Discipline Course-I Semester-II

Maxwell's equation: "Introduction of Displacement Current" Discipline Course-I Semester-II Paper No: Electricity and Magnetism Lesson: Maxwell's equation: "Introduction of Displacement Current" Lesson Developer: Dr. Narmata Soni College/ Department: Hans Raj College, University of Delhi Institute of Lifelong Learning, University of Delhi

Lesson 3: Introduction of Displacement Current LEARNING OBJECTIIVES Going through this chapter, the reader would know 1. How the basic laws in electrostatics and magneto statics get modified, when time is introduced? 2. The contribution of Maxwell in the field of electromagnetic. 3. The importance of displacement current. 4. The physical significance of Maxwell equations. Institute of Life Long Learning, University of Delhi Page 1

1. Introduction Just as Newton s three basic, fundamental laws, namely Newton's laws of motion are sufficient to describe all of the phenomena of classical mechanics; Maxwell s four basic, fundamental equations explain all of the phenomena discovered by various eminent scientists like Oersted, Ampere et al of classical electricity and magnetism. We describe these four equations in this chapter, an, in passing. Maxwell s contribution to field of Physics is very useful even today. His work not only united the field of electricity, magnetism and light but also is very helpful in reforming the field of communication. Though his work was purely theoretical, yet has logic and the verification of his theoretical work comes in 1885, when hertz with his experiment demonstrated the presence of electromagnetic waves. Displacement current We have study the following laws in electrostatics and magneto statics in terms of divergence and curl of electric and magnetic fields. Here, we have shown that these laws hold good for static electric and magnetic field where charge is not piling up anywhere. Gauss law for electrostatics: (2.1) Gauss law for magneto statics: (2.2) Faraday s law: (2.3) Ampere-circuital law: (2.4) These equations summarize electromagnetic theory till the time and moreover the electrostatic and magneto static were considered as two different fields. The only link between them was that the current flowing through the wire gives rise to magnetic field and change in the flux linked with the coil generates emf. But when time is introduced in these fields, these fields no longer remain independent. From equation 2.3, Faraday s illustrated that the magnetic field varying in time gives rise to electric field. It happens to Maxwell that as nature supports symmetry, so vice versa, that is time varying electric field should also generate magnetic field, must exist. The idea was purely theoretical and he fixed it theoretically. He found that there exists a flaw in one of the equations mentioned above. Since divergence of curl of a vector is always zero, so there must be something missing in equation number (2.4). As we can see i.e. if we take divergence of number (2.3) on either side, We have everything right, as the left hand side vanishes because of vector property that divergence of curl of a vector is always zero; and the right hand side vanishes from the equation (2.2). Now if we take the divergence of equation number (2.4), we will have something missing: (2.5) As again the left hand side of equation (2.5) will be zero by virtue of curl of divergence of a vector is always zero. On the other hand, the right side of the equation (2.5) is not always zero (example for non-steady currents). Since is zero for steady currents (where charge is not piling up anywhere), the equation (2.5) holds good for magneto static. But when we enter into the region where we have non steady current or say charge is getting accumulated or piling up somewhere, the divergence Institute of Life Long Learning, University of Delhi Page 2

of is not zero. So it is obvious to say that Ampere s law need not be right if we go away from magneto statics. Did You Know James Clerk Maxwell (1831-1879) Prof J C Maxwell is the Physicist from Scotland whose master work in the field of electrodynamics integrated the fields of electricity, magnetism and light. The gem born on 13 June, 1831 in Edinburgh, Scotland. At the age of 25, he became Professor in Physics at Marischal College in Aberdeen. In 1855, he joined King's College in London, followed by University at Cambridge. There he supervises the Cavendish Laboratory and did a good amount of research. In 1861, he got credit for developing first permanent colored photograph. He died in 1879. James Clerk Maxwell himself (in 1864) said: "We have strong reason to conclude that light itself - including radiant heat and other radiation, if any - is an electromagnetic disturbance in the form of waves propagated through the electro-magnetic field according to electro-magnetic laws." Albert Einstein said: "The special theory of relativity owes its origins to Maxwell's equations of the electromagnetic field." //Ref: http://soulconection.net/glossary_indepth/maxwell.html How Maxwell fixed the problem As Maxwell found that there is inconsistency in equation (2.5), as the right hand side of equation should be zero otherwise. To fix this flaw, he makes use of equation of continuity and Gauss law of electrostatics. From equation of continuity we have And now using Gauss s law in the above equation, we have Maxwell combined with, in Ampere s law, to kill off the extra divergence. Institute of Life Long Learning, University of Delhi Page 3

i.e. (2.6) The adjustment made by Maxwell fixes the flaw in Amperes law, which has become universal with Maxwell modification. The extra term added by Maxwell did not change Amperes law for magneto static, as for static fields we will have to be constant and therefore,. But when we go far from static fields where might not be constant, the modified Amperes law will hold. The experimental verification of Maxwell s theoretical correction to Amperes law comes in 1888, when German Physicist Hertz prove the existence of electromagnetic waves through his experiment. Did You Know Heinrich Rudolf Hertz (1857-1894) was a German physicist who clarified and expanded James Clerk Maxwell's electromagnetic theory of light, which was first demonstrated by David Edward Hughes using non-rigorous trial and error procedures. Hertz is distinguished from Maxwell and Hughes because he was the first to conclusively prove the existence of electromagnetic waves by engineering instruments to transmit and receive radio pulses using experimental procedures that ruled out all other known wireless phenomena. The scientific unit of frequency cycles per second was named the "hertz" in his honor. Reference: http://en.wikipedia.org/wiki/heinrich_hertz Institute of Life Long Learning, University of Delhi Page 4

Did You Know Heinrich Hertz's Wireless Experiment (1887) In the 1880s many were seeking experimental evidence to establish the equivalence of light and electromagnetic propagation. James Clerk Maxwell's mathematical theory of 1873 had predicted that electromagnetic disturbances should propagate through space at the speed of light and should exhibit the wave-like characteristics of light propagation. In 1883 Hertz became a lecturer in theoretical physics at the University of Kiel and two years later he was appointed professor of physics at Karlsruhe Polytechnic. In 1887 Hertz designed a brilliant set of experiments tested Maxwell's hypothesis. He used an oscillator made of polished brass knobs, each connected to an induction coil and separated by a tiny gap over which sparks could leap. Hertz reasoned that, if Maxwell's predictions were correct, electromagnetic waves would be transmitted during each series of sparks. To confirm this, Hertz made a simple receiver of looped wire. At the ends of the loop were small knobs separated by a tiny gap. The receiver was placed several yards from the oscillator. According to theory, if electromagnetic waves were spreading from the oscillator sparks, they would induce a current in the loop that would send sparks across the gap. This occurred when Hertz turned on the oscillator, producing the first transmission and reception of electromagnetic waves. Hertz also noted that electrical conductors reflect the waves and that they can be focused by concave reflectors. He found that nonconductors allow most of the waves to pass through. Conceptual Schematic of Hertz's Experiment Reference: http://people.seas.harvard.edu/~jone s/cscie129/nu_lectures/lecture6/hert z/hertz_exp.html Thus Maxwell s correction not only makes the Ampere s law to be true beyond magneto statics but also it gives a symmetry to the four Maxwell equations, i.e, just as faraday s showed that a time varying magnetic field is responsible for the generation of electric field, we do have magnetic field which is the result of time varying electric field. Now, let us see how a time varying electric field generates a magnetic field by considering an example of charging up of a capacitor with a time dependent electric field say. Now to find the value of magnetic field at some point P shown in the figure 1 below, because of the time dependent current flowing in the wire, let us consider a plane circular loop of radius OP= r. The current carrying wire is passing through the center of the Institute of Life Long Learning, University of Delhi Page 5

loop considered and therefore, the plane of the loop so chosen is perpendicular to the wire. The magnetic field at the said point P can be obtained using Ampere s circuital law in integral form given as (2.7) Figure 1. Charging up of a parallel plate capacitor C through which a time dependent current is flowing, piercing a plane circular surface of radius OP to find the value of magnetic field. Now from equation 2.7 and from the given chose Gaussian surface, the direction of the magnetic field will be along the circumference of the plane loop and has the same magnitude for all points lying on its circumference. Therefore, if B is the magnitude of the magnetic field and is the circumference of the loop considered, then from the equation (2.7), we have Here is the current piercing the chosen Gaussian surface. At this point, we will see what will happen to equation (2.7), if we choose a different Gaussian surface? Let us consider another Gaussian surface, say for example a balloon shaped surface (shown in figure 2 below), such that the mouth of the balloon is behaving like that a plane circular loop (considered above) with the body of the balloon not touching the current and has its top in between the parallel plates of the capacitor. P C Source Figure 2. shows the charging up of a parallel plate capacitor through which a time dependent current is flowing, with a balloon like surface, as the chosen Gaussian surface, to find the value of magnetic field at point P outside the capacitor. Our aim is to find the value of magnetic field at point P outside the parallel plate capacitor, but now we have chosen different Gaussian surface. Again, we can make use of Ampere s circuital law given in equation (2.7) to find the value of magnetic field. We will see that the right hand side of the equation for the chosen Gaussian surface is still the same i.e.. But as no current flowing through wire is piercing the chosen surface (figure 2), therefore, right hand side of equation (2.7) is now zero. Institute of Life Long Learning, University of Delhi Page 6

i.e. for a balloon shaped Gaussian surface we have from Ampere s law This implies that there must be some ambiguity, as we cannot have different value of magnetic field at the same point P, for different Gaussian surface. That is the value of the field at a point should be independent of the choice of Gaussian surface. Again this ambiguity is the consequence of the Ampere s circuital law for magneto statics. So we can conclude that there is something missing in this law, which if incorporated, will give us the same value of magnetic field at point P, independent of the choice of Gaussian surface. We can see this obscurity arises on the right hand side of the equation (2.7) as no conduction current is piercing the surface. If there is no conduction current, then what else is actually passing through the surface? This obscurity can be fixed, if we know what else is there in place of conduction current? Do you guess electric field? Figure 3. shows the existence of electric field lines in between the parallel plates of the capacitor. Yes, of course, it is right. We know that, an electric field which has magnitude equals to, (where Q is the total charge on the plates and A is the area of the plates of parallel plate capacitor) exist in between the plates of a parallel plate capacitor. The field in between the plates of capacitor is perpendicular to the surface S of the balloon as shown in figure 2 and 3. So presence of electric field will result in transmission of electric flux,, through the surface S and its value is given by Gauss s law as Since the field applied to the plates of capacitor varies with time, so will the charge Q on the plates of the capacitor, thus there is a current I given by the plates of capacitor. Therefore, from above equation we can see must be there in between This suggests that to remove any inconsistency, we should have So, it might occur to you that if we add this term to the right hand side of the equation (2.7), i.e. to the total conduction current through the surface, then the said equation will hold good for all the surfaces. Thus addition of this term generalizes the Ampere s circuital law, and the total current now is the sum of the conduction current (that is because of flow Institute of Life Long Learning, University of Delhi Page 7

of charges) and current due to rate of change of electric field. The latter is known as the displacement current. Thus, Maxwell contribution in the modification of Ampere s circuital law is as follows. (2.8) Here, above equation says that the conduction current because of flow of charges as well as the displacement current due to the changing electric field can result in the production of magnetic field. Such that the conduction current density denoted by, and the displacement current density denoted by density So we have sum up to give the total current Hence, Maxwell contribution introduces a new term called displacement current. Now, let us applied the results we got so far, to the following circuit: The above circuit consist of a parallel plate capacitor, C, in parallel with a resistance R. The arrangement is in connected across the alternating emf source with voltage given by The plates of the capacitor are circular with radius k and area A. The plates are separated by a distance d. You are now supposed to find the following things. (i) the current through the resistance R? (ii) the current through the capacitor C? (iii) the current arriving at the outside terminals of the capacitor? (iv) the magnetic field between the capacitor plates at a distance k from the axis? (i) We can see that the current through the resistance R in the given circuit can be obtained using Ohm s law as Institute of Life Long Learning, University of Delhi Page 8

(ii) Now to find the current through the capacitor means we have to find the value of displacement current that exist in between the plates of a capacitor. To find that, we will first find the value of electric flux through the capacitor and that is equal to The voltage across the capacitor is equal to the external emf. The electric field between the capacitor plates is therefore equal to Therefore, the value of displacement current through the capacitor is Thus the current arriving outside the terminals of the capacitor is the total of the current charging the capacitor and the current passing through the resistor R. Now the charge on the capacitor can be found as Therefore the charging current is The total current then is Now to find the magnetic field at radial point k, let us consider the magnetic lines of force inside the capacitor. These lines of force are shown in figure below. They form concentric circles and now if we apply the Ampere s circuital law around a circular path of radius k, the left hand side of the law will be Figure: shows the Amperian loop considered to find the value of magnetic field inside the capacitor Institute of Life Long Learning, University of Delhi Page 9

For the considered surface the value of electric flux through the disk of radius k is Therefore, the displacement current intercepted by this surface is equal to Thus from Ampere s law we Thus magnetic field will be: Now let us see the characteristics of displacement current. 1. We call Displacement current, a current as it can produce a magnetic field just as conduction current does. Otherwise, it has no other property of current and has finite value even in perfect vacuum. Now the question arises: If the displacement current has the same unit as conduction current? Let us check it by substituting the S I units of and and simplify the resultant expression as The answer is yes, as it should be. 2. The value of magnitude of displacement current is given by the time rate of electric displacement. It s value is therefore zero, in the regions where electric field is not changing with time. Say for example, if we consider a circuit of LC oscillator, then Institute of Life Long Learning, University of Delhi Page 10

the value of displacement current is zero at the moment capacitor becomes fully charged. Because when the capacitor becomes fully charged, we have maximum electric flux at that moment and consequently, the value of displacement current in between the plates of the capacitor is zero. 3. Whenever there exist rate of change of electric flux, we have finite displacement current. 4. Displacement current helps in bridging the gap that exist in the medium and hence make the total current continuous through the medium. 5. The ratio of displacement current to conduction current is found to be equal to. And thus we can say that the value of displacement current compared to conduction current is negligible in good conductors for frequencies lower than the optical frequencies. 6. Amalgamation of electric and magnetic phenomena is the result of addition of displacement current to Ampere's law. Importance of Displacement Current So, in nutshell if we interpret the importance of displacement current, we will find that 1. It is because of the displacement current that electromagnetic signals can propagate through a material or more precisely electromagnetic radiation is possible. 2. Thus, it might occur to you that the one of the reasons for life to exist on planet Earth is presence of displacement current. Yes, it is right to think so, as, we all know that we receive heat (that warms our planet and is very important to carry out various activities necessary for sustaining the life on it) from Sun. This heat transmission takes place through radiative heat transfer rather than by conduction and convection mechanisms, as heat radiation is also an electromagnetic wave. Had it not been associated with the displacement current, we would not be able to receive any heat from the Sun and subsequently, there will be no life on Earth. Maxwell Equations With the inclusion of Maxwell s correction to the Ampere s law, the four equations that describe the fundamental laws of electricity and magnetism, discussed before will now become or (3.1) or (3.2) or (3.3) or (3.4) Institute of Life Long Learning, University of Delhi Page 11

It is worth to note that all the properties of the electric and magnetic fields can be derived from the above mentioned four Maxwell equations by doing some mathematical manipulations. Say for example we can prove the charge conservation mathematically using Maxwell s equations. It means that if there is no flow of electric current into or out a given surface S enclosing a volume V, then the electric charge enclosed by this volume will remain constant. Consider Equation 3.1, which shows how the enclosed charge is related to the electric flux Differentiating the above equation with respect to time will give us the rate of change of the charge enclosed in the considered volume V. Now the from the equation 3.4, the line integral over considered closed surface (say a bag with its mouth shrinks to zero) of the magnetic field is zero and will give Using above two equations we will have Above equation means that if there is no flow of electric current ( surface S enclosing a volume V, then ) into or out a given The electric charge enclosed by this volume will remain constant. This implies Charge conservation. We can also derive the well-known coulomb s law in electrostatics and Biot Savart s law in Magneto static from the Maxwell equations as following. Let us consider a point charge q and we are supposed to calculate the value of electric field at a distance say r from it. To find the field, we will consider first Maxwell equation (3.1) with sphere of radius r as the Gaussian surface (shown in figure below). Therefore from integral form of Maxwell s equation (3.1), we have Institute of Life Long Learning, University of Delhi Page 12

Or This is nothing but coulomb s law. Now to get Biot Savart s law in Magneto static from the Maxwell equations, let us calculate the value of magnetic field at a distance r, around a straight conductor of infinite length carrying current I. To find the field, we will consider fourth Maxwell equation (3.4) with circle of radius r has its center passing through the wire, as the Amperian loop. Therefore from integral form of Maxwell s equation (3.4), we have Or Now, if you compared the result thus obtained with the one you obtained for magnetic field of the straight infinite current using Biot Savart s Law directly, you got the same answer. Just like we have used these equations to derive fundamental law of charge conservation and coulomb law, etc, Maxwell, mathematically manipulated these equations to derive the wave equation in terms of Electric and Magnetic intensity vector and showed that the electromagnetic waves travel with the velocity of light in free space. We can see how he did that, as following: Consider Maxwell equation (3.3) and take curl on its either side, then we will have Institute of Life Long Learning, University of Delhi Page 13

Now for the charge free region, we have, hence equation (3.1) will give, and therefore above equation will become Using Maxwell equation (3.4) and substitute the value of in above equation we have Again for free space, we have Therefore above equation will reduce to Or This is wave equation in terms of field vector in free space. Similarly we can have wave equation in terms of magnetic intensity vector, by taking curl of Maxwell equation (3.4). Where we have used and for free space. or using Maxwell equation (3.2) and (3.3) in above we have Or Institute of Life Long Learning, University of Delhi Page 14

This is the wave equation in terms of field vector. Now we will see with what velocity these wave travels in free space? If you compare the equations obtained above with the standard classical wave equation, you will find the velocity with which these waves travels in free space can be calculated as Or Now for free space we have and This is nothing but velocity of light. Therefore, one of the important results of Maxwell formulation is that electromagnetic waves travel with the speed of light in free space. Thus light wave is an electromagnetic wave. He further showed that electromagnetic waves are transverse in nature (i.e. the field vector and (or ) are mutually perpendicular as well perpendicular to the direction of propagation of the wave represented by in the above figure. Therefore, electromagnetic waves can be polarized. By polarization, we mean the restricting the direction of either of electric or magnetic field vector to a plane. Institute of Life Long Learning, University of Delhi Page 15

Physical Significance of Maxwell Equations (1) This law tells that the net outward electric flux enclosed by a closed surface S enclosing volume V, in an electric field is equal to times the total charge enclosed by the surface. Or we can say, if we consider a spherically symmetric charge distribution, then electric field at a distance r from it is given by: Where q is the total charge within the distance r and is the unit vector in the direction of r. (2) Institute of Life Long Learning, University of Delhi Page 16

Above law emphasis that the magnetic flux through any closed surface is zero. It means that the total flux entering a bounded region is equal to the total flux leaving it. Or more precisely, magnetic lines of force forms a closed loop as shown in the figure above. This means that magnetic monopoles do not exist. Had they been than the above law would have the form similar to that of Gauss law in electrostatic and will be Where is the free magnetic charge density. (3) This law tells us that the line integral of electric intensity around a closed path is equal to the negative rate of change of magnetic flux linked with the path. OR magnetic field changing with time produces electric field. (4) This law tells us that the line integral of magnetic field around a closed path is equal to the conduction current plus displacement current. OR electric field changing with time and conduction current produces electric field. Summary 1. In this chapter, we learnt how Maxwell theoretically fixes the flaw in one of the four equations governing all the fundamental laws of electricity and magnetism. 2. The following are the four Maxwell equations that describe the fundamental laws of electricity and magnetism, or or or or 3. The fourth equation is known as the modified Amperes Law with Maxwell correction. 4. The additional term in the fourth equation is known as Displacement current. 5. The introduction of displacement current by Maxwell, gives these equations symmetry. That is, just as changing magnetic field generates electric field, a changing electric field would also generate magnetic field. 6. Displacement current has nothing to do with the motion of charges as in conduction current, but we call it current because just as conduction current produces magnetic field, it can also result in the generation of magnetic field. It helps in making total current continuous across the discontinuities in the medium. Institute of Life Long Learning, University of Delhi Page 17

7. The first equation, tells that the net outward electric flux enclosed by a closed surface S enclosing volume V, in an electric field is equal to times the total charge enclosed by the surface. 8. The second equation, tells that the magnetic flux through any closed surface is zero. This means that magnetic monopoles do not exist. 9. The third equation, tells us that the line integral of electric intensity around a closed path is equal to the negative rate of change of magnetic flux linked with the path. OR magnetic field changing with time produces electric field. 10. The fourth equation, tells us that the line integral of magnetic field around a closed path is equal to the conduction current plus displacement current. OR electric field changing with time and conduction current produces electric field. Questions Fill in the blanks: Answers (i) The value of magnitude of displacement current is equal to the time rate of -. (ii) The, makes the total current continuous, at the points of discontinuity through medium. (iii) The verification of Maxwell s theory is demonstrated by experiment. (iv) The value of displacement current becomes increasingly important as the frequency. (v) Maxwell applies correction to the existing law of magneto-static. (i) Electric displacement (ii) Displacement current (iii) Hertz (iv) increases (v) Ampere s law True of False State whether the following statements are true or False. (i) (ii) (iii) (iv) Faraday s law states that time varying electric field give rise to magnetic field. The ratio of conduction current and displacement current for time varying electric field is independent of frequency. A Phasor is a time dependent quantity. Maxwell s modification to Ampere s law is the Displacement current. Answers Institute of Life Long Learning, University of Delhi Page 18

(i) False (Faraday s law states that a time varying magnetic field gives rise to an electric field). (ii) False (the displacement current becomes increasingly important as the frequency increases). (iii) False (A Phasor is a scalar or vector or more precisely complex but time independent quantity). (iv) True (This is correct statement). Multiple Choice Questions Select the best alternative in each of the following: (i) (ii) (iii) (iv) (v) The concept of displacement current was introduced by (a) Amperes (b) Faraday (c) Maxwell (d) Lorentz When a capacitor is connected to an ac source: (a) No current flows. (b) Current flows through it, due to motion of charges. (c) Current flows but no charge is transported between the plates. (d) Conduction current as well as displacement current flows. Which of the following is not a Maxwell s equation? (a). (b) (c) (d) What does (a) Faraday s law (b) Magnetic monopoles do not exist. (c) Ampere s circuital law. (d) Gauss s law. Continuity equation is (a) Integral form of charge conservation law. (b) Differential form of charge conservation law. (c) Both of the above. (d) None of the above Answers 1. (c) Justification/Feedback for the correct answer: (a) Amperes showed that conduction current is the source of electric field. (b) Faraday showed that changing magnetic field give rise to electric field. (c) It was Maxwell who introduces the displacement current. (d) Lorentz gives the value of force experience by the charge particle in magnetic field. 2. (c) Justification/Feedback for the correct answer: Institute of Life Long Learning, University of Delhi Page 19

(a) As there is a gap (insulation in between the plates of the capacitor), Current flows because of changing electric field, but no charge is transported between the plates. (b) As there is a gap (insulation in between the plates of the capacitor), Current flows because of changing electric field, but no charge is transported between the plates. (c) As there is a gap (insulation in between the plates of the capacitor), Current flows because of changing electric field, but no charge is transported between the plates. (d) As there is a gap (insulation in between the plates of the capacitor), Current flows because of changing electric field, but no charge is transported between the plates. 3. (d) Justification/Feedback for the correct answer: (a) This is correct equation i.e. Ampere s law with Maxwell correction. (b) This is correct equation. (c) This is correct equation i.e. Faraday law illustrating that changing magnetic field give rise to electric field. (d) This equation is incorrect. 4. (b) Justification/Feedback for the correct answer: (a) Faraday s law illustrates that changing magnetic field give rise to electric field. e Faraday law illustrating that changing magnetic field give rise to electric field. (b) This is the correct explanation of the given equation. (c) Ampere s circuital law states that the electric field is generated by either conduction current or changing electric field. (d) Gauss s law states that rate of change of electric flux through a surface S enclosing volume V is times the total charge enclosed by it. 5. (c) Justification/Feedback for the correct answer: (a) The continuity equation is the differential form of law of charge conservation. (b) The continuity equation is the differential form of law of charge conservation. (c) The continuity equation is the differential form of law of charge conservation. (d) The continuity equation is the differential form of law of charge conservation. Exercise 1. Show how Maxwell modified Ampere s law to make it consistent with the equation of continuity. 2. Explain the Physical significance of the term displacement current. 3. Calculate the ratio of J c to J D for the electric field dependence of. 4. Derive equation of continuity from Maxwell s equations. (Hint Ans 4. Some materials are neither good conductors nor perfect dielectrics, so that both conduction and displacement current exist. Assuming the time dependence for, we have / From which Institute of Life Long Learning, University of Delhi Page 20

As expected, the displacement current becomes increasingly important as the frequency increases.) Institute of Life Long Learning, University of Delhi Page 21