Maxwell s Equations Introduction In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are D = ρ () E = 0 (2) B = 0 (3) H = J (4) In the integral form, these equations can be read as D. d = Qenclosed (5) E. dl = 0 (6) C B. d = 0 (7) H. dl = J. d (8) C Now we move on to the system where the elctric and magnetic fields are time varying. 2 Faraday s Law Oersted experimentally discovered that a steady current produces a magnetic field. Later, Faraday discovered that a time varying magnetic flux through a coil can give rise to a potential in the coil, and empirically found that V emf = N dψ dt where ψ is the magnetic flux through each turn of the coil, N is the number of turns and V emf is the induced electric potential (also known as Electro-Motive Force (EMF)). The negative sign shows that the induced voltage is such that, it opposes the flux producing the voltage. This is Lenz s Law. In other words, the direction of current flow is such (9)
that, the induced magnetic field produced by the induced current will oppose the change in original magnetic field. The equation can be modified as L L d E. dl = dt E. dl = B. d (0) B. d () for the case where the magnetic field is time varying. Applying stokes theorem, we have ( E ). B d =. d (2) E = B This is one of Maxwell s equations for time varying fields. 3 Equation of Continuity Equation of continuity is a consequence of the principle of conservation of charges. (3) The time rate of decrease of charge within a given volume must be equal to the net outward current flow through the surface of the volume. The current coming out of the closed surface, I out = J. d = dq enc dt (4) where Q enc is the total charge enclosed by a volume V, bounded by a surface. From divergence theorem, J. d =. J dv (5) V 2
and dq enc dt = d dt V ρdv = ρ dv (6) Combining the two equation, we have ρ. J dv = dv (7) V ince this relation must be true irrespective of the volume considered, J = ρ The above equation is known as the equation of continuity. (8) 4 Maxwell s Equations We now need to reconsider the curl equation for H to be suitable for time varying fields. From magnetostatics, we H = J (9) Taking divergence on both sides, we have ( H ) = J (20) We know that divergence of a curl is always zero. Thus, J = 0 (2) Clearly this statement is not true because from the continuity equations, we have J = ρ Thus the Ampere s law is to be suitably changed as follows (22) H = J + Jd (23) where J d is the additional term so that Ampere s law is applicable for time varying cases. To find J d, we take divergence on both sides of eqn.(23) J + Jd = 0 (24) Jd = J (25) Jd = ρ (26) 3
ubstituting for ρ from Gauss law, Jd = ( D) (27) Jd = D J d = D J d is known as displacement current and Ampere s law gets modified as (28) (29) H = J + D The corresponding equation in integral form is C H. dl = (30) ( J ) D +. d (3) Thus the final point form of the four Maxwell s equations are as follows D = ρ (32) B = 0 (33) B E = D H = J + Each of these equations have their own integral forms which were derived in the previous sections. 5 Boundary Conditions ince the Maxwell s equations true for any point in space, they are useful to understand the electric and magnetic fields at the interface of materials. 5. Electric Fields Consider an interface between two media as shown (34) (35) 4
Medium 2 E 2 Medium 2 Medium w E h Medium D 2 D Fig.(a) Figure : Boundary conditions for electric fields Fig.(b) The electric fields in each media can be resolved into components that are parallel to the boundary (E ) and that perpendicular to the boundary (E ). We can use Faraday s law in the loop shown in fig.(a). Let the width of the loop be w and the height be h. Then, E. dl = 0 (36) In the limit where h 0, we we 2 = 0 (37) E = E 2 (38) Thus the two parallel electric fields at the interface of two materials are identical. imilarly we can assume a Gaussian surface of height h and surface area as shown in fig.(b). Applying Gauss law to the surface in the limit as h 0, D. d = Qenclosed (39) where ρ is the charge density at the surface. ince D = ɛe, D 2 D = ρ (40) D 2 D = ρ (4) ɛ 2 E 2 ɛ E = ρ (42) where ɛ and ɛ 2 represent the permittivity of media and 2 respectively. Thus we have the boundary condition for perpendicular electric fields. 5.2 Magnetic Fields As we did in the case of electric fields, we can use the other two of Maxwell s equations to find the boundary conditions for magnetic fields. Consider an interface between two media as shown 5
Medium 2 H 2 Medium 2 Medium w K H h Medium B 2 B Fig.(a) Fig.(b) Figure 2: Boundary conditions for magnetic fields The magnetic fields in each media can be resolved into components that are parallel to the boundary (H ) and that perpendicular to the boundary (H ). We can use Ampere s law in the loop shown in fig.(a). Let the width of the loop be w and the height be h. Then, H. dl = Ienc (43) In the limit where h 0, wk wk 2 = I enc (44) ince I enc = Kw, where K is the surface current density, we can write the above equation as wh wh 2 = wk (45) H H 2 = K (46) Thus we have the boundary condition for parallel magnetic field. imilarly we can assume a Gaussian surface of height h and surface area as shown in fig.(b). Applying Gauss law for magnetic field to the surface in the limit as h 0, B. d = 0 (47) ince B = µh, B 2 B = 0 (48) B 2 =B (49) µ 2 H 2 =µ H (50) where µ and µ 2 represent the permeability of media and 2 respectively. Thus we have the boundary condition for perpendicular magnetic fields. 6
6 Helmholtz Equations We will now derive the wave equation in a linear, isotropic, homogeneous, lossless dielectric medium that is charge free. Maxwell s equations for this system are E = 0 (5) H = 0 (52) B E = D H = If we assume that the solution is a time harmonic and substituting for B and D, we have E = jωµ H (55) (53) (54) H = jωɛ E (56) Taking curl on both sides of eqn. (9), E = jωµ( H ) (57) (. E ) 2 E = jωµ( H ) (58) 2 E ω 2 µɛ E = 0 (59) imilarly, We can write the two equations as 2 H ω 2 µɛ H = 0 (60) 2 E β 2 E = 0 (6) 2 H β 2 H = 0 (62) where β 2 = ω 2 µɛ. β is known as the wave number. The velocity of the wave is given by v = ω β (63) = µɛ (64) The velocity is equal to 3 0 8 for a wave in free space (µ = µ 0, ɛ = ɛ 0 ). Eqn. () and (2) are knows as Helmholtz equations or wave equations. Without loss of generality, if we assume that the electro-magnetic wave is traveling in the positive z-direction and E has only x-component, we have 2 E z 2 β2 E = 0 (65) E = E 0 e j(ωt βz) â x (66) 7
ubstituting this in eqn. (9) H = H0 e j(ωt βz) â y (67) where H 0 = E 0 η and η = µ. Thus we have the equation of a plane electro-magnetic ɛ wave in a linear, isotropic, homogeneous, lossless dielectric medium that is charge free. 8