PPT No. 35
Ampere’s Law and the Conservation of Charge
Ampere’s original law in differential form states that
the curl of Magnetic field equals Current density .
Symbolically
B is the magnetic field,
j is the current densityIt represents the flow of electric charge.
Ampere’s Law and the Conservation of Charge
Consider the divergence of both sides of the equation.
The divergence of the curl of any vector field is identically zero.
LHS = 0
LHS =
=> RHS is zero
Ampere’s Law and the Conservation of Charge
However, According to the Continuity equation
≠ 0
Thus there is inconsistency between original circuital law
(as stated by Ampère) & the law of conservation of charge.
Ampère's law seems to be applicable only
in limited situations where charge density is constant.
Maxwell concluded that this equation was incomplete.
He did the needed amendment to the RHS.
Maxwell’s Modification in Ampere’s Law
According to Coulomb’s law
Taking the partial derivative with respect to time and
using commutative property of partial differentiation
However, According to Continuity equation
Combining above two equations =>
Maxwell’s Modification in Ampere’s Law
Since the combination of
charge conservation and Coulomb’s law implies that
the divergence of (j + ∂E/∂t) is zero,
the term on RHS (vector j) in equation 1
should be replaced by the vector j + ∂E/∂t.
The extra term in Ampere’s law gives
the divergenceless field as required.
Maxwell’s Modification in Ampere’s Law
In material media,
Since
But
(According to the Continuity equation)
Gauss's Law
converted to differential form using Divergence theorem is
=>
Maxwell’s Modification in Ampere’s Law
The Continuity equation is rewritten as
or
where
D = Displacement vector and
Maxwell’s Modification in Ampere’s Law
The additional term
or ε0∂E/∂t
by which Maxwell augmented the equation
is called as the displacement current
as it involves
the rate of change of
the dielectric displacement vector D.
Maxwell’s Modification in Ampere’s Law
The Statement of corrected form of Ampère's circuital law
Differential form
Integral form
The integral of the magnetic field B
around a closed circuit path ∂S is equal to
the sum of terms due to current density J
through any surface spanning the loop and
the displacement current ε0 ∂E / ∂t through the surface.
Maxwell’s Argument for the Displacement Current
Maxwell postulated the concept of displacement current in
part III of his paper 'On Physical Lines of Force' (1861).
According to Maxwell's concept of the vacuum, the aether,
had dielectric properties, and there was motion of charges in
the vacuum similar to that in the dielectric material
which gave rise to the displacement current.
This explanation caused tremendous confusion and
misunderstanding among physics community.
Maxwell’s Argument for the Displacement Current
Presently, his concept of aether is discarded and
Maxwell’s explanation is not accepted.
Nonetheless, the equation modified by him by
adding the term of displacement current is
one of the set of four of Maxwell’s equations
Explanation using Charging of Capacitor
The current density j at a given location
does not actually represent the total current flow
at that location
(even though that is essentially its definition).
The process of linear polarization of a dielectric mediumA dielectric medium can be considered to consist of
atoms having pairs of positive and negative charges.
Explanation using Charging of Capacitor
In the presence of electric field E,
these charges are pulled in opposite directions, stretching
the links between them until they achieve equilibrium.
If the strength of the field is increased,
the charges are pulled further apart,
so during periods when the electric field is changing
there is movement of the electric charge elements
of the dielectric medium.
Explanation using Charging of Capacitor
These moving charges contribute to
current flow in the direction of the applied electric field.
This current must produce magnetic field,
and hence should be added to the conduction current.
This movement of charge is proportional to ε0∂E/∂t.
It is called as the displacement current.
This is the mechanism of creation
a magnetic field due to a time-varying electric field.
Importance of the Displacement Current
Maxwell utilized the correction
to Ampère's circuital law in his paper (1864) entitled
“A Dynamical Theory of the Electromagnetic Field”.
The inclusion of the “displacement current”
in Ampere’s formula was the key contribution
by Maxwell in his electromagnetic field theory
Importance of the Displacement Current
By adding the term due to the displacement current
to the real current,
the Ampere’s law becomes complete and
self-consistent mathematically and physically.
It is one of the four
Maxwell’s field equations of electromagnetism.
Importance of the Displacement Current
The modification of Ampere’s equation
by adding the term of
Displacement current ε0∂E/∂t
leads to the existence of
transverse electromagnetic waves
propagating in a vacuum at the speed of lightIt is a result of tremendous importance.