Electromagnetic Waves and Antennas Exercise book Sophocles J. Orfanidis1 Davide Ramaccia2 Alessandro Toscano2 1Department of Electrical & Computer Engineering Rutgers University, Piscataway, NJ 08854 [email protected] www.ece.rutgers.edu/~orfanidi/ewa 2Department of Applied Electronics, University "Roma Tre" via della Vasca Navale, 84 00146, Rome, Italy [email protected] [email protected] S.J. Orfanidis – Electromagnetic Waves and Antennas Exercises Chapter 1 Table of Contents Chapter1 Maxwell's Equations...................................................................................1 1.1 Exercise..........................................................................................................1 1.2 Exercise..........................................................................................................6 1.3 Exercise........................................................................................................12 1.4 Exercise........................................................................................................16 1.5 Exercise........................................................................................................18 1.6 Exercise........................................................................................................20 1.7 Exercise........................................................................................................25 1.8 Exercise........................................................................................................29 1.9 Exercise........................................................................................................30 1.10 Exercise........................................................................................................32 1.11 Exercise........................................................................................................42 1.12 Exercise........................................................................................................54 1.13 Exercise........................................................................................................56   D. Ramaccia and A. Toscano S.J. Orfanidis – Electromagnetic Waves and Antennas Exercises Chapter 1 Chapter1 Maxwell's Equations 1.1 Exercise Prove the vector algebra identities: a) A×(B×C) =B(A⋅C)-C(A⋅B) It is possible to write the vectors in the form: ⎧A=A xˆ +A yˆ +A zˆ x y z ⎪⎪ ⎨B=Bxxˆ +Byyˆ +Bzzˆ (1.1.1) ⎪ C=C xˆ +C yˆ +C zˆ ⎪⎩ x y z and to use the follow relationship: xˆ yˆ zˆ U×V = U U U = x y z (1.1.2) V V V x y z = xˆ(U V −U V )−yˆ(U V −U V )+zˆ(U V −U V ) y z z y x z z x x y y x Now we can prove the algebra identities with simply mathematical substitutions: A×(B×C)=A×(xˆ(B C −B C )−yˆ(B C −B C )+zˆ(B C −B C ))= y z z y x z z x x y y x = xˆ(A (B C −B C )+A (B C −B C )) y x y y x z x z z x (1.1.3) ( ( ) ( )) −yˆ A B C −B C −A B C −B C x x y y x z y z z y +zˆ(−A (B C −B C )−A (B C −B C )) x x z z x y y z z y Expanding the terms in (1.1.3), we have: A×(B×C)= ( ) +xˆ A B C −A B C +A B C −A B C y x y y y x z x z z z x ( ) +yˆ A B C −A B C +A B C −A B C (1.1.4) x y x x x y z y z z z y ( ) +zˆ A B C −A B C −A B C +A B C x z x x x z y y z y z y Let us write eq. (1.1.4) in matrix form, separating the terms with the minus sign and the terms with the plus sign: D. Ramaccia and A. Toscano    Pag. 1 S.J. Orfanidis – Electromagnetic Waves and Antennas Exercises Chapter 1 ⎡ 0 B A C B A C ⎤ ⎡ 0 C A B C A B ⎤ y x x z x x y x x z x x ⎢ ⎥ ⎢ ⎥ A×(B×C)= ⎢B A C 0 B A C ⎥−⎢C A B 0 C A B ⎥(1.1.5) x y y z y y x y y z y y ⎢ ⎥ ⎢ ⎥ B A C B A C 0 C A B C A B 0 ⎢⎣ x z z y z z ⎥⎦ ⎢⎣ x z z y z z ⎥⎦ Note that the elements of the diagonal of each matrix are zero. Each term can be filled with the product of the three component with the same subscript (a = A B C ): ii i i i ⎡B A C B A C B A C ⎤ ⎡C A B C A B C A B ⎤ x x x y x x z x x x x x y x x z x x ⎢ ⎥ ⎢ ⎥ A×(B×C)=⎢B A C B A C B A C ⎥−⎢C A B C A B C A B ⎥= x y y y y y z y y x y y y y y z y y ⎢ ⎥ ⎢ ⎥ B A C B A C B A C C A B C A B C A B ⎢⎣ x z z y z z z z z⎥⎦ ⎢⎣ x z z y z z z z z⎥⎦ ( ) ( ) ( ) =+A C B xˆ+B yˆ+B zˆ +A C B xˆ+B yˆ+B zˆ +A C B xˆ+B yˆ+B zˆ − x x x y z y y x y z z z x y z −A B (C xˆ+C yˆ+C zˆ)−A B (C xˆ+C yˆ+C zˆ)−A B (C xˆ+C yˆ+C zˆ)= (1.1.6) x x x y z y y x y z z z x y z ( ) ( ) =B A C +A C +A C −C A B +A B +A B = x x y y z z x x y y z z =B(A⋅C)−C(A⋅B) b) A⋅(B×C)= B⋅(C×A)= C⋅(A×B) Using relationships (1.1.1) and (1.1.2), we can write: A⋅(B×C)= A⋅(xˆ(B C −B C )−yˆ(B C −B C )+zˆ(B C −B C ))= y z z y x z z x x y y x (A B C −A B C )−(A B C −A B C )+(A B C −A B C )= (1.1.7) x y z x z y y x z y z x z x y z y x ( ) ( ) A B C +A B C +A B C − A B C +A B C +A B C x y z y z x z x y x z y y x z z y x B⋅(C×A)= B⋅(xˆ(C A −C A )−yˆ(C A −C A )+zˆ(A C −A C ))= y z z y x z z x x y y x ( ) ( ) ( ) B C A −B C A − B C A −B C A + B A C −B A C = x y z x z y y x z y z x z x y z y x (1.1.8) ( ) ( ) B C A +B C A +B C A − B C A +B C A +B C A = x y z y z x z x y x z y y x z z y x ↑ order them ( ) ( ) A B C +A B C +A B C − A B C +A B C +A B C x y z y z x z x y x z y y x z z y x C⋅(A×B)=C⋅(xˆ(A B −A B )−yˆ(A B −A B )+zˆ(A B −A B ))= y z z y x z z x x y y x ( ) ( ) ( ) C A B −C A B − C A B −C A B + C A B −C A B = x y z x z y y x z y z x z x y z y x (1.1.9) ( ) ( ) C A B +C A B +C A B − C A B +C A B +C A B = x y z y z x z x y x z y y x z z y x ↑ order them ( ) ( ) A B C +A B C +A B C − A B C +A B C +A B C x y z y z x z x y x z y y x z z y x If we compare the last row of each expression, we note that they are identical so the algebra identity is verified. D. Ramaccia and A. Toscano    Pag. 2 S.J. Orfanidis – Electromagnetic Waves and Antennas Exercises Chapter 1 2 2 2 2 c) A×B + A⋅B = A B Using relationships (1.1.1) and (1.1.2), we can write: 2 A×B2 + A⋅B2 = xˆ(A B −A B )−yˆ(A B −A B )+zˆ(A B −A B ) + y z z y x z z x x y y x ( )2 + A B +A B +A B = x x y y z z 2 ⎛⎜ (AyBz −AzBy)2 +(AxBz −AzBx)2 +(AxBy −AyBx)2 ⎞⎟ +(AxBx +AyBy +AzBz)2 = ⎝ ⎠ (A B −A B )2 +(A B −A B )2 +(A B −A B )2 +(A B +A B +A B )2 = y z z y x z z x x y y x x x y y z z 2 2 2 2 2 2 2 2 A B +A B −2A B A B +A B +A B −2A B A B + y z z y y z z y x z z x x z z x 2 2 2 2 ( )2 A B +A B −2A B A B + A B +A B +A B = x y y x x y y x x x y y z z 2 2 2 2 2 2 2 2 A B +A B −2A B A B +A B +A B −2A B A B + y z z y y z z y x z z x x z z x 2 2 2 2 2 2 2 2 2 2 A B +A B −2A B A B +A B +A B +A B + x y y x x y y x x x y y z z 2A B A B +2A B A B +2A B A B = x y y x x z z x x y y x ↑ cancel the opposites 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 A B +A B +A B +A B +A B +A B +A B +A B +A B = y z z y x z z x x y y x x x y y z z ( 2 2 2)( 2 2 2) 2 2 A +A +A B +B +B = A B x y z x y z d) A=nˆ×A×nˆ +(nˆ⋅A)nˆ Does it make a difference whether nˆ ×A×nˆ is taken to mean (nˆ ×A)×nˆ or nˆ ×(A×nˆ)? The unit vector nˆcan be expressed as follow: ⎧nˆ = n xˆ +n yˆ +n zˆ ⎪ x y z (1.1.10) ⎨ ⎪nˆ = n2 +n2 +n2 =1 ⎩ x y z Let us begin considering the first case: D. Ramaccia and A. Toscano    Pag. 3 S.J. Orfanidis – Electromagnetic Waves and Antennas Exercises Chapter 1 (nˆ×A)×nˆ = ⎡xˆ(n A −n A )− yˆ(n A −n A )+zˆ(n A −n A )⎤×nˆ = ⎣ y z z y x z z x x y y x ⎦ +xˆ⎡(n A −n A )n −(n A −n A )n ⎤+ ⎣ z x x z z x y y x y⎦ − yˆ ⎡(n A −n A )n −(n A −n A )n ⎤+ ⎣ y z z y z x y y x x⎦ +zˆ⎡(n A −n A )n −(n A −n A )n ⎤ = (1.1.11) ⎣ y z z y y z x x z x⎦ +xˆ⎡n2A −n n A −n n A +n2A ⎤+ ⎣ z x x z z x y y y x⎦ − yˆ ⎡n n A −n2A −n2A +n n A ⎤+ ⎣ y z z z y x y y x x⎦ +zˆ⎡n2A −n n A −n n A +n2A ⎤ ⎣ y z z y y z x x x z⎦ And now consider the second case: nˆ×(A×nˆ)=nˆ×⎡xˆ(A n −A n )− yˆ(A n −A n )+zˆ(A n −A n )⎤ = ⎣ y z z y x z z x x y y x ⎦ +xˆ⎡n (A n −A n )−n (A n −A n )⎤+ ⎣ y x y y x z z x x z ⎦ −yˆ ⎡n (A n −A n )−n (A n −A n )⎤+ ⎣ x x y y x z y z z y ⎦ +zˆ⎡n (A n −A n )−n (A n −A n )⎤ = (1.1.12) ⎣ x z x x z y y z z y ⎦ +xˆ⎡A n2 −A n n −A n n +A n2⎤+ ⎣ x y y y x z z x x z⎦ −yˆ ⎡A n n −A n2 −A n2 +A n n ⎤+ ⎣ x x y y x y z z z y⎦ +zˆ⎡A n2 −A n n −A n n +A n2⎤ ⎣ z x x x z y y z z y⎦ It is very easy to show that (nˆ × A)×nˆ = nˆ ×(A×nˆ). The second term of the identity can be written as: D. Ramaccia and A. Toscano    Pag. 4 S.J. Orfanidis – Electromagnetic Waves and Antennas Exercises Chapter 1 ( )( ) (nˆ ⋅A)nˆ = n A +n A +n A n xˆ +n yˆ +n zˆ = x x y y z z x y z +xˆ⎡n (n A +n A +n A )⎤+ ⎣ x x x y y z z ⎦ +yˆ ⎡n (n A +n A +n A )⎤+ ⎣ y x x y y z z ⎦ +zˆ⎡n (n A +n A +n A )⎤ = ⎣ z x x y y z z ⎦ (1.1.13) +xˆ⎡n2A +n n A +n n A ⎤+ ⎣ x x x y y x z z⎦ +yˆ ⎡n n A +n2A +n n A ⎤+ ⎣ y x x y y y z z⎦ +zˆ⎡n n A +n n A +n2A ⎤ ⎣ z x x z y y z z⎦ Adding the two results, we obtain: nˆ×A×nˆ +(nˆ ⋅A)nˆ = +xˆ⎡A n2 −A n n −A n n +A n2⎤+ ⎣ x y y y x z z x x z⎦ −yˆ ⎡A n n −A n2 −A n2 +A n n ⎤+ ⎣ x x y y x y z z z y⎦ +zˆ⎡A n2 −A n n −A n n +A n2⎤+ ⎣ z x x x z y y z z y⎦ +xˆ⎡n2A +n n A +n n A ⎤+ ⎣ x x x y y x z z⎦ +yˆ ⎡n n A +n2A +n n A ⎤+ ⎣ y x x y y y z z⎦ +zˆ⎡n n A +n n A +n2A ⎤ = ⎣ z x x z y y z z⎦ ↑ change signs in parentheses at first yˆ and add +xˆ⎡A n2 −A n n −A n n +A n2 +n2A +n n A +n n A ⎤+ ⎣ x y y y x z z x x z x x x y y x z z⎦ +yˆ ⎡A n2 +A n2 −A n n −A n n +n n A +n2A +n n A ⎤+ ⎣ y x y z x x y z z y y x x y y y z z⎦ +zˆ⎡A n2 −A n n −A n n +A n2 +n n A +n n A +n2A ⎤ = ⎣ z x x x z y y z z y z x x z y y z z⎦ (1.1.14) +xˆA ⎡n2 +n2 +n2⎤++yˆA ⎡n2 +n2 +n2⎤++zˆA ⎡n2 +n2 +n2⎤ = A x ⎣ y z x⎦ y ⎣ x z y⎦ z ⎣ x y z⎦ D. Ramaccia and A. Toscano    Pag. 5 S.J. Orfanidis – Electromagnetic Waves and Antennas Exercises Chapter 1 1.2 Exercise Prove the vector analysis identities: 1. ∇×(∇φ)= 0 2. ∇⋅(φ∇ψ)= φ∇2ψ+∇φ⋅∇ψ (Green's first identity) 3. ∇⋅(φ∇ψ−ψ∇φ)=φ∇2ψ−ψ∇2φ (Green's second identity) 4. ∇⋅(φA)= (∇φ)⋅A +φ∇⋅A 5. ∇×(φA)= (∇φ)×A + φ∇×A 6. ∇⋅(∇×A)= 0 7. ∇ ⋅A ×B = B⋅(∇× A)− A ⋅(∇×B) 8. ∇×(∇×A)=∇(∇⋅A)−∇2A First of all we have to express the operator ∇in general orthogonal coordinates in four common applications. All vector components are presented with respect to the normalized base (eˆ ,eˆ ,eˆ ): 1 2 3 ⎧ eˆ ∂φ eˆ ∂φ eˆ ∂φ 1 2 3 ∇φ= + + ⎪ h ∂q h ∂q h ∂q ⎪ 1 1 2 2 3 3 ⎪ 1 ⎡ ∂ ⎛h h ∂φ ⎞ ∂ ⎛h h ∂φ ⎞ ∂ ⎛h h ∂φ ⎞⎤ ⎪∇2φ= ⎢ ⎜ 2 3 ⎟+ ⎜ 1 3 ⎟+ ⎜ 1 2 ⎟⎥ ⎪ h1h2h3 ⎢⎣∂q1⎝ h1 ∂q1⎠ ∂q2 ⎝ h2 ∂q2 ⎠ ∂q3 ⎝ h3 ∂q3 ⎠⎥⎦ ⎪ ⎪ 1 ⎡ ∂ ∂ ∂ ⎤ ∇⋅F = (Fh h )+ (F h h )+ (F h h ) ⎪ ⎢ 1 2 3 2 1 3 3 1 2 ⎥ h h h ∂q ∂q ∂q ⎪ 1 2 3 ⎣ 1 2 3 ⎦ ⎪⎪ h eˆ h eˆ h eˆ 1 1 2 2 3 3 ⎨ ⎪ 1 ∂ ∂ ∂ ∇×F = = ⎪ h h h ∂q ∂q ∂q 1 2 3 1 2 3 ⎪ ⎪ h F h F h F 1 1 2 2 3 3 ⎪ ⎪ + eˆ1 ⎡ ∂ (h F )− ∂ (h F )⎤+ eˆ2 ⎡ ∂ (h F )− ∂ (h F )⎤+ ⎪ ⎢ 3 3 2 2 ⎥ ⎢ 1 1 3 3 ⎥ h h ∂q ∂q h h ∂q ∂q ⎪ 2 3 ⎣ 2 3 ⎦ 1 3 ⎣ 3 1 ⎦ ⎪ eˆ ⎡ ∂ ∂ ⎤ ⎪ + 3 ⎢ (h2F2)− (h1F1)⎥ ⎪⎩ h1h2 ⎣∂q1 ∂q2 ⎦ (1.2.1) where (h ,h ,h )are the metric coefficients. For common geometries they are defined as follow: 1 2 3 D. Ramaccia and A. Toscano    Pag. 6 S.J. Orfanidis – Electromagnetic Waves and Antennas Exercises Chapter 1 ⎧h =1, h =1, h =1 (rectangular coordinates) 1 2 3 ⎪ ⎨h1 =1, h2 = r, h3 =1 (cylindrical coordinates) (1.2.2) ⎪ h =1, h = r, h = rsinϑ (spherical coordinates) ⎩ 1 2 3 For simplicity, the proves are done using rectangular coordinates (h =1, h =1, h =1): 1 2 3 • Identity n° 1 eˆ eˆ eˆ 1 2 3 ∂ ∂ ∂ ∇×(∇φ)= = ∂q ∂q ∂q 1 2 3 ⎛ ∂φ ⎞ ⎛ ∂φ ⎞ ⎛ ∂φ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂q ∂q ∂q ⎝ 1⎠ ⎝ 2 ⎠ ⎝ 3 ⎠ ⎡ ⎛ ∂ ∂φ ∂ ∂φ ⎞ ⎛ ∂ ∂φ ∂ ∂φ ⎞ ⎤ ⎢eˆ1⎜ − ⎟−eˆ2⎜ − ⎟+⎥ ∂q ∂q ∂q ∂q ∂q ∂q ∂q ∂q ⎢ ⎝ 2 3 3 2 ⎠ ⎝ 1 3 3 1⎠ ⎥ = =0 ⎢ ⎥ ⎛ ∂ ∂φ ∂ ∂φ ⎞ ⎢+eˆ3⎜ − ⎟ ⎥ ⎢ ∂q ∂q ∂q ∂q ⎥ ⎣ ⎝ 1 2 2 1⎠ ⎦ ∂ ∂ ∂ ∂ For the property of linearity of the derivate operator φ = φ, so each term in the ∂q ∂q ∂q ∂q i j j i parentheses vanishes and also the result. • Identity n° 2 ⎛ ∂ψ ∂ψ ∂ψ ⎞ ∇⋅(φ∇ψ)=∇⋅⎜eˆ1φ +eˆ2φ +eˆ3φ ⎟ = ∂q ∂q ∂q ⎝ 1 2 3 ⎠ ∂ ⎛ ∂ψ ⎞ ∂ ⎛ ∂ψ ⎞ ∂ ⎛ ∂ψ ⎞ = ⎜φ ⎟+ ⎜φ ⎟+ ⎜φ ⎟ = ∂q ∂q ∂q ∂q ∂q ∂q 1⎝ 1 ⎠ 2 ⎝ 2 ⎠ 3 ⎝ 3 ⎠ ⎛ ∂φ ∂ψ ∂2ψ⎞ ⎛ ∂φ ∂ψ ∂2ψ⎞ ⎛ ∂φ ∂ψ ∂2ψ⎞ =⎜ +φ ⎟+⎜ +φ ⎟+⎜ +φ ⎟ = ⎜∂q ∂q ∂q ⎟ ⎜∂q ∂q ∂q ⎟ ⎜∂q ∂q ∂q ⎟ ⎝ 1 1 1 ⎠ ⎝ 2 2 2 ⎠ ⎝ 3 3 3 ⎠ ⎛∂2ψ ∂2ψ ∂2ψ⎞ ⎛ ∂φ ∂ψ ∂φ ∂ψ ∂φ ∂ψ ⎞ 2 =φ⎜ + + ⎟+⎜ + + ⎟ = φ∇ ψ+∇φ⋅∇ψ ⎜⎝ ∂q1 ∂q2 ∂q3 ⎟⎠ ⎝∂q1 ∂q1 ∂q2 ∂q2 ∂q3 ∂q3 ⎠ • Identity n° 3 First of all we expand the sum inside parentheses: ⎧ ∂ψ ∂ψ ∂ψ φ∇ψ =φeˆ +φeˆ +φeˆ ⎪ 1 2 3 ⎪ ∂q1 ∂q2 ∂q3 ⎨ ∂φ ∂φ ∂φ ⎪ψ∇φ=ψeˆ +ψeˆ +ψeˆ 1 2 3 ⎪ ∂q ∂q ∂q ⎩ 1 2 3 D. Ramaccia and A. Toscano    Pag. 7