mathematically, it is expressed as ∇⋅E = -∂ρ/∂t, where ∇ is the divergence operator, E is the field, ρ is the charge density, and ∂/∂t represents the partial derivative with respect to time. This equation ensures that the field and charge distributions are consistent over time, in accordance with the principle of charge conservation.