Maxwell’s Equations in Differential Form ∂B ∇ × E = − − M = −M − M ∂t i d i ∂D ∇ × H = Ji + Jc + = Ji + Jc + J d ∂t ∇ ⋅ D = ρ ev ∇ ⋅ B = ρ mv ∂B ∂D M = , J = d ∂t d ∂t E ≡ Electric field intensity [V/m] B ≡ Magnetic flux density [Weber/m2 = V s/m2 = Tesla] 2 M i ≡ Impressed (source) magnetic current density [V/m ] 2 M d ≡ Magnetic displacement current density [V/m ] H ≡ Magnetic field intensity [A/m] 2 Ji ≡ Impressed (source) electric current density [A/m ] D ≡ Electric flux density or electric displacement [C/m2] 2 Jc ≡ Electric conduction current density [A/m ] 2 J d ≡ Electric displacement current density [A/m ] 3 ρev ≡ Electric charge volume density [C/m ] 3 ρ mv ≡ Magnetic charge volume density [Weber/m ] Remarks: 1. Impressed magnetic current density ( M i ) and magnetic charge density ( ρ mv ) are unphysical quantities introduced through “generalized” current to balance Maxwell’s equations. 2. Although unphysical, M i , ρ mv similar to J i and ρev can be considered as energy sources that generate the fields. J c ∂D J d = J i dt Current source J i generates the electric displacement current 3. Through “equivalent principle” M i and ρ mv can be used to simplify the solutions to some boundary value problems. ∂B 4. M = (magnetic displacement current density [V/m2]) is introduced d ∂ t ∂D analogous to J = (electric displacement current density [A/m2]) d ∂t Integral form of Maxwell’s Equations Elementary vector calculus: S C Stokes’ Theorem: ∫∫(∇× A)⋅ds = ∫ A⋅dl S C • It says that if you want to know what is happening in the interior of a surface bounded by a curve just go around the curve and add up the field contributions. volume V surface S Divergence Theorem: ∫∫∫(∇ ⋅ A) dV = ∫∫ A⋅ds V S • In simple words, divergence theorem states that if you want to know what is happening within a volume of V just go around the surface S (bounding volume V ) and add up the field contributions. Null Identities: ∇ ⋅ (∇ × A) = 0 ⇔ ∇ ⋅ B = 0 ⇔ B = ∇ × A ∇ × (∇φ) = 0 ⇔ ∇ × E = 0 ⇔ E = −∇φ (electrostatic) • The Divergence and Stokes’ theorems can be used to obtain the integral forms of the Maxwell’s Equations from their differential form. ∂B ∂ ∇ × = − − ⇒ ∇ × ⋅ = − ⋅ − ⋅ • E M i ∫∫ E ds ∫∫ B ds ∫∫ M i ds ∂t S ∂t S S S C ∂ ⇒ E ⋅ dl = − B ⋅ ds − M ⋅ ds ∫ ∂t ∫∫ ∫∫ i C S S ∂D ∂D ∇ × = + σ + ⇔ ∇ × ⋅ = ⋅ + σ ⋅ + ⋅ • H E J i ∫∫ H ds ∫∫ ds ∫∫ E ds ∫∫ J i ds ∂t S S ∂t S S ∂ ⇒ H ⋅ dl = D ⋅ ds + J ⋅ ds + J ⋅ ds ∫ ∂t ∫∫ ∫∫ c ∫∫ i C S S S Where Jc = σ E ∇ ⋅ = ρ ⇔ ∇ ⋅ = ρ ⇒ ⋅ = ρ = • D ev ∫∫∫ D dv ∫∫∫ ev dv ∫∫ D ds ∫∫∫ ev dv Qe V V S V S ∇ ⋅ = ρ ⇔ ∇ ⋅ = ρ ⇒ ⋅ = = ρ • B mv ∫∫∫ Bdv ∫∫∫ mvdv ∫∫ B ds Qm ∫∫∫ mvdv V V V = ⇒ ⋅ = Since Qm 0 ∫∫ B ds 0 S V Helmholtz Theorem τ • Traditionally, Newtonian mechanic is formulated in terms of force ( F ) and torque ( ), dP dL F = , τ = where L is the angular momentum. dt dt However such an approach to classical electromagnetism will be unnecessarily cumbersome. Instead, the description of electromagnetics starts with Maxwell’s equations which are written in terms of curls and divergences. The question is then whether or not such a description (in terms of curls and divergences) is sufficient and unique? The answer to this question is provided by Helmholtz Theorem • A vector field is determined to within an additive constant if both its divergence and its curl are specified everywhere. • Equivalent statement: A vector field is uniquely specified by giving its divergence and its curl within a region and its normal component over the boundary, that is if: S and C are known and given by ∇ ⋅ = M S, ∇ × = M C and M n (the normal component of M on the boundary) is also known; then M is uniquely defined. Remark: Helmholtz’s theorem allows us to appreciate the importance of the Maxwell’s equations in which E and H are defined by their divergence and curl. ∂ ρ Ex.: ∇ × E = − B and ∇ ⋅ E = ev ∂t ε Irrotational & Solenoidal Fields (Use of Helmholtz Theorem) Definition: • A field is irrotational if its curl is zero ∇ × Fi = 0 ≡ Fi is irrotational • A field is solenoidal (divergenceless) if its divergence is zero ∇ ⋅ Fs = 0 ≡ Fs is solenoidal Theorem: • A vector field which its divergence and curl vanishes at infinity can be written as the sum of an irrotational & a solenoidal fields. • According to the theorem stated above, the vector field M can be written as (1) M = Fi + Fs • Since F is irrotational then ∇× F = 0 ⇒ F = −∇V whereV is a scalar function. i i i • Since Fs is solenoidal then ∇ ⋅ Fs = 0 ⇒ Fs = ∇ × A then (1) ⇒ M = −∇V + ∇× A Constitutive Relations D = ε E B = µ H ε = ε r ε 0 µ = µr µ0 ε ≡ permittivity [F/m] -12 ε o ≡ vacuum permittivity = 8.85 ×10 [F/m] ε r ≡ Relative permittivity or dielectric constant [#] µ ≡ permeability [H/m] -7 µ0 ≡ free space permeability = 4π × 10 [H/m] µr ≡ relative permeability [#] • We also write (1) µr = 1+ χ m (2) ε r = 1+ χ e Where χ m and χ e are the magnetic and electric susceptibility, respectively. χ m , χ e are dimensionless. • Index of refraction is defined as (3) n = ε r µr n ≡ index of refraction or phase index [#] • If we are mostly concerned with non-magnetic materials then µr ≈ 1 ⇒ µ = µ0 ⇒ n = ε r Polarization and Magnetization • Polarization vector P and magnetization vector M are related to D and E and B and H according to: (4) D = P + ε 0 E (5) B = µ0 H + µ0 M = µ0 (H + M ) • Assuming P = ε χ E , then: 0 e D = ε E + P = ε E + ε χ E = ε (1+ χ )E = ε ε E ⇒ 0 0 0 e 0 e 0 r D = εE • Assuming M = χ H then m B = µ H + µ M = µ (1+ χ )H = µ µ H ⇒ 0 0 0 m o r B = µH • ε and µ describe the macroscopic response of the media.ε characterizes the electric response while µ describes the magnetic response. In the following we assume our medium is nonmagnetic. Homogeneous vs. Inhomogeneous, Isotropic vs. Anisotropic, Linear vs. Non-Linear • If ε depends on position, i.e.ε (r), media is non-homogeneous. • If ε depends on the direction of the applied field, i.e., D and E are not co-linear, then the medium is said to be anisotropic. Examples of anisotropic materials are Calcite (uniaxial) or topaz (biaxial). • In the case of anisotropic mediumε is a tensor (for our purposes a matrix of rank 2). We then write: (1) D = ε E where Dx ε xx ε xy ε xz Ex = ε ε ε ε (2) Dy 0 yx yy yz Ey ε ε ε Dz zx zy zz Ez • If ε depends on the magnitude of the applied field, i.e. ε (E ), we say medium is nonlinear. Note that in this case even though permittivity is a function of the filed strength, it can still be a scalar function. • An example of non-linear medium is when −1/ 2 ε 1 = + 2 2 − 2 (3) 1 2 (c B E ) , ε 0 b Whereb is the maximum field strength. • Interesting thing about (3) is the fact that it describes the response of the vacuum, (proposed by Born & Infeld) in order to address the problem of vacuum infinite self- energy. Infinite Self Energy • A charge particle can be thought as the localization of the charge density. As a charge distribution localizes to a point charge, its electromagnetic energy grows more and more and becomes unbounded1. To avoid this infinite self-energy we can think that some saturation of field strength takes place, i.e., field strength has an upper bound. This 1/ 2 ε 1 = + 2 2 − 2 classical non-linear effect is given by 1 2 (c B E ) ε 0 b • However, there are few problems with Born & Infeld classical non-linear vacuum response. (1) The theory suffers from arbitrariness in the manner in which the nonlinearities occur. (2) There are problems with transitions to the quantum domain. (3) So far, there has been no experimental evidence of the existence of this kind of classical nonlinearities. • As to the last point, we may note that in the orbits of electrons in atoms, field strengths of 1011-1017 V/m are present. For heavier atoms, these fields can be even as large as 1021 V/m at the edge of the nucleus; yet ordinary quantum theory with linear superposition is sufficient to describe the observed phenomena with a high degree of accuracy. HW: Consider a hydrogen atom unexcited and in thermal equilibrium. Calculate the magnitude of the electric field due to its nucleus at the site of its electron. Temporal dispersion • If ε depends on frequency, i.e. ε (ω), we say the medium is dispersive (frequency dispersion) ε ω 2 ε = = + p (1) r 1 2 2 ε 0 ω0 + j γ ω −ω • Note that from (1) we can write (2) ε = ε ′ − jε ′′ • Remarks: Temporal dispersion means that the parameters describing the medium response (e.g. ε and µ ) are functions of time derivatives. Spatial dispersion means that the parameters describing the medium response (e.g. ε and µ ) are functions of space derivatives. • If a medium is linear, homogeneous, and isotropic, we say the medium is simple. 1 Recall that the potential energy (U) corresponding to two charges q1 and q2 separated by a distance r is given by: U = q1 q2 /4 π ε0 r Electric Field • Electric field due to a point charge in origin q aˆ q r q r E = 1 r = 1 = 1 where 2 3 3 z A 4πε 0 r 4πε 0 r 4πε 0 r r Observation Point r r aˆ aˆ = = and we use the shorthand notation r r r r q r = r .