ECE 350 Lecture Notes - Maxwell's Equations PDF

ECE 350 Lecture Notes — Summer 10/11 - Sp 23 Erhan Kudeki Maxwell’s equations: ρ...

ECE 350 Lecture Notes — Summer 10/11 - Sp 23 Erhan Kudeki Maxwell’s equations: ρ ∇·E = ϵo ∇·B = 0 ∂B ∂t ∇×E = − 1 Overview, Maxwell’s equations ∂E = µo J + µo ϵ o. ∂t ∇×B ECE 329 introduced the Maxwell’s equations and examined their such that F = q(E + v × B), circuit implications (inductance, capacitance) and TEM plane-wave so- with lutions in homogeneous media and on “two-wire” transmission lines. H , m µo ≡ 4π × 10−7 and In ECE 350 we continue our study of the solutions and applications 1 1 F ϵo = ≈ , of Maxwell’s equations with a focus on: µo c 2 36π × 109 m in mksA units, where 1 m 1. Radiation of spherical TEM waves from practical compact antennas (e.g., c= √ µo ϵ o ≈ 3 × 108 s used in cell phones and wireless links). is the speed of light in free space. 2. Propagation, reflection, and interference of TEM waves in 3D geometries. (In Gaussian-cgs units Bc is used 1 in place of B above, while ϵo = 4π 3. Antenna reception and link budgets in communication applications. and µo = ϵo1c2 = 4π c2.) 4. Dispersion effects in frequency dependent propagation media. 5. Guided waves in TEM, TE, and TM modes. 6. Field fluctuations in enclosed cavities + thermal noise in fields and circuits ECE 350 completes the introductory description of electromagnetic (EM) effects in our curriculum and prepares the student for specialization courses in EM (ECE 447, 452, 453, 454, 455, 457, 458, etc.) and applications. 1 Review: ✬ ✬ ✩ ✩ Microscopic applications: Maxwell’s Equations: ρ and J describe compact (pointlike) sources, ∇·D = ρ Gauss’s law D = ϵoE and B = µoH ∇·B = 0 ∂B Macroscopic applications: Faraday’s law ∂t ∇×E = − ∂D ρ and J describe smooth sources com- Ampere’s law ∂t ∇×H = J+ posed of free charge carriers, where ⇒ D = ϵE and B = µH ✫ ✪ specified in the in frequency domain with ω dependent – permittivity ϵ and ✫ – permeability µ. ✪ Fields E and B determine how a “test charge” q with mass m, position dt accelerates in accordance with r, and velocity v ≡ ṙ = dr Lorentz force F = q(E + v × B) and Newton’s 2nd law d F= mv. dt 2  D+ n̂ ✬ ✩✬ ✩ w D− Maxwell’s Equations: Boundary Conditions: = ρs Units in mksA sys- ∇·D = ρ n̂ · [D+ − D−] tem: ∇·B = 0 n̂ · [B+ − B−] = 0 ∂B q[=]C=sA, n̂ × [E+ − E−] = 0 ∂t ∇×E = − = Js ρ[=]C/m3 , ∂D n̂ × [H+ − H−] J[=]A/m2 , ∂t ∇×H = J+ where n̂ is a unit normal to the E[=]N/C=V/m, boundary surface pointing from − ✫ ✪to + side. D[=]C/m2 [=]ρs , B[=]V.s/m2 =Wb/m2 =T, ✫ ✪ Note: the same units for H[=]A/m[=]Js – Displacement D and surface charge density ρs, where – Magnetic field intensity H and surface current density Js. C, N, V, Wb, and T are abbreviations for Coulombs, Newtons, Volts, In right-handed Cartesian coordinates div, grad, and curl are pro- Webers, and Teslas, respectively. duced by applying the del operator ∂ ∂ ∂ ∂ ∂ ∂ Charge q is quantized in units of ∂x ∂y ∂z ∂x ∂y ∂z ∇ ≡ ( , , ) = x̂ + ŷ + ẑ e = 1.602 × 10−19 C, a relativistic invariant. on vector or scalar fields as appropriate. 3  Vectors and vector functions can be expressed in terms of mutually z r = (x, y, z) orthogonal unit vectors x̂, ŷ, and ẑ as in = xx̂ + y ŷ + z ẑ y r = (x, y, z) = xx̂+y ŷ+z ẑ and E = (Ex, Ey , Ez ) = Exx̂+Ey ŷ+Ez ẑ etc., ẑ ŷ where x x̂ UNIT VECTORS AND A POSITION VECTOR IN RIGHT-HANDED " CARETESIAN COORDINATES ! – |r| ≡ x2 + y 2 + z 2 and |E| ≡ Ex2 + Ey2 + Ez2 etc., are vector magnitudes, Right handed con- r E – r̂ ≡ and Ê ≡ etc., are associated unit vectors, with vention: cross product vec- tor points in the direction indi- |r| |E| cated by the thumb of your right Dot products: Cross products: hand when you rotate your fin- ✬ ✬ ✩ ✩ gers from vector A toward vector x̂ × ŷ = ẑ, B through angle θ you decide to r̂ · r̂ = 1, Ê · Ê = 1, x̂ · x̂ = 1, etc., ŷ × ẑ = x̂, use. but ẑ× x̂ = ŷ in a right-handed system. B = |B|b̂ x̂ · ŷ = x̂ · ẑ = ŷ · ẑ = 0. Cross product A × B is a vector with a magnitude the product of |A| and |B| and |B| sin θ the sine of angle θ between A and B and a θ n̂ â Dot product A · B is a scalar which is the direction orthogonal to A and B in a right- |B| cos θ A = |A|â product of |A| and |B| and the cosine of an- gle θ between A and B. handed sense. A · B = |A||B| cos θ Dot product is zero when angle θ is 90◦ , as Cross product is zero when the vectors cross DOT PRODUCT:product of projected vector lengths in the case of x̂ and ŷ, etc. multiplied are collinear (θ = 0◦ ) or anti- linear (θ = 180◦ ). A × B = |A||B| sin θâ × n̂ ✫ ✪ CROSS PRODUCT: right-handed ✫ ✪ perpendicular area vector of the parallelogram formed by co-planar vectors 4  Having three non-collinear force measurements Fi cor- Example 1: A particle with charge q = 1 C passing through the origin r = (x, y, z) = 0 of the lab frame is observed responding to three distinct to accelerate with forces test particle velocities vi is F1 = 2x̂, F2 = 2x̂ − 6ẑ, F3 = 2x̂ + 9ŷ N sufficient to determine the when the velocity of the particle is fields E and B at any location m v1 = 0, v2 = 2ŷ, v3 = 3ẑ , in space produced by distant s in turns. Use the Lorentz force equation sources as illustrated by this F = q(E + v × B) example. to determine the fields E and B at the origin. Solution: Using the Lorentz force formula first with F = F1 and v =v1 , we note that z 2x̂ = (1)(E + 0 × B), which implies that y N V E = 2x̂ = 2x̂. C m Next, we use v1 = 0 F F x q q v × B = − E = − 2x̂ F1 = 2x̂ with F2 = 2x̂ − 6ẑ and v2 = 2ŷ, as well as E = 2x̂ V/m, to obtain z 2ŷ × B = −6ẑ ⇒ ŷ × B = −3ẑ; likewise, with F3 = 2x̂ + 9ŷ and v3 = 3ẑ, y 3ẑ × B = 9ŷ ⇒ ẑ × B = 3ŷ. Substitute B = Bx x̂ + By ŷ + Bz ẑ in above relations to obtain v2 = 2ŷ ŷ × (Bx x̂ + By ŷ + Bz ẑ) = −Bx ẑ + Bz x̂ = −3ẑ x and F2 = 2x̂ − 6ẑ ẑ × (Bx x̂ + By ŷ + Bz ẑ) = Bx ŷ − By x̂ = 3ŷ. z Matching the coefficients of x̂, ŷ, and ẑ in each of these relations we find that Wb Bx = 3 , and By = Bz = 0. y m2 Hence, vector Wb v3 = 3ẑ F3 = 2x̂ + 9ŷ B = 3x̂. m2 x 5 Conservation laws In HW1 you are asked to derive the continuity equation ∂ρ ∂t +∇·J=0 by taking the divergence of Ampere’s Law and combining it with Gauss’ Law. – This equation expresses the conservation of electrical charge by putting a constraint on charge density ρ and current density J as it was first explained in ECE 329 (this is just a review, recall). Another conservation law derived in ECE 329 from Maxwell’s equations was Poynting Theorem, namely ∂w ∂t + ∇ · S = −J · E, where 1 1 2 2 w = ϵoE · E + µoH · H EM energy density, S ≡ E × H Poynting vector, −J · E power produced per unit volume, – expressing the conservation of electromagnetic energy. 6  All conservation laws found in nature can be expressed mathematically in the forms given above in terms of a time-derivative of the volumetric density of the conserved quantity, the divergence of the flux of the conserved quantity (the so-called transport term), and a production term on the right (zero in case of charge conservation). The above conservation laws account for the increase/decrease of the conserved quantity density in terms of local transport and production effects. Hence charge conservation, for instance, is a local conservation principle. – If charge density decreases at a location, it will increase at a neigh- boring location because of local transport between the locations — charge cannot disappear in one volume and appear simultane- ously in another volume (satisfying a so-called global conservation principle) without having traveled between the volumes. – All conservation laws observed in nature are local (as opposed to global ) in the sense just described — the proof for this very broad statement can be based on the principle of relativity1. 1 Note that if charge could travel between the volumes with an infinite speed, then “global conservation” as opposed to “local conservation” could have been a viable idea — however no object can travel faster than light according to the principle of relativity and thus conservation laws have to be necessarily local and have mathematical expressions similar to those given in the continuity equation. A more general (but simple) proof of the local nature of all conservation laws (based on special relativity) is given by Feynman (see “The character of physical law”, 1965, MIT Press). 7