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Electric potential Previously, it shown that any vector field whose curl is zero, is equal to the gradient of some scalar field: If Then Because V x E = 0, the line integral of E around any closed loop is zero (that follows from Stokes' theorem), ∮ Then the line integral of E from point a to point b (over an open path) is the same for all paths (otherwise you could go out along path (i) and return along path (ii)- as in Fig. 2.3Q- ∫ Because the line integral is independent of path, we can define a function (scalar field) () ∫ The infinity is a reference point, in which the function is zero. then depends only on the point r. It is called the electric potential. Evidently, the potential difference between two points a and b is ∫ (∫ )∫ ∫ ∫∫ ∫

From the fundamental theorem of gradient ∫( ) Then ∫( ) ∫ Since, finally, this is true for any points a and b, the integrands must be equal: This equation shows that the electric field is the gradient of a scalar potential. Poisson’s equation and Laplace’s equation Previously it is found the electric field can be written as the gradient of a scalar potential. From the fundamental equations for electrostatic fieldE, Then ()

This is known as Poisson's equation. In regions where there is no charge, so that p = 0, Poisson's equation reduces to Laplace's equation, On the other hand, take the curl of electrostatic field E () This equation is true where the left side, curl of gradient of scalar field is zero. This satisfy the fundamental property of electrostatic field which it is curl-less vector field (xE=0). This equation satisfy the property that, if xE=0, it must be E=- . It takes only one differential equation (Poisson's) to determine , because is a scalar; for E we needed two, the divergence and the curl. Work and energy in electrostatics The work done to move the charge Suppose you have a stationary configuration of source charges, and you want to move a test charge q from point a to point b (Fig. 2.39).

Question: How much work will you have to do? At any point along the path, the electric force on q is F = qE; the force you must exert, in opposition to this electrical force, is - qE. (If the sign bothers you, think about lifting a brick: Gravity exerts a force mg downward, but you exert a force mg upward. Of course, you could apply an even greater force-then the brick would accelerate, and part. The work done by the electric field to transfer the charge q from a to b over an open path l, is therefore ∫∫ ∫ The potential difference is the work per unit charge, then () () ∫ In words, the potential difference between points a and b is equal to the work per unit charge required to carry a particle from a to b. Then the electric potential energy is differ from the electric potential, where the electric potential is the electric potential energy per unit charge; () While, he electric field is the electric force per unit charge ()

Conductors electrostatic properties of conductor In an insulator, such as glass or rubber, each electron is attached to a particular atom. In a metallic conductor, by contrast, one or more electrons per atom are free to roam about at will through the material. (In liquid conductors such as salt water it is ions that do the moving.) A perfect conductor would be a material containing an unlimited supply of completely free charges. In real life there are no perfect conductors, but many substances come amazingly close. From this definition the basic electrostatic properties of ideal conductors immediately follow: (1) E =0 inside a conductor. Why? Because if there were any field, those free charges would move, and it wouldn't be electrostatics any more. Well ... that's hardly a satisfactory explanation; maybe all it proves is that you can't have electrostatics when conductors are present. We had better examine what happens when you put a conductor into an external electric field Eo (Fig. 2.42). Initially, this will drive any free positive charges to the right, and negative ones to the left. (In practice it's only the negative charges-electrons-that do the moving, but when they depart the right side is left with a net positive charge-the stationary nuclei-so it doesn't really matter which charges move; the effect is the same.) When they come to the edge of the material, the charges pile up: plus on the right side, minus on the left. Now, these induced charges produce a field of their own, El, which, as you can see from the figure, is in the opposite direction to Eo. That's

the crucial point, for it means that the field of the induced charges tends to cancel off the original field. Charge will continue to flow until this cancellation is complete, and the resultant field inside the conductor is precisely zero. The whole process is practically instantaneous. (2)  = 0 inside a conductor. This follows from Gauss's law: So also is . There is still charge around, but exactly as much plus charge as minus, so the net charge density in the interior is zero. (3) Any net charge resides on the surface. That's the only other place it can be. (4) A conductor is an equipotential. For if a and b are any two points within (or at the surface of) a given conductor, V(b) – V(a) = - J~b E· dl = 0, and hence Yea) = V(b). (5) E is perpendicular to the surface, just outside a conductor. Otherwise, as in (i), charge will immediately flow around the surface utiti! it kills off the tangential component (Fig. 2.43). (Perpendicular to the surface, charge cannot flow, of course, since it is confined to the conducting object.)



Chapter 4 electric field in matter Maxwell’s equations in matter Polarization of dielectric In this chapter we shall study electric fields in matter. Matter, of course, comes in many varieties-solids, liquids, gases, metals, woods, glasses-and these substances do not all respond in the same way to electrostatic fields. Nevertheless, most everyday objects belong (at least, in good approximation) to one of two large classes: conductors and insulators (or dielectrics). We have already talked about conductors; these are substances that contain an \"unlimited\" supply of charges that are free to move about through the material. In practice what this ordinarily means is that many of the electrons (one or two per atom in a typical metal) are not associated with any particular nucleus, but roam around at will. In dielectrics, by contrast, all charges are attached to specific atoms or molecules-they're on a tight leash, and all they Can do is move a bit within the atom or molecule. Such microscopic displacements are not as dramatic as the wholesale rearrangement of charge in a conductor, but their cumulative effects account for the characteristic behavior of dielectric materials. There are actually two principal mechanisms by which electric fields can distort the charge distribution of a dielectric atom or molecule: stretching and rotating. In the next two sections I'll discuss these processes. Induced dipoles What happens to a neutral atom when it is placed in an electric field E? Your first guess might well be: \"Absolutely nothing-since the atom is not charged, the field has no effect on it.\" But that is incorrect. Although the atom as a whole is electrically neutral, there is a positively charged core (the nucleus) and a negatively charged electron cloud surrounding it. These two regions of charge within the atom are influenced by the field: the nucleus is pushed in the direction of the field, and the electrons the opposite way. In principle, if the field is large enough, it can pull the atom apart completely, \"ionizing\" it (the substance then becomes a conductor). With less extreme fields, however, an equilibrium is soon established, for if the center of the electron cloud does not coincide with the nucleus, these positive and negative charges attract one another, and this holds the atoms together. The two opposing forces-E pulling the electrons and nucleus apart, their mutual attraction drawing them together-reach a balance, leaving the atom polarized, with plus charge shifted slightly one way, and

minus the other. The atom now has a tiny dipole moment p, which points in the same direction as E. Typically, this induced dipole moment is approximately proportional to the field (as long as the latter is not too strong): The constant of proportionality a is called atomic polarizability. Its value depends on the detailed structure of the atom in question. For molecules the situation is not quite so simple, because frequently they polarize more readily in some directions than others. Alignment of polar molecules The neutral atom had no dipole moment to start with-p was induced by the applied field. Some molecules have built-in, permanent dipole moments and are called polar molecules. What happens when such molecules (called polar molecules) are placed in an electric field? Polarization In the previous two sections we have considered the effect of an external electric field on an individual atom or molecule. We are now in a position to answer (qualitatively) the original question: What happens to a piece of dielectric material when it is placed in an electric field? If the substance consists of neutral atoms (or nonpolar molecules), the field will induce in each a tiny dipole moment, pointing in the same direction as the field. 2 If the material is made up of polar molecules, each permanent dipole will experience a torque, tending to line it up along the field direction. (Random thermal motions compete with this process, so the alignment is never complete, especially at higher temperatures, and disappears almost at once when the field is removed.) Notice that these two mechanisms produce the same basic result: a lot of little dipoles pointing along the direction of the field-the material becomes polarized. A convenient measure of this effect is

which is called the polarization. From now on we shall not worry much about how the polarization got there. Actually, the two mechanisms I described are not as clear-cut as 1 tried to pretend. Even in polar molecules there will be some polarization by displacement (though generally it is a lot easier to rotate a molecule than to stretch it, so the second mechanism dominates). It's even possible in some materials to \"freeze in\" polarization, so that it persists after the field is removed. But let's forget for a moment about the cause of the polarization and study the field that a chunk of polarized material itselfproduces. Then. in Sect. 4.3, we'll put it all together: the original field, which was responsible for P, plus the new field, which is due to P. The field of polarized object Free charges and bounded charges The electric displacement: Gauss’s law in presence dielectric It is found that the effect of polarization is to produce accumulations of bound charge, within the dielectric and b on the surface. The field due to polarization of the medium is just the field of this bound charge. We are now ready to put it all together: the field attributable to bound charge plus the field due to everything else (which, for want of a better term, we call free charge). The free charge might consist of electrons on a conductor or ions embedded in the dielectric material or whatever; any charge, in other words, that is not a result of polarization. Within the dielectric, then, the total charge density can be written:

Applying Gauss’s law (first Maxwell’s equation) ) ( )( () () () The expression in parentheses, designated by the letter D, is known as the electric displacement. Then the Gauss’s law (first Maxwell’s equation) in matter (dielectric) is given by For the divergence alone is insufficient to determine a vector field; you need to know the curl as well. One tends to forget this in the case of electrostatic fields because the curl of E is always zero. But the curl of D is not always zero. () There is no reason, in general, to suppose that the curl of P vanishes. Sometimes it does. If there is no free charge anywhere, so if you really believe that the only source of D is pf, you will be forced to conclude that D = 0 everywhere, and hence that

That is inside and E = 0 outsidet, which is obviously wrong. Because Moreover. D cannot be expressed as the gradient of a scalar-there is no \"potential\" for D. Linear dielectrics: susceptibility, permittivity and dielectric constant The polarization of a dielectric ordinarily results from an electric field, which lines up the atomic or molecular dipoles. For many substances, in fact, the polarization is proportional to the field, provided E is not too strong: The constant of proportionality, Xe, is called the electric susceptibility of the medium (a factor of o has been extracted to make Xe dimensionless). The value of Xe depends on the microscopic structure of the substance in question (and also on external conditions such as temperature). The materials that obey this equation is called linear dielectrics. Note that E in equation is the total field; it may be due in part to free charges and in part to the polarization itself. If, for instance, we put a piece of dielectric into an external field Eo, we cannot compute P directly; the external field will polarize the material, and this polarization will produce its own field, which then contributes to the total field, and this in turn modifies the polarization, which breaking out of this infinite regress is not always easy. The simplest approach is to begin with the displacement, at least in those cases where D can be deduced directly from the free charge distribution. In linear dielectric, the electric displacement D is

() () Where () This new constant  is called the permittivity of the material. (In vacuum, where there is no matter to polarize, the susceptibility is zero, and the permittivity is o. That's why it is called the permittivity of free space. I dislike the term, for it suggests that the vacuum is just a special kind of linear dielectric, in which the permittivity happens to have the value 8.85 x 10-12 C 2/N.m2. The relative permittivity, or dielectric constant, of the material is given by () In a homogeneous linear dielectric the bound charge density b) is proportional to the free charge density Pf, ( )( ) () This equation gives the relationship between bound charges and free charges, ()

Energy in dielectric systems Previously, the energy W stored in electrostatic field for parallel-plate capacitor of capacitance Co, is given by Where C is the capacitance of capacitor in presence of dielectric between the plates, and V is the voltage difference between the two plates. If Co and C are the capacitances of parallel-plate capacitor in presence of air and dielectric respectively, then Then () The volume V=Ad of the dielectric material in capacitor, then

∫ ∫( ) ……………………………. ………………………………….. …………………….

Magnetostatics Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. Magnetic force One feature of the magnetic force law warrants special attention: magnetic force do not work. For if q moves an amount dl = v dt, the work done is () () ( ) This follows because (v x B) is perpendicular to v, so (v x B). v = 0. ( ) (( )( )) ( ) ……………………………… …………………………… Magnetic forces may alter the direction in which a particle moves, but they cannot speed it up or slow it down. The fact that magnetic forces do no work is an elementary and direct consequence of the Lorentz force law, but there are many situations in which it appears so manifestly false that one's confidence is bound to waver. When a magnetic crane lifts the carcass of a junked car, for instance, something is obviously doing work, and it seems perverse to deny that the magnetic force is responsible. Well, perverse or not, deny it we must, and it can be a very subtle matter to figure out what agency does deserve the credit in such circumstances. I'll show you several examples as we go along. ………………………………….. …………………………………… ………………………………. ………………………………………

Biot-Savart law: magnetic field due to steady current Steady current Stationary charges produce electric fields that are constant in time; hence the term electrostatics. 4 Steady currents produce magnetic fields that are constant in time; the theory of steady currents is called magnetostatics. Stationary charges create constant electric fields (electrostatics), while steady currents create constant magnetic fields (Magnetostatics). By steady current I mean a continuous flow that has been going on forever, without change and without charge piling up anywhere. (Some people call them \"stationary currents\"; to my ear, that's a contradiction in terms.) Of course, there's no such thing in practice as a truly steady current, any more than there is a truly stationary charge. In this sense both electrostatics and magnetostatics describe artificial worlds that exist only in textbooks. However, they represent suitable approximations as long as the actual fluctuations are reasonably slow; in fact, for most purposes magnetostatics applies very well to household currents, which alternate 60 times a second! When a steady current flows in a wire, its magnitude I must be the same all along the line; otherwise, charge would be piling up somewhere, and it wouldn't be a steady current. By the same token, And hence the continuity equation becomes

Comparison of magnetostatics and electrostatics Thinking about the properties of electric and magnetic fields as vector fields, and the meaning divergence and curl, where it is known the lines of electric field is diverged and curl-less, while the lines of magnetic field is divergence-less and curl. These properties could be expressed as These equations are written based on the values rather than zero does not know. Using physical meaning, these equations could be, in electrostatics and magnetostatics, written as The first two equations are the divergence and curl of the electrostatic field (Maxwell's equations for electrostatics). Maxwell's equations determine the field, if the source charge density p is given; they contain essentially the same information as Coulomb's law plus the principle of superposition. The second two equations are the divergence and curl of the magnetostatic field (Maxwell's equations for magnetostatics). Maxwell's equations determine the magnetic field: they are equivalent to the Biot-Savart law (plus superposition). Maxwell's equations and the force law

() constitute the most elegant formulation of electrostatics and magnetostatics. In electromagnetics and electrodynamics (time-varying fields): The electric field diverges away from a (positive) charge; the magnetic field line curls around a current (Fig. 5.44). Electric field lines originate on positive charges and terminate on negative ones; magnetic field lines do not begin or end anywhere-to do so would require a nonzero divergence. They either form closed loops or extend out to infinity.

To put it another way, there are no point sources for B, as there are for E; there exists no magnetic analog to electric charge. This is the physical content of the statement  . B = 0. Coulomb and others believed that magnetism was produced by magnetic charges (magnetic monopoles, as we would now call them), and in some older books you will still find references to a magnetic version of Coulomb's law, giving the force of attraction or repulsion between them. It was Ampere who first speculated that all magnetic effects are attributable to electric charges in motion (currents). As far as we know, Ampere was right; nevertheless, it remains an open experimental question whether magnetic monopoles exist in nature (they are obviously pretty rare, or somebody would have found oneI2), and in fact some recent elementary particle theories require them. For our purposes, though, B is divergenceless and there are no magnetic monopoles. It takes a moving electric charge to produce a magnetic field, and it takes another moving electric charge to \"feel\" a magnetic field. Magnetic vector potential In electrostatics, it is shown that if the curl of electrostatic field, as a vector field, is zero, it must be equal the gradient of the scalar electric potential, as a scalar field. The electric potential is called scalar potential; if Then So that () The curl of the gradient of scalar field is equal to zero. So in magnetoststics, if the divergence of magnetic field is zero, as a vector field, it must be equal to the curl of vector magnetic potential, as vector field. The magnetic potential is called vector magnetic potential; if Then () So that

The divergence of the curl of vector field is equal to zero. The vector field A is called a vector magnetic potential. The potential formulation automatically takes care of  . B = 0 (since the divergence of a curl is always zero); there remains Ampere's law, in magnetostatics: ( ) () () The electric potential had a built-in ambiguity: you can add to V any function whose gradient is zero (which is to say, any constant), without altering the physical quantity E. Likewise, you can add to the magnetic potential any function whose curl vanishes (which is to say, the gradient of any scalar), with no effect on B. We can exploit this freedom to eliminate the divergence of A:

Scalar electric potential and vector magnetic potential Scalar electric potential Vector magnetic potential A These equations are Poisson’s equations, their solutions are () ∫ () () ∫() It must be said that A is not as useful as . For one thing, it's still a vector, It would be nice if we could get away with a scalar potential, Where () () this is incompatible with Ampere's law, since the curl of a gradient is always zero. (A magnetostatic scalar potential can be used, if you stick scrupulously to simply-connected. current-free regions, but as a theoretical tool it is oflimited interest. See Prob. 5.28.) Moreover, since magnetic forces do no work, A does not admit a simple physical interpretation in terms of potential energy per unit charge. (In some contexts it can be interpreted as momentum per unit charge. 14) Nevertheless, the vector potential has substantial theoretical importance, as we shall see in Chapter 10.



Chapter 6: magnetic field in matter Magnetization All magnetic phenomena are due to electric charges in motion, and in fact, if you could examine a piece of magnetic material on an atomic scale you would find tiny currents: electrons orbiting around nuclei and electrons spinning about their axes. For macroscopic purposes, these current loops are so small that we may treat them as magnetic dipoles. Ordinarily, they cancel each other out because of the random orientation of the atoms. But when a magnetic field is applied, a net alignment of these magnetic dipoles occurs, and the medium becomes magnetically polarized, or magnetized. Unlike electric polarization, which is almost always in the same direction as E, some materials acquire a magnetization parallel to B (paramagnets) and some opposite to B (diamagnets). A few substances (called ferromagnets, in deference to the most common example, iron) retain their magnetization even after the external field has been removed for these the magnetization is not determined by the present field but by the whole magnetic \"history\" of the object. Permanent magnets made of iron are the most familiar examples of magnetism, though from a theoretical point of view they are the most complicated. In the presence of a magnetic field, matter becomes magnetized; that is, upon microscopic examination it will be found to contain many tiny dipoles, with a net alignment along some direction. We have discussed two mechanisms that account for this magnetic polarization: (l) paramagnetism (the dipoles associated with the spins of unpaired electrons experience a torque tending to line them up parallel to the field) and (2) diamagnetism (the orbital speed of the electrons is altered in such a way as to change the orbital dipole moment in a direction opposite to the field). Whatever the cause, we describe the state of magnetic polarization by the vector quantity. M is called the magnetization; it plays a role analogous to the polarization P in electrostatics. Bound currents

() The magnetic field inside the matter Like the electric field, the actual microscopic magnetic field inside matter fluctuates wildly from point to point and instant to instant. When we speak of \"the\" magnetic field in matter. we mean the macroscopic field: the average over regions large enough to contain man) atoms. (The magnetization M is \"smoothed out\" in the same sense.). The auxiliary field H Ampere’s law in magnetic materials The field due to magnetization of the medium is just the field produced by these bound currents. We are now ready to put everything together: the field attributable to bound currents, plus the field due to everything else-which I shall call the free current. The free current might flow through wires imbedded in the magnetized substance or, if the latter is a conductor, through the material itself. In any event, the total current can be written as There is no new physics in Eq. 6.17; it is simply a convenience to separate the current into these two parts because they got there by quite different means: the free current is there because somebody hooked up a wire to a battery-it involves actual transport of charge; the bound current is there because of magnetization-it results from the conspiracy of many aligned atomic dipoles. Ampere’s law ( )( ) ( )( )

( )( ) () The quantity in parentheses is designated by the letter H: In terms of H, then, Ampere's law reads H plays a role in magnetostatics analogous to D in electrostatics: Just as D allowed us to write Gauss's law in terms of the free charge alone, H permits us to express Ampere's law in terms of the free current alone-and free current is what we control directly. Bound current, like bound charge, comes along for the ride-the material gets magnetized, and this results in bound currents; we cannot turn them on or off independently, as we can free currents. In applying Eq. 6.20 all we need to worry about is the free current, which we know about because we put it there. In particular, when symmetry permits, we can calculate H immediately from Eq. 6.20 by the usual Ampere's law methods. As it turns out, H is a more useful quantity than D. In the laboratory you will frequently hear people talking about H (more often even than B), but you will never hear anyone speak of D (only E). The reason is this: To build an electromagnet you run a certain (free) current through a coil. The current is the thing you read on the dial, and this determines H (or at any rate, the line integral of H); B depends on the specific materials you used and even, if iron is present, on the history of your magnet. On the other hand, if you want to set up an electric field, you do not plaster a known free charge on the plates of a parallel plate capacitor; rather, you connect them to a battery of known voltage. It's the potential difference you read on your dial, and that determines E (or at any rate, the line integral of E); D depends on the details of the dielectric you're using. If it were easy to measure charge, and hard to measure potential, then you'd find experimentalists talking about D instead of E. So the relative familiarity of H, as

contrasted with D, derives from purely practical considerations; theoretically, they're all on equal footing. Many authors call H, not B, the \"magnetic field.\" Then they have to invent a new word for B: the \"flux density,\" or magnetic \"induction\" (an absurd choice, since that term already has at least two other meanings in electrodynamics). Anyway, B is indisputably the fundamental quantity, so I shall continue to call it the \"magnetic field,\" as everyone does in the spoken language. H has no sensible name: just call it \"H\". ( ) () That is only when the divergence of M vanishes is the parallel between B and oH faithful. Ifyou think I'm being pedantic, consider the example ofthe barmagnet-a short cylinder ofiron that carries a permanent uniform magnetizationM parallel to its axis. (See Probs. 6.9 and 6.14.) In this case there is no free current anywhere, and a na'ive application ofEq. 6.20 might lead you to suppose that H = 0, and hence that B = fLoM inside the magnet and B = °outside, which is nonsense. It is quite true that the curl of H vanishes everywhere, but the divergence does not. (Can you see where V .M = 07) Linear and nonlinear media Magnetic susceptibility and permeability In paramagnetic and diamagnetic materials, the magnetization is sustained by the field; when B is removed, M disappears. In fact, for most substances the magnetization is proportional to the field, provided the field is not too strong.

The constant of proportionality Xm is called the magnetic susceptibility; it is a dimensionless quantity that varies from one substance to another-positive for paramagnets and negative for diamagnets. Materials that obey this equation are called linear media. ( )( )( ) Where () Where  is called the permeability of the materia1. In a vacuum, where there is no matter to magnetize, the susceptibility Xm vanishes, and the permeability is o. That is called the permeability of free space. The relative permeability is () The volume bound current density in a homogeneous linear material is proportional to the free current density: () In particular, unless free current actually flows through the material, all bound current will be at the surface.

Electrodynamics Ohm’s law To make a current flow, you have to push on the charges. How fast they move, in response to a given push, depends on the nature of the material. For most substances, the current density J is proportional to the force per unit charge, () The proportionality factor  is an empirical constant that varies from one material to another; it's called the electrical conductivity of the medium. The reciprocal of , called the resistivity: =1/ (not to be confused with charge density). Notice that even insulators conduct slightly, for most purposes metals can be regarded as perfect conductors, with =. In principle, the force that drives the charges to produce the current could be an electromagnetic force. Then the current density is Ordinarily, the velocity of the charges is sufficiently small ( 0) that the second term can be ignored: This Equation is called Ohm's law, E = 0 inside a conductor at electrostatic equilibrium. But that's for stationary charges (J = 0). Moreover, for perfect conductors E = J/ = 0 even if current is

flowing. In practice, metals are such good conductors that the electric field required to drive current in them is negligible. Thus we routinely treat the connecting wire in electric circuits (for example) as equipotentials. Resistors, by contrast, are made from poorly conducting materials. Experimentally, Ohm’s law is given by The total current flowing from one electrode to the other is proportional to the potential difference between them. This, of course, is the more familiar version of Ohm's law. The constant of proportionality R is called the resistance; it's a function of the geometry of the arrangement and the conductivity of the medium between the electrodes. If L and A are the length and cross section area of wire respectively, then Where From Ohm’s law Take the divergence of this eq

() For steady current and uniform electrical conductivity, the divergence of E equals to zero, .E=0, then (Eq. 5.31), and therefore the charge density is zero; any unbalanced charge resides on the surface. (We proved this long ago, for the case of stationary charges, using the fact that E = 0; evidently, it is still true when the charges are allowed to move.) It follows, in particular, that Laplace's equation holds within a homogeneous ohmic material carrying a steady current, so all the tools and tricks of Chapter 3 are available for computing the potential. 7.3.4 Magnetic charges There is a pleasing symmetry about Maxwell's equations; it is particularly striking in free space, where  and J vanish, =0 and J=0:

Conservation laws Charge conservation – energy conservation – momentum conservation Charge conservation law The charge continuity equation The charge continuity equation is the conservation law of charge, the total charge of the universe is constant, that's global conservation of charge; but local conservation of charge is a much stronger statement: If the total charge in some volume changes, then exactly that amount of charge must have passed in or out through the surface. The tiger can't simply rematerialize outside the cage; if it got from inside to outside it must have found a hole in the fence. Formally, the charge in a volume V encloses by closed surface A, is ∫ The current flowing out through the boundary of closed surface A is ∮ (∫ ) ∮ ∫∮ ∮ ∫

This is the continuity equation-the precise mathematical statement of local conservation of charge. conservation of charge is not an independent assumption, but a consequence of the laws of electrodynamics. Derivation of the continuity equation from Maxwell’s equations The charge continuity equation can be derived from Maxwell's equations as follows: The Maxwell’s equations in differential form are Take the divergence o fourth equation, then ) ( The left side is the divergence of curl and equals zero. Substituting about .E from first equation, then ()

This is charge continuity equation derived from Maxwell’s equations. Energy conservation law Pointing theorem: work-energy theorem of electrodynamics The Poynting Theorem is in the nature of a statement of the conservation of energy for a configuration consisting of electric and magnetic fields acting on charges. Consider a volume V encloses with a closed surface A. Then, the time rate of change of electromagnetic energy within the volume V plus the net energy flowing out of V through A per unit time is equal to the negative of the total work done on the charges within V. Consider first a single particle of charge q traveling with a velocity vector . Let E and B be electric and magnetic fields external to the particle; i.e., E and B do not include the electric and magnetic fields generated by the moving charged particle. The force on the particle is given by the Lorentz formula  The work done by the electric field on that particle is equal to The power  The work done by the magnetic field on the particle is zero because the force due to the magnetic field is perpendicular to the velocity vector v.

For a vector field of current density J the work done on the charges within a volume V, electric power density, is ∫ For a single particle of charge q traveling with velocity v the above quantity reduces to qv·E. One form of the Ampere-Maxwell's Law says that Then Then ( ) Using the identity Then And substituting from third Maxwell’s equation

() () Substituting in equation of E.J, then ) () ) ( () ( Using the rules Then

() ) ( () ) ( ) ( ( )() () ∫ ) ) ∫( ( ∫( )∫ Using divergence theorem

Then ∮∫ Then ∮∫ ∫( )∫ ---------------------------------------------------------------------------------------- Previously, it is found the energy density of electric and magnetic fields are The power density. electric power per unit volume, Then

∫ ∫ ∫ Therefore, the total energy stored in electromagnetic field is ) ∫ ∫ ∫( This is the energy conservation law for electrodynamics.   From the fourth Maxwell’s equation Then ( )

Momentum conservation law Newton’s third law of electrodynamics Maxwell’s stress tensor Let's calculate the total electromagnetic force on the charges in volume V:  The volume charge density is Then ∫ ∫ ∫ The current density J is given by Then  ∫( ∫∫ ) This is the force per unit volume Find the charge density  from first Maxwell’s equation, and the current density from fourth Maxwell’s equation:

First Maxwell’s equation Then Fourth Maxwell’s equation Then Substituting by  and J in the force equation, it gives () Now ( )( ) Then ( )( ) From third Maxwell’s equation

Then ()

Electromagnetic waves Wave equation A wave is a disturbance of a continuous medium that propagates with a fixed shape at constant velocity. In the presence of absorption, the wave will diminish in size as it moves; if the medium is dispersive different frequencies travel at different speeds; in two or three dimensions, as the wave spreads out its amplitude will decrease; and of course standing waves don't propagate at all. But these are refinements. let's start with the simple case: fixed shape, constant speed (Fig. (--). How would you represent such an object mathematically? In the figure I have drawn the wave at two different times, once at t = 0, and again at some later time t- each point on the wave form simply shifts to the right by an amount vt, where v is the velocity. Maybe the wave is generated by shaking one end of a taut string; f (z, t) represents the displacement of the string at the point z, at time t. Given the initial shape of the string, what is the subsequent form, f(x,t)? Evidently, the displacement at point x, at the later time t, is the same as the displacement a distance vt to the left (i.e. at x - vt), back at time t = 0:

That statement captures (mathematically) the essence of wave motion. It tells us that the function f(x, t), which might have depended on x and t in any old way, in fact depends on them only in the very special combination x - vt; when that is true, the function f(x, t) represents a wave of fixed shape traveling in the x direction at speed v. Wave equation for stretched string Imagine a very long string under tension T. If it is displaced from equilibrium, the net transverse force on the segment between x and x+ x as in fig (--) ……………………………….. ……………………………………….. …………………………………………….

 √ Sinusoidal waves Of all possible wave forms, the sinusoidal one Where A is the amplitude of the wave (it is positive, and represents the maximum displacement from equilibrium). The argument of the sine is called the phase, and is the phase constant (obviously, you can add any integer multiple of 2 to without changing f(x, t); ordinarily, one uses a value in the range Notice that at x = vt - /k, the phase is zero; let's call this the \"central maximum.\" If = 0, the central maximum passes the origin at time t = 0; more generally, /k is the distance by which the central maximum (and therefore the entire wave) is \"delayed.\" Finally, k is the wave number; it is related to the wavelength  by the equation 

for when x advances by 2/ k, the sine executes one complete cycle. As time passes, the entire wave train proceeds to the right, at speed . At any fixed Point x, the string vibrates up and down, undergoing one full cycle in a period time  Where is the frequency (the number of oscillations per unit time) The speed of wave is given by  For our purposes, a more convenient unit is the angular frequency ev, so-called because in the analogous case of uniform circular motion it represents the number of radians swept out per unit time: ()   A sinusoidal oscillation of wave number k and (angular) frequency ev traveling to the left would be written

Electromagnetic waves in vacuum Energy and momentum in electromagnetic waves Previously, it is showed that the energy density stored in electric and magnetic fields are Then, the energy stored in electromagnetic field is ) ( In the case of a monochromatic plane wave Then

Substitute about the sinusoidal wave equation of E as Then Electromagnetic waves in vacuum Electromagnetic waves in matter

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√ √ √ Absorption and dispersion of electromagnetic waves Electromagnetic waves in conductors: skin depth Ohm’s law in a conductor is Then the Maxwell’s equations in a conductor are And the continuity equation is Together with Ohm's law and first Maxwell’s equation, the continuity equation is () Then it gives