Continuity equation, Displacement current

1. The continuity equation and Maxwell’s equations
Dimpal Boro
Tezpur University
November 4, 2022

2. Conservation law
In physics, a conservation law states that a particular measurable
property of an isolated physical system does not change as the
system evolves over time.

3. Global conservation
The total amount of some conserved quantity in the universe could
remain unchanged if an equal amount were to appear at one point
and simultaneously disappear from another point in the universe.
This is global conservation and it does not occur in nature.

4. Local conservation
A stronger form of conservation law requires that, for the amount
of a conserved quantity at a point to change, there must be a flow,
or flux of the quantity into or out of the point.
This is called a local conservation law and it occurs in nature.
Local conservation also implies global conservation; not vice versa.

5. Conservation of charge
In physics, charge conservation is the principle that the total
electric charge in an isolated system never changes.
The change in the amount of electric charge in any volume of
space is exactly equal to the amount of charge flowing into the
volume minus the amount of charge flowing out of the volume.

6. The continuity equation
The concepts of local charge conservation can be expressed
mathematically as
∂ρ
∂t
+ ∇ · ⃗
J = 0
This equation is called the continuity equation.

7. The Ampere’s law
The Ampere’s law relates the line integral of magnetic field around
a closed loop to the electric current passing through the loop.
I
⃗
B.dl = µ0Ienc
In differential form
⃗
∇ × ⃗
B = µ0
⃗
J
Figure: Magnetic
field around a wire
Figure: Electric current penetrating a surface enclosed by a
curve C

8. The enclosed electric current Ienc
Figure: Currents enclosed (and not enclosed) by paths.
Figure: Membranes stretched across paths.
Figure: Alternative surfaces with boundaries C1, C2, and C3.

9. An important concept
The enclosed current is exactly the same irrespective of the shape
of the surface we choose, provided that the path of integration is a
boundary (edge) of that surface.

10. Modification by Maxwell
The Ampere’s law is I
⃗
B.dl = µ0Ienc
In differential form
⃗
∇ × ⃗
B = µ0
⃗
J
Maxwell and his contemporaries realized that the Ampere’s law
applies only to steady electric currents. Maxwell modified the
Ampere’s law as following
⃗
∇ × ⃗
B = µ0
⃗
J + µ0ϵ0
∂ ⃗
E
∂t

11. The need for modification
The Ampere’s law is
⃗
∇ × ⃗
B = µ0
⃗
J
Taking divergence on both side
⃗
∇.(⃗
∇ × ⃗
B) = µ0
⃗
∇. ⃗
J
In steady current situation there is no inconsistency but if the
current is not steady:
⃗
∇. ⃗
J ̸= 0
Ampere’s law cannot be applied in the non-steady current
situation.

12. The need for modification
Figure: Charging capacitor.

13. The need for modification
Figure: Surface to determine the enclosed current

14. The need for modification
Figure: Alternative surfaces for determining enclosed current.

15. Maxwell’s modification
Maxwell’s modification:
⃗
∇ × ⃗
B = µ0
⃗
J + µ0ϵ0
∂ ⃗
E
∂t
Taking divergence on both side
⃗
∇.(⃗
∇ × ⃗
B) = µ0
⃗
∇. ⃗
J + µ0ϵ0
∂(⃗
∇.⃗
E)
∂t
= µ0
⃗
∇. ⃗
J + µ0
∂ρ
∂t
= µ0
⃗
∇. ⃗
J + µ0
∂ρ
∂t

16. Maxwell’s modification
From continuity equation
⃗
∇. ⃗
J = −
∂ρ
∂t
⃗
∇.(⃗
∇ × ⃗
B) == µ0
⃗
∇. ⃗
J + µ0(−⃗
∇. ⃗
J) = 0
After modification by Maxwell the new equation, is consistent in
all situations: Steady and Non steady current.

17. Displacement current
The term ϵ0
∂ ⃗
E
∂t is called the displacement current density.
⃗
JD = ϵ0
∂ ⃗
E
∂t

18. Fixing the capacitor problem
The new equation fixes the capacitor problem as following:
Figure: The capacitor paradox
⃗
∇ × ⃗
B = µ0
⃗
J + µ0ϵ0
∂ ⃗
E
∂t
I
⃗
B.dl = µ0Ienc + µ0ϵ0
Z
∂ ⃗
E
∂t
!
.da

19. Fixing the capacitor problem
E =
σ
ϵ0
=
Q
Aϵ0
∂E
∂t
=
I
Aϵ0
I
⃗
B.dl = µ0Ienc + µ0ϵ0
Z
∂ ⃗
E
∂t
!
.da

20. Maxwell’s equations
Gauss’s law
∇ · E =
ρ
ε0
No name
∇ · B = 0
Faraday’s law
∇ × E = −
∂B
∂t
Ampere’s law
∇ × B = µ0
J + ε0
∂E
∂t

21. THE END.