Download A Look at Discharging of a Capacitor and Maxwell Equations

A Look at Charging and Discharging of a Capacitor and Maxwell’s Equations
Aniruddh Singh
Department. of Applied Sciences, Ajay Kumar Garg Engineering College, P.O. Adhyatmic Nagar, Ghaziabad. 201009
[email protected]
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract --Traditional way of introducing the Maxwell’s
correction to Ampere’s circuital law in most undergraduate texts
is by analyzing a RC circuit. The capacitor is allowed to
discharge through a resistor and the current is measured. Then
one invokes the classic argument where the line integral of the
magnetic field is calculated along a closed curve which binds two
different surfaces, one through which the current carrying wire
passes and the other balloon shaped surface which escapes the
resistive wire and passes through the gap between the capacitor
plates. Then with the help of Stokes theorem one introduces the
displacement current between the gap and argues that it is equal
to the conduction current and proves it to be so by explicit
calculation. In this paper we show that the above usual argument
is inconsistent and the conduction current is not exactly equal to
the displacement current between the plates. It is also shown that
the argument reduces to tautology when logical gap is accounted
for.
Keywords:
Capacitor
Displacement current
discharging,
Maxwell’s
equations,
I. INTRODUCTION
It is well known that Maxwell provided a complete
description of Electromagnetism with the help of four
equations known by his name. Almost all the four equations
were known before hand except the Ampere’s law which was
incomplete. Maxwell fixed the Ampere’s law by introducing
an extra term known as the Displacement current density
ε0dE/dt. With Maxwell’s correction the Ampere’s law reads:


 
  B = 0 J   0 0E / t
The usual way of introducing this law is by picking up an
example of a RC circuit.
As shown in figure 1, the two plates of a parallel plate
capacitor are connected end to end by a resistive wire. The
charged capacitor starts discharging by transferring charge
from positive plate to the negative plate. In this process a
current flows in the wire. This current is time varying as the
driving force which is the capacitor voltage is dependent on
time. The line integral of the field B is calculated along the
curve C. This according to Stokes theorem is equal to the
surface integral of curl of B calculated on any surface bound
by C. As shown in the figure the two surfaces S1 and S2 are
bound by the same closed curve C. According to Ampere’s
circuital law the line integral of B along curve C is equal to µ0
times I, the total current passing through any surface bound by
C. As the current present in the wire is passing through
surface S1 but not through S2, the line integral is µ0I in one
case and zero in another. To account for this theoretical
anomaly Maxwell introduced an extra term in the Ampere’s
law. He called it the Displacement current density which was
present in the space between the capacitor plates. The current
associated with this density was shown to be equal to the total
conduction current present in the wire. Thus the two surfaces
S1 and S2 gave the same result for the line integral.
The above argument does not involve the time varying fields
produced by accelerating charges or time varying currents.
Indeed the current in the wire is time varying and depending
on the values of capacitance and resistance the variation of
current with time can be very high. Thus equating the
displacement current between the plates with the conduction
current in the wire by using simple arguments will be
erroneous. Looking from another angle the displacement
current is directed perpendicular to the plates. Taking this
direction as x-axis one gets in between the plates:
 
 E  0
This gives according to Faraday’s law:

B / t  0
And according to Ampere’s law with Maxwell’s correction :
Fig.1
 

  B   0  0 E / t



  CB / t   0  0  2 E / t 2  0
49
The above contradiction can only be resolved if we consider
e-m waves being produced by time varying current densities
so that curl of E is no where zero. These e-m wave fields will
have there own displacement currents and one cannot
conclude from simplistic theoretical considerations that
conduction current in the wire is equal to displacement current
between the plates. We consider a model to account for this
logical gap in what follows.
II. MODEL
Consider a parallel plate capacitor with its two plates
connected to two infinitely long wires which extend in the xdirection. Let there be some mechanism by which a sinusoidal
current is driven in the wires. Thus the current in the wire is of
the type : I0sin(ωt). The surface charge density on the plates
also follows the same time dependence: ±σ0sin(ωt). As the
current is time varying it will produce e-m waves. Let us pick
the Coulomb gauge to solve the time varying problem.
According to the Coulomb gauge:


where  = | r  r  | and t r is the retarded time. It is
important to note here that the above equation involves the
Maxwell term.
To solve the above equation for the specific charge and
current configuration described above we note that the current

 
I0sin(ωt) is equal to J .da

inside the wire and is equal to


 0 V / t.da , which is calculated inside the parallel
plates
of
the
capacitor.
Interestingly
though





V / t.da is equal to the conduction current Ic , the
0





E
/ t.da
displacement
current
is
not
as
0




E  V  A / t in non- static cases. This fact is
overlooked whenever the example of charging or discharging
of capacitors is picked up while introducing
 
.A  0
Maxwell equations. Let us now solve our model problem of a
capacitative circuit with
Under this the equations for the scalar and vector potentials
derived from Maxwell’ eq. become:
conducting wires of infinite length as shown in the figure 2.
 2V    /  0
and




 2 A   0 0  2 A / t 2  0 J   0 0(V / t ) .
The first equation for scalar potential is similar to the one
encountered in electrostatics. The solution is straightforward
and the scalar potential is conveyed instantaneously.
V  1 / 40  (  / )d .
For a given charge distribution (constant surface charge
density ±σ on the two plates) this gives the usual solution such
that:

V    /  0 x̂ between the plates and zero elsewhere.
A on the other hand is given by:



A(r , t )   0 / 4  [( J (r , t r )   0 V (r , t r ) / t ) / ]d
.
Fig.2

Here, the current is in the x direction. As J and V / t are
both in the x direction A will be in x direction too. In the
radiation zone r as well as  will be very large, much larger
than the dimensions of the capacitor and the scalar potential V
will be zero in this zone. The vector potential A on the other
hand will be given by:



A(r , t )   0 / 4  ( I 0 sin  (t   / c) xdx) / 

Here we neglect the surface currents present on the capacitor
plates as their contribution will be negligible in the radiation
zone due to mutual cancellation because of parallel plate
geometry and their symmetric nature around the x-axis.
To calculate the E and B field in the radiation zone we rewrite
the above integral in the following form:
50

of displacement current beforehand, even before presenting


A(r , t )   0 / 4  [( I 0 sin  (t  r 2  x 2 / c) / r 2  x 2 ]xdx logical argument in favor of it. We can maintain the train of

One can calculate the above integral and from it the electric
and magnetic fields. The solution shows that mutually
perpendicular E and B field exist in the radiation zone and the
ratio of the magnitudes of the two fields is equal to the speed
of light, an essential condition for the existence of e-m waves.
The displacement current due to e-m waves is equal for the
two surfaces for this highly symmetric example as A is
directed and constant along the x-axis. This can be generalized
to any arbitrary case by noting that Divergence of A is zero
everywhere. Therefore the radiation fields do not diverge for
an arbitrary RC configuration and will give zero flux
contribution for any closed surface thus giving equal
contributions to any partition (like S1-S2) of the closed
surface (fig 1). Although we note that the validity of the
above statements for our infinitely long circuit (fig.2) in
regions close to the circuit needs to be checked as A will be in
x- direction but not constant. This is because the differences in
the retarded time present in the integral originating from JD at
different locations in between the capacitor plates will be
significant and will depend on the dimensions of the capacitor.
Therefore one will not be able to approximate the volume
integral to a one dimensional integral as we have done to
calculate A.
 
This will have an effect on . A as it will turn out to be not
equal to zero and this will have implications on the Gauge
freedom. Similarly the surface currents will make a small but
significant contribution to A inside the plates.
Tragically the whole improved argument discussed above
although depicts the working of Maxwell equation and how it
satisfies Stokes theorem in the radiation zone approximation,
reduces to tautology as Maxwell term is already required for
the existence of e-m waves.
Discussion.
From the above discussion it is clear that one cannot by
applying Stokes theorem to the two surfaces bound by curve C
(fig.1) equate the conduction current in the wire to the
displacement current as the total current passing through
surface S1 is the sum of the conduction current due to electron
flow in the wire and displacement current produced by the e-m
waves in the vicinity of the conducting wire. Only this sum
can be equated with the total displacement current passing
through the surface S2. In the hindsight this sum could never
be calculated for S1 and equated with some other calculable
thing passing through S2 unless one knew about the existence
logic by including the currents produced by e-m waves. But
that already requires that Maxwell term be known as the e-m
wave equations are derived from it. Therefore the usual
argument for introducing the displacement current as a
correction to the Ampere’s law is not logically secure as it is
hinged upon the IC=ID equality.
IV. CONCLUSION
The discussion above shows a gap in the usual logical
argument put forward for the Maxwell correction. This
argument is made to support the more compelling argument
for the inclusion of Maxwell correction, namely the savior of
the continuity equation. In this context we would like to add
that that saving of the continuity equation does not
conclusively ensures the existence of the displacement current
but is merely suggestive. It requires only the Divergence of
the field to be equal to ρ/Є0 and puts no restriction on the Curl.
Therefore the continuity equation can be saved in many
alternative ways.
It has been shown in this article that the usual pedagogic
arguments in defense of the Maxwell term are highly
suggestive but not logically foolproof and have also shown the
way to make one of them so. Also we have shown that the
process of making it logically secure reduces the argument to
mere tautology. Therefore like all other fundamental laws of
Physics one cannot derive the Maxwell term from first
principles. As a byproduct of this analysis we have also
suggested that Gauge symmetry is an approximate symmetry
and becomes exact only in the limit of c→infinity.
V. REFERENCE
[1]. “Introduction to Electrodynamics”, by D.J.Griffiths pp - 304-306.
Dr. Aniruddh Singh is Ph.D in Theoretical
Nuclear Physics from Jamia Millia Islamia, New
Delhi.
He obtained BSc Hons in Physics from Delhi
University and MSc Physics from IIT Kanpur.
His PhD thesis is in the field of Variational
Monte Carlo methods as applied to light nuclei
and hypernuclei. He has 3.5 years of teaching
experience at various universities and about 1.5
years research experience in industry.
Currently, he is an assistant professor with the Department of Applied
Sciences, Ajay Kumar Garg Engineering College, Ghaziabad.
51