Conservation of Current is universal and exact independent of the polarization properties of matter, from inside atoms to between stars.

1. Conservation of Current is
Universal and Exact
in five slides
Bob Eisenberg
January 11, 2018
Department of Applied Mathematics
Illinois Institute of Technology Chicago
and
Department of Physiology and Biophysics
Rush University, Chicago

2. 2
Maxwell’s Magnetism
Current is Conserved
PERFECTLY
𝐝𝐝𝐝𝐝𝐝𝐝 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 ≡ 0 is an identity
𝐝𝐝𝐝𝐝𝐝𝐝 𝐉𝐉 𝑥𝑥, 𝑡𝑡 + 𝜀𝜀0
𝜕𝜕𝐄𝐄 𝑥𝑥, 𝑡𝑡
𝜕𝜕𝑡𝑡
= 𝟎𝟎
𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 ⁄𝐁𝐁 𝑥𝑥, 𝑡𝑡 𝜇𝜇0 = 𝐉𝐉 𝑥𝑥, 𝑡𝑡 + 𝜀𝜀0
𝜕𝜕𝐄𝐄 𝑥𝑥, 𝑡𝑡
𝜕𝜕𝜕𝜕
Current
Current

3. ‘Charge’ has
Very different Physics
in different systems
but Continuity of Current is
Exact
No matter what carries the current!
*
i E t=∂ ∂ =∂ ∂i D t
D = permittivity × E
Ag AgClAg AgCl

4. Conservation of Current is Exact and Universal
even though
Physics of Charge Flow
Varies Profoundly
𝐝𝐝𝐝𝐝𝐝𝐝 𝐉𝐉 𝑥𝑥, 𝑡𝑡 + 𝜀𝜀0
𝜕𝜕𝐄𝐄 𝑥𝑥, 𝑡𝑡
𝜕𝜕𝑡𝑡
= 𝟎𝟎
Current

5. 5
Displacement Current
is
Different in Each Device
because
𝐄𝐄 𝑥𝑥, 𝑡𝑡 is Different in every Device
so the
TOTAL Current is exactly equal
at every time in every location and every device
Total Current = Displacement Current + Device Current
Device 1 Device 2
𝐝𝐝𝐝𝐝𝐝𝐝 𝐉𝐉 𝑥𝑥, 𝑡𝑡 + 𝜀𝜀0
𝜕𝜕𝐄𝐄 𝑥𝑥, 𝑡𝑡
𝜕𝜕𝑡𝑡
= 𝟎𝟎
Current

6. Page 6
Electric Field takes on the
Value that Conserves Current
𝐄𝐄 𝑥𝑥, 𝑡𝑡 = − 1
𝜀𝜀0
� 𝐉𝐉 𝑥𝑥, 𝑡𝑡 𝑑𝑑𝑑𝑑
Details and PROOF
including quantum mechanics at
https://arxiv.org/abs/1609.09175
Specifically,
E changes the displacement current ε0 ⁄𝜕𝜕𝑬𝑬 𝜕𝜕𝜕𝜕
So total current 𝐉𝐉 𝑥𝑥, 𝑡𝑡 + ε0 ⁄𝜕𝜕𝑬𝑬 𝜕𝜕𝑡𝑡 is always conserved
(that Maxwell called the polarization of the vacuum)