Editing Displacement current (section)

====Mathematical formulation====
In a more mathematical vein, the same results can be obtained from the underlying differential equations. Consider for simplicity a non-magnetic medium where the [[Magnetic permeability#Relative permeability|relative magnetic permeability]] is unity, and the complication of [[Magnetization current#Magnetization current|magnetization current]] (bound current) is absent, so that <math>\mathbf{M} = 0</math> and {{nowrap|1=<math>\mathbf{J} = \mathbf{J}_\mathrm{f}</math>.}}
The current leaving a volume must equal the rate of decrease of charge in a volume. In differential form this [[Current density#Continuity equation|continuity equation]] becomes:

<math display=block>\nabla \cdot \mathbf{J}_\mathrm{f} = -\frac {\partial \rho_\mathrm{f}}{\partial t}\,,</math>

where the left side is the divergence of the free current density and the right side is the rate of decrease of the free charge density. However, [[Ampère's law]] in its original form states:

<math display=block>\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_\mathrm{f}\,,</math>

which implies that the divergence of the current term vanishes, contradicting the continuity equation. (Vanishing of the ''divergence'' is a result of the [[Vector calculus identities#Divergence of the curl|mathematical identity]] that states the divergence of a ''curl'' is always zero.) This conflict is removed by addition of the displacement current, as then:<ref name=Cloude>
{{cite book
 |first1=Raymond |last1=Bonnett
 |first2=Shane   |last2=Cloude
 |name-list-style=amp
 |year=1995
 |title=An Introduction to Electromagnetic Wave Propagation and Antennas
 |page=16
 |publisher=Taylor & Francis
 |isbn=978-1-85728-241-2
 |url=https://books.google.com/books?id=gME9zlyG304C&pg=PA16
 |via=Google Books
}}
</ref><ref name=Slater>
{{cite book
 |first1=J.C. |last1=Slater
 |first2=N.H. |last2=Frank
 |name-list-style=amp
 |year=1969   |orig-year=1947
 |title=Electromagnetism   |edition=reprint   |page=84
 |publisher=Courier Dover Publications
 |isbn=978-0-486-62263-7
 |url=https://books.google.com/books?id=GYsphnFwUuUC&pg=PA83
 |via=Google Books
}}
</ref>

<math display=block>\nabla \times \mathbf{B} = \mu_0 \left(\mathbf{J} + \varepsilon_0 \frac{ \partial \mathbf{E} }{ \partial t }\right) = \mu_0 \left( \mathbf{J}_\mathrm{f}  + \frac{\partial \mathbf{D}}{\partial t }\right)\,,</math>

and

<math display=block>\nabla \cdot \left( \nabla  \times \mathbf{B} \right) = 0 = \mu_0 \left( \nabla \cdot \mathbf{J}_\mathrm{f} +\frac {\partial }{\partial t} \nabla \cdot \mathbf{D} \right)\,,</math>

which is in agreement with the continuity equation because of [[Gauss's law]]:

<math display=block>\nabla \cdot \mathbf{D} = \rho_\mathrm{f}\,.</math>