Electromagnetic Theory PDF

Safia Ahmad Electromagnetic Theory If there are some electric charges, and we want to find the force on a test charge Q due to these charges then that can be done by using the principle of superposition which states that the interaction between any two charges is not...

Safia Ahmad Electromagnetic Theory If there are some electric charges, and we want to find the force on a test charge Q due to these charges then that can be done by using the principle of superposition which states that the interaction between any two charges is not affected by the presence of others. So, if we want to find a force on some test charge Q due to several source charges qi , then this can be done by determining forces on Q due to individual charges, q1 , q2 ,.. and taking the vector sum of them. We first consider a case of electrostatics where all the source charges are at rest. Electric Field The force on a test charge Q due to a single point charge q which is at rest a distance r away is given by the Coulomb’s Law, 1 qQ F= 4πϵ0 2 r r̂ (1) where, ϵ0 = 8.85 × 10−12 C2 /(Nm2 ) → permittivity of free space. Thus, the force is pro- portional to the product of charges and inversely proportional to the square of separation distance. The force is repulsive if q and Q have the same sign and attractive if their signs are opposite. r is the separation vector from r (the location of q) to r (the location of Q) : ′ r=r−r ′ 1  Safia Ahmad If there are several point charges q1 , q2 ,..., qn , at distances r , r ,..., r 1 2 n from Q, the total force on Q is   1 q1 Q q2 Q F = F1 + F2 +... = 4πϵ0 1 2 r r̂ + r r̂ 1 2 2 2 +... = Q  q1 1 q2 2 r̂ + 2 +... r̂ 4πϵ0 2 1 r 2 r Thus, F=QE where, N 1 X qi E(r) ≡ 4πϵ0 i=1 2i r r̂ i with E being the electric field of the source charges. Electric field is a vector quantity that varies from point to point and is determined by the configuration of the source charges. Physically, E(r) is the force per unit charge, produced by the source charges,qi that would be exerted on a test charge Q, if it were to place at P. Continuous Charge Distributions So far it was assumed that the source charges are the collection of discrete point charges. If the charge is distributed continuously over a region, the sum is replaced by an integral, Z 1 1 E(r) = 4πϵ0 2 dq r r̂ So, if the charge is distributed along a line with charge per unit length λ then dq = λ dl. If the charge is spread out over a surface with charge per unit area σ then dq = σ da where da is the area element. And if the charge is distributed over a volume with charge per unit volume ρ then dq = ρ dτ where dτ is the volume element. 2  Safia Ahmad 1 Gauss’s Law Electric field of a single point charge q situated at the origin is 1 q E (r) = r̂ (2) 4πϵ0 r2 If the point charge q is positive then electric field is directed radially outwards and since the field falls off as 1/r2 , the electric field vector gets shorter as you go farther away from origin. Instead of connecting arrows, the field can be represented by the field lines. The field lines emanate from a point charge symmetrically in all directions. And the magnitude of the field is indicated by the density of field lines; it’s strong near the center where the field lines are close together, and weak farther out where they are relatively far apart. If the charge q is negative then field lines are directed radially inwards. Thus, field lines originate on positive charges and end on negative ones. Also, field lines can never intersect as it is not possible for the electric field to have two different directions at a point. The strength of electric field is proportional to the density of field lines i.e. the number of field lines per unit area. The flux of electric field E through a surface S, Z ΦE ≡ E. da S is a measure of the number of field lines passing through S. Hence, E. da is proportional to the number of lines passing through the infinitesimal area da; the dot product picks out the component of area vector da along the direction of E. 3  Safia Ahmad In the case of a point charge, q, at the origin, the flux of E through a sphere of radius r is I Z   1 q 2
q E. da = 2 r̂. r sin θ dθ dϕ r̂ = 4πϵ0 r ϵ0 where we have used the fact that area vector on the surface of sphere is da = (r2 sin θ dθ dϕ r̂) and electric field E due to point charge q is given by (2). Note that the electric flux through a sphere enclosing charge q is independent of the radius of the sphere. This makes sense as the number of field lines passing through any sphere (or any closed surface for that matter) would be same regardless of its size. This is the Gauss’s law which states that: The flux through any surface enclosing the charge is q/ϵ0. Now if instead of a single charge at the origin, there are a bunch of charges scattered about then according to the principle of superposition, the total field is the vector sum of all the individual fields, n X E= Ei i=1 Thus, flux through a surface that encloses all the charges is I n I  X n   n X 1 1 X E. da = Ei. da = qi = qi i=1 i=1 ϵ0 ϵ0 i=1 For any closed surface I 1 E. da = Qenc (3) S ϵ0 where, Qenc is the total charge enclosed within the surface. This is the quantitative statement of Gauss’s law in integral form. This can be written in differential form by applying the divergence theorem (also known as Gauss’s theorem or Green’s theorem) which states that: Z I (∇.v) dτ = v. da V S That is, it says integral of a divergence over a volume is equal to the value of the function at the boundary i.e. the surface that bounds the volume. Therefore, on applying the divergence theorem, we have I Z E. da = (∇.E) dτ S V 4  Safia Ahmad Also, in terms of charge density ρ, Qenc can be rewritten as Z Qenc = ρ dτ V Thus, Gauss’s law becomes Z Z   Qenc ρ (∇.E) dτ = = dτ V ϵ0 V ϵ0 And because this is valid for any volume, the integrands must be equal, ρ ∇.E= (4) ϵ0 This is Gauss’s law in differential form. The Curl of E The electric field of a point charge q situated at the origin is 1 q E= r̂ 4πϵ0 r2 The line integral of this field from a to b is Z b E. dl a In spherical coordinates, the line element is dl = dr r̂ + r dθ θ̂ + r sin θ dϕ ϕ̂, so 1 q E. dl = dr 4πϵ0 r2 Z b Z b   1 q −1 q rb 1 q q ∴ E. dl = dr = = − a 4πϵ0 a r2 4πϵ0 r ra 4πϵ0 ra rb where ra is the distance from the origin to point a and rb is the distance from the origin to point b. The integral around a closed path would be zero as ra = rb , I E. dl = 0 (5) Using Stoke’s theorem which states that Z I (∇ × v). da = v. dl (6) S P 5  Safia Ahmad i.e. the integral of a curl over a surface is equal to the value of function at the boundary i.e. the perimeter of the surface. Therefore, we have I Z E. dl = (∇ × E). da = 0 S Therefore, ∇×E=0 (7) This holds for any static charge distribution. Electric Fields in Matter In conductors, many of the electrons (one or two per atom in a typical metal) are not associ- ated with any particular nucleus, but roam around at will. While in dielectrics (insulators), all charges are attached to specific atoms or molecules; they are tightly attached to the nucleus and can only move a bit within the atom or molecule. Although the atom as a whole is electrically neutral; there is a positively charged nucleus and a negatively charged electron cloud that surrounds it. So, when it is placed in an external electric field E, the nucleus is pushed in the direction of the field and the electrons in the opposite way. If the field is large enough, it can pull the atom apart completely, ionizing it. While for moderate fields, equilibrium is soon reached and the centre of the electron cloud no longer coincides with the nucleus. So, the two opposing forces – E pulling the electrons and nucleus apart and the mutual attraction between the nucleus and electrons drawing them together – reach a balance, leaving the atom polarized, with plus charge shifted one way and the minus the other way. Thus, the atom has a tiny dipole moment p which points in the same direction as E (as the direction of dipole moment is from negative to positive). This induced dipole moment is approximately proportional to the field, p = αE where α is the constant of proportionality known as atomic polarizability. So, when a dielectric material consisting of neutral atoms is placed in an electric field, the field will induce tiny dipole moments in each atom, pointing in the same direction as 6  Safia Ahmad the electric field. In a material made up of polar molecules, each permanent dipole will experience a torque which tend to line them up along the field direction. Thus, there are lots of tiny dipoles pointing along the direction of the field making the material polarized. To, measure this effect, we define the polarization, P ≡ dipole moment per unit volume The effect of polarization in a dielectric material is to produce accumulations of volume bound charge, ρb = −∇. P within the dielectric and surface bound charge, σb = P. n̂ on the surface. The field due to polarization of the medium is just the field of this bound charge. The total electric field will be the field due to bound charge plus the field due to everything else (which we call the free charge). The free charge might consist of electrons on a conductor or ions embedded in the dielectric material i.e. any charge which is not a result of polarization. So, the total charge density within a dielectric can be written as ρ = ρb + ρf where, ρf is the free charge density. Therefore, Gauss’s law becomes ρ ∇.E= =⇒ ϵ0 ∇. E = ρ = ρb + ρf ϵ0 ∇. E = −∇. P + ρf where E is the total field. The above equation may be rewritten as ∇. (ϵ0 E + P) = ρf (8) where the expression in the parenthesis, D ≡ ϵ0 E + P (9) 7  Safia Ahmad is known as the electric displacement. Thus, in terms of D, Gauss’s law is ∇. D = ρf (10) This is the Gauss’s law in the presence of dielectrics. In integral form Z Z (∇. D) dτ = ρf dτ I =⇒ D. da = Qf enc Z I Z ∵ (∇. D) dτ = D. da, and ρf dτ = Qf enc where Qf enc is the total free charge enclosed in a volume. Linear Dielectrics For many substances, the polarization is proportional to the electric field provided that the electric field E is not very strong, therefore P = ϵ0 χe E (11) where χe is the constant of proportionality known as the electric susceptibility of the medium. χe is dimensionless. Thus, in linear media, D = ϵ0 E + P = ϵ0 E + ϵ0 χe E = ϵ0 (1 + χe )E (12) so D is also proportional to E, D=ϵE where, ϵ ≡ ϵ0 (1 + χe ) is the permittivity of the material. The relative permittivity or dielectric constant, ϵ ϵr ≡ 1 + χe = ϵ0 8  Safia Ahmad Moving Charges and Magnetism In 1820, Hans Christian Oersted discovered that the flow of current in a straight wire produces magnetic field around it. The direction of magnetic field can be found by right hand thumb rule. If the thumb points in the direction of current flow then the curled fingers give the direction of magnetic field. Motion of a charged particle in magnetic field The magnetic force in a charge q, moving with velocity v in a magnetic field B is Fmag = q(v × B) (13) This is known as the Lorentz force law. The magnitude of magnetic force is Fmag = q v B sin θ where θ is the angle between v and B. In the presence of both electric and magnetic fields, the net force on q would be F = q[E + (v × B)] (14) If a charge q moves a distance dl = v dt in time dt, the work done by the magnetic force is dWmag = Fmag. dl = q(v × B). v dt = 0 9  Safia Ahmad where we have used the fact that since the curl (v × B) is perpendicular to the velocity v, therefore the dot product ((v×B). v) is zero i.e. magnetic forces do no work. And therefore, there is no change in kinetic energy. Thus, magnetic forces may alter the direction in which a particle moves but they cannot speed it up or slow it down. If a charged particle q is moving with velocity v in a uniform magnetic field B such that the velocity v is perpendicular to B, then the magnetic force, qv × B which is perpendicular to velocity acts as a centripetal force. Thus, the charged particle moves in a circular path perpendicular to the magnetic field. So, if θ = 90◦ , mv 2 mv = q v B =⇒ R = R qB where m is the mass of a charged particle and R is the radius of the circle in which the charged particle moves. If velocity is not perpendicular to the magnetic field then it has a component along B and a component perpendicular to B. The motion in a plane perpendicular to B is as before a circular motion. However, the component of velocity in the direction of B remains unchanged as the magnetic force due to this component is zero. Thus, the particle will move in a helical path. Velocity Selector Now, if a positive charged particle q is moving with velocity v in the presence of both electric and magnetic field then the net force on the charged particle is F = q[E + (v × B)] (15) Consider a case where electric and magnetic fields are perpendicular to each other and these fields are also perpendicular to the velocity of the particle. The directions are chose in a manner such that the electric force and the magnetic force are in opposite directions. So if v = v î, B = B k̂, E = E ĵ then Fe = q E ĵ, Fmag = q(v î × B k̂) = −q v B ĵ 10  Safia Ahmad so that, F = q(E − vB) ĵ The path that the charged particle takes in such a situation depends on which force dominates. So, if the magnetic force dominates over electric force, charged particle will move downwards in a circular path. And, if the electric force dominates, the charge particle will move upwards in a circular path. If more than one charged particles are moving with different velocities, there may be some particles whose speed v is such that the magnitude of magnetic and electric forces are exactly equal, i.e. Fmag = Fe (16) and since they are in the opposite directions, they exactly cancel each other and the charge particle will experience no force and will continue to move in a straight line in a positive x-direction. The velocity of these charged particles, v, can be obtained as E Fmag = Fe =⇒ q v B = q E =⇒ v = (17) B We can therefore select a velocity of a charge particle moving in an electric and magnetic field by tuning the magnitude of electric and magnetic field so that the charge particle has that particular velocity v. So, if we have a large number of charged particles, only those particles will come out of this setup undeflected which have the velocity equal to E/B. So, we have found a way to select the charged particles moving with very specific velocity. The crossed E and B fields, therefore serve as velocity selector. Currents The current in a wire is a charge per unit time passing a given point. Current is measured in amperes (A) or coulombs-per-second. If a line charge λ (which is charge per unit length) is travelling down a wire at a speed v then as the infinitesimal line element dl = v dt carries a charge dq = λ dl = λ v dt, the current flowing through the wire is dq I= =λv dt 11  Safia Ahmad Since the direction of current flow is dictated by the shape of the wire, the vectorial character of I is not required to be mentioned explicitly for the case of line charge and therefore we can simply write the current as I = λ v. The charge density λ refers only to the moving charges here. If the flow of charge is distributed throughout a three di- mensional region, the volume current density is defined as the current dI flowing through a tube of infinitesimal cross section da⊥ , running parallel to the flow. Thus, the volume current density is dI J= da⊥ So, J is the current per unit area-perpendicular-to-the-flow. Therefore, the current flowing out of a surface S can be written as Z Z I= J da⊥ = J. da S since the dot product picks out the appropriate component of da. Continuity Equation The total current flowing out through the boundary S is Z I = J. da Thus, the total charge per unit time leaving a volume V can be written using the divergence theorem as I Z I= J. da = (∇. J) dτ S V Because charge is conserved therefore whatever flows out of the surface S must be exactly equal to the change in total charge enclosed by the volume V which is bounded by the surface S, Z Z Z   dQenc d ∂ρ (∇. J) dτ = − =− ρ dτ = − dτ V dt dt V V ∂t 12  Safia Ahmad The minus sign reflects the fact that an outward flow decreases the charge left in volume V. Since this applies to any volume, therefore we conclude that ∂ρ ∇.J=− (18) ∂t This is the continuity equation; a mathematical statement of local charge conservation which is If the total charge in some volume changes, then exactly that amount of charge must have passed in or out through the surface. Biot-Savart Law As stationary charges produce electric fields that are constant in time hence the theory of stationary charges are given the name electrostatics. Similarly, steady currents produce magnetic fields that are constant in time and therefore the theory of steady currents is called magnetostatics. Steady current means a continuous flow of charge that has been going on forever without change and without charge piling up anywhere. So, when a steady current flows in a wire, its magnitude I must be same along the length of the wire because otherwise charge would be piling up somewhere and it wouldn’t be a steady current. Thus, ∂ρ/∂t = 0 in magnetostatics and therefore the continuity equation becomes ∇. J = 0, (in magnetostatics) The magnetic field of a steady line current at a distance r is given by the Biot-Savart law: B(r) = µ0 Z I× r̂ dl = µ ′ 0 I Z dl′ × r̂ 4π r2 4π r2 The integration is along the direction of current flow. dl′ is a length element along the wire and r is the vector from the source r’ to the point r. The constant, µ0 = 4π × 10−7 N/A2 is known as the permeability of free space. The unit of magnetic field is in newtons per ampere-meter or Tesla. 13  Safia Ahmad Ampere’s Law The Ampere’s law states that the line in- tegral of magnetic field B around a closed loop is equal to µ0 times the total current passing through the sur- face. The current outside of the closed loop does not contribute to the line integral. I B. dl = µ0 Ienc (19) where Ienc stands for the total current enclosed by the loop. This is a qualitative statement of Ampere’s law. For example, the magnetic field of an infinite wire at a distance s from the current carrying wire is µ0 I B= 2πs If the current I is coming out of the page, then B curls around it in anticlockwise direction i.e. the curl of B is non-zero. The direction of magnetic field at every point on this circle is tangent at that point. I I I µ0 I µ0 I µ0 I B. dl = dl = dl = 2πs = µ0 I (20) 2πs 2πs 2πs If the flow of charge is represented by a volume current density J, the enclosed current is Z Ienc = J. da (21) where the integral is over the surface bounded by the loop so that the area vector is perpen- dicular to the loop at every point. Substituting this in (19) we have I Z B. dl = µ0 J. da Applying Stoke’s theorem to the line integral, Z Z (∇ × B). da = (µ0 J). da ∴ ∇ × B = µ0 J 14  Safia Ahmad This is the Ampere’s law in differential form. Magnetic flux The flux of magnetic field B through the loop is given as, Z ΦB ≡ B. da This gives the number of magnetic field lines passing through a surface da. The magnetic field lines for a closed loop look very similar to the magnetic field lines of a bar magnet. They also look similar to the electric field lines of an electric dipole if we look at some distance away from the dipole. Gauss’s law in electrostatics tells us that the closed surface integral of the electric field is the charge enclosed divided by ϵ0. That is closed surface integral of electric flux is not zero because there is a net charge inside the closed surface. However, if an electric dipole is enclosed by the surface then the net charge enclosed by the surface would be zero and therefore the electric flux through the surface would be zero. For the case of magnetic fields, no matter where in the magnetic field we make a closed surface, the net magnetic flux going through the surface is always zero because magnetic monopole does not exist. Thus, the magnetic flux i.e. the magnetic field lines through the surface is Z B. da = 0 unless there exists a magnetic monopole. This is the second Maxwell’s equation in integral form. The differential form of this Maxwell’s equation can be obtained by applying the divergence theorem, Z Z B. da = (∇. B) dτ = 0 S S =⇒ ∇. B = 0 This is the second Maxwell’s equation in differential form. 15  Safia Ahmad Figure 1: Caption Electromagnetic Induction We have seen that if a current is passed through a wire or a coil, it generates magnetic field around it i.e. Electric current generates magnetic field. Michael Faraday carried out a series of experiment to know whether the reverse is also possible or not i.e. whether magnetic field can also generate electric currents. It turned out that steady magnetic field i.e. the magnetic fields which are constant in time, cannot generate electric currents. In 1831, Michael Faraday then reported a series of experiments, including three that can be characterized as: Experiment 1: He pulled a loop of wire to the right through a magnetic field. A current flowed in the loop. Experiment 2: He moved the magnet to the left, holding the loop still. Again, a current flowed on the loop. Experiment 3: With both the loop and magnet at rest, he changed the strength of the field. Again, a current flowed on the loop. 16  Safia Ahmad Faraday concluded that A changing magnetic field induces an electric field. That is, whenever the magnetic flux through circuit changes, an emf is induced in the circuit. This phenomenon is called Faraday’s Law of Electromagnetic Induction. So, whenever and for whatever reason the magnetic flux through the loop changes, an emf is induced in the loop whose magnitude is equal to the time rate of change of the magnetic flux, i.e. dΦB E =− dt. To find the total emf induced around the whole circuit, we sum up the emf produced at each point over the length of the wire. This is known as a line integral. However, an emf implies the existence of electric field. Thus, emf is also given as I E = E. dl i.e. total emf around the circuit is equal to summing up electric field around the length of the circuit. Therefore, we have I dΦB E = E. dl = − dt I Z Z d =⇒ E. dl = − B. da ∵ ΦB = B. da dt I Z ∂B =⇒ E. dl = −. da ∂t This is the Faraday’s law in integral form. The Faraday’s law in differential form can be obtained by applying the Stoke’s theorem to the line integral, Z Z ∂B ∇ × E. da = −. da ∂t ∂B =⇒ ∇ × E = − ∂t This is the Faraday’s law in differential form. For the static case i.e. constant B, Faraday’s H law reduces to the old rule E. dl = 0. 17  Safia Ahmad Maxwell’s equations Electrodynamics Before Maxwell So far, the divergence and curl of electric and magnetic fields have given us the following equations: 1 (i) ∇.E= ϵ0 ρ, (Gauss’s law), (ii) ∇. B = 0, (no name), (22) (iii) ∇ × E = − ∂B ∂t , (Faraday’s law), (iv) ∇ × B = µ0 J, (Ampere’s law) These were the equations of electromagnetic theory before Maxwell’s. Maxwell found out that there is inconsistency in one of these formulas. As the divergence of curl is always zero, if we take the divergence of equation (iii), we find that   ∂B ∂ ∇. (∇ × E) = ∇. − = − (∇. B) = 0 ∵∇.B=0 ∂t ∂t So, this equation is consistent with the fact that the divergence of curl is always zero. If we now take the divergence of equation (iv), we have ∇. (∇ × B) = µ0 (∇. J) Here, the left side must be zero as the divergence of curl is always zero, but the right side is, generally, not zero (because continuity equation is ∇. J + ∂ρ/∂t = 0). So, the Ampere’s law is not valid for non-steady currents. Inconsistency in Ampere’s law If we take the divergence of equation (iv), we have ∇. (∇ × B) = µ0 (∇. J) Here, the left side must be zero as the divergence of curl is always zero, but the right side is in general not zero (because continuity equation is ∇. J + ∂ρ/∂t = 0). So, the Ampere’s law is not valid for non-steady currents. 18  Safia Ahmad Maxwell fixed the inconsistency in Ampere’s law as follows. From continuity equation, we have   ∂ρ ∂ ∂E ∇.J=− = − (ϵ0 ∇. E) = −∇. ϵ0 ∂t ∂t ∂t where, we have used the Gauss’s law, ∇. E = ρ/ϵ0. From the above equation, we have   ∂E ∇. J + ϵ0 =0 ∂t So, Maxwell suggested that if we combine ϵ0 ∂E/∂t with J in Ampere’s law, then the fourth equation will be consistent. Thus, ∂E ∇ × B = µ0 J + µ0 ϵ0 ∂t This is also consistent with magnetostatics: when E is constant in time, we have ∇×B = µ0 J. This is the Ampere’s law with Maxwell’s correction. Maxwell called this extra term, the displacement current, ∂E Jd =≡ ϵ0 ∂t The modified Ampere’s law in integral form can be obtained by taking the surface integral Z Z Z   ∂E (∇ × B). da = µ0 J. da + µ0 ϵ0. da ∂t Applying Stoke’s theorem, I Z   ∂E B. dl = µ0 Ienc + µ0 ϵ0. da ∂t This is modified Ampere’s law in integral form. Maxwell’s Equations In differential form, Maxwell’s equations are 1 (i) ∇.E= ϵ0 ρ, (Gauss’s law), (ii) ∇. B = 0, (no name), (iii) ∇ × E = − ∂B ∂t , (Faraday’s law), (23) ∂E (iv) ∇ × B = µ0 J + µ0 ϵ0 ∂t , (Ampere’s law with Maxwell’s correction) 19  Safia Ahmad In integral form, Maxwell’s equations are E. da = ϵ10 Qenc , H (i) (Gauss’s law), H (ii) B. da = 0, (no name), H R ∂B
(iii) E. dl = − ∂t. da, (Faraday’s law), (24) H R ∂E
(iv) B. dl = µ0 Ienc + µ0 ϵ0 ∂t. da, (Ampere’s law with Maxwell’s correction) Propagation of plane electromagnetic waves in Free Space Free space or non-conducting or lossless or in general perfect dielectric medium has following characteristics. (i) σ = 0 =⇒ J = 0, ∵ J = σE. (ii) No charges: ρ = 0. In regions of space where there is no charge or current, i.e. ρ = 0 and J = 0, the Maxwell Equations become (i) ∇. E = 0, (ii) ∇. B = 0, (25) (iii) ∇ × E = − ∂B ∂t , ∂E (iv) ∇ × B = µ0 ϵ0 ∂t , Third and fourth Maxwell equations are coupled first order differential equations for E and B. They can be decoupled by taking the curl. The curl of equation (iii) gives   ∂B ∇ × (∇ × E) = ∇ × − , [using (iii)] ∂t ∂ ∇(∇.E) − ∇2 E = − (∇ × B) ∂t   2 ∂ ∂E =⇒ 0 − ∇ E = − µ0 ϵ0 [using (i) and (iv)] ∂t ∂t (26) ∂ 2E ∇2 E = µ0 ϵ0 (27) ∂t2 20  Safia Ahmad Similarly, curl of (iv) gives   ∂E ∇ × (∇ × B) = ∇ × µ0 ϵ0 , [using (iv)] ∂t ∂ =⇒ ∇(∇.B) − ∇2 B = µ0 ϵ0 (∇ × E)  ∂t ∂ ∂B =⇒ 0 − ∇2 B = µ0 ϵ0 − [using (ii) and (iii)] ∂t ∂t ∂ 2B =⇒ ∇2 B = µ0 ϵ0 (28) ∂t2 Equations (27) and (28) are three dimensional wave equations of the form 1 ∂ 2f ∇2 f = v 2 ∂t2 This implies that the electromagnetic waves propagate with speed, 1 1 v=√ ∵ = ϵ0 µ0 ϵ0 µ0 v2 1 4π ∵ = 9 × 109 N m2 /C 2 , µ0 = 4π × 10−7 =⇒ = 107 4πϵ0 µ0 1 4π 1 =⇒ = 9 × 109 × 107 =⇒ = 9 × 1016 4πϵ0 µ0 ϵ0 µ0 1 =⇒ v = √ = 3 × 108 m/s (29) ϵ0 µ0 This is precisely the speed of light, c which implies that light is an electromagnetic wave. Propagation of plane electromagnetic waves in non-conducting medium In matter, in terms of free charges and currents, Maxwell’s equations are ∇. D = ρf ∇.B=0 ∂B ∇×E=− ∂t ∂D ∇ × H = Jf + ∂t 21  Safia Ahmad where, ρf is the free charge density and Jf is the free current density in a dielectric. Also, D is electric displacement and H is the magnetic intensity. For a linear media, 1 D = ϵ E, and H = B µ where ϵ is the permittivity of the material and µ is the permeability of the material. Thus, in case of non-conducting medium i.e. a perfectly dielectric medium, ρf = Jf = 0 and Maxwell’s equations become, (i) ∇. D = 0 =⇒ ∇. E = 0 (ii) ∇. B = 0 ∂B (iii) ∇ × E = − ∂t ∂E ∂E (iv) ∇ × H = ϵ =⇒ ∇ × B = µ ϵ ∂t ∂t Third and fourth Maxwell equations are coupled first order differential equations for E and B. They can be decoupled by taking the curl. The curl of equation (iii) gives   ∂B ∇ × (∇ × E) = ∇ × − , [using (iii)] ∂t Since, for any vector A, ∇ × (∇ × A) = ∇(∇.A) − ∇2 A Therefore, we have ∂ ∇(∇.E) − ∇2 E = − (∇ × B) ∂t   2 ∂ ∂E =⇒ 0 − ∇ E = − µϵ [using (i) and (iv)] ∂t ∂t 2 ∂ 2E =⇒ ∇ E = µ ϵ (30) ∂t2 Similarly, curl of (iv) gives   ∂E ∇ × (∇ × B) = ∇ × µ ϵ , [using (iv)] ∂t ∂ =⇒ ∇(∇.B) − ∇2 B = µ ϵ (∇ × E)  ∂t ∂ ∂B =⇒ 0 − ∇2 B = µ ϵ − [using (ii) and (iii)] ∂t ∂t ∂ 2B =⇒ ∇2 B = µ ϵ (31) ∂t2 22  Safia Ahmad Equations (30) and (31) are the three dimensional wave equation of the form 1 ∂ 2f ∇2 f = v 2 ∂t2 Thus, the wave equations (27) and (28) represents the propagation of electromagnetic waves in a non-conducting medium with speed, √ 1 1 ϵ0 µ0 1 v=√ =√ √ ∵ =ϵµ ϵµ ϵ µ ϵ0 µ0 v2 c 1 =⇒ v = ∵c= n ϵ0 µ0 r ϵµ where, n = (32) ϵ0 µ0 is the index of refraction for the material. 23