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A Look at Charging and Discharging of a Capacitor and Maxwell’sEquations
Aniruddh Singh
Department. of Applied Sciences, AjayKumarGargEngineeringCollege, P.O. Adhyatmic Nagar, Ghaziabad.201009
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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 discharging, Maxwell’s equations, Displacement current
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 beforehand 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:
=
The usual way of introducing this law is by picking up an example of a RC circuit.
Fig.1
C
As shown infigure 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 ofB 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 Balong 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:
This gives according to Faraday’s law:
And according to Ampere’s law with Maxwell’s correction:
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 x-direction. 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:
Under this the equations for the scalar and vector potentials derived from Maxwell’ eq. become:
and
.
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.
.
For a given charge distribution (constant surface charge density ±σ on the two plates) this gives the usual solution such that:
between the plates and zero elsewhere.
A on the other hand is given by:
.
where = || and 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 inside the wire and is equal to , which is calculated inside the parallel plates of the capacitor. Interestingly though is equal to the conduction current Ic , the displacement current is not as in non- static cases. This fact is overlooked whenever the example of charging or discharging of capacitors is picked up while introducing
Maxwell equations. Let us now solve our model problem of a capacitative circuit with
conducting wires of infinite length as shown in the figure 2.
Fig.2
Here, the current is in the x direction. As J andare 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:
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:
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 JDat different locations in between the capacitor plateswill 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 as it will turn out to benot 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 thingpassing through S2 unless one knew about the existence of displacement current beforehand, even before presenting logical argument in favor of it. We can maintain the train of 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 DelhiUniversityand 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, AjayKumarGargEngineeringCollege, Ghaziabad.
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