MAXWELL EQUATION IN DIFFERENTIAL FORM E M WAVE
- Slides: 93
MAXWELL EQUATION IN DIFFERENTIAL FORM & E. M. WAVE EQUATIONS CHAPTER 2 Dr. Ir. Suwarno
DIFFERENSIASI VEKTOR If A (u) is a vector and u independent variabel then difference in A (u) , A can be expressed as A = A(u + u) - A(u) For small u 0
For position vector from a point at Cartesian r(u) = x(u)ax + y(u)ay + z(u)az
GRADIENT OF A SCALAR FUNCTION For describing gradient following illustration of temperature change is given. At P 1 (x, y, z) temperature is T 1 (x, y, z). At P 2 (x + x, y + y, z + z) temperature is T 2 (x + x, y + y, z + z). Using Taylor expansion T 2 can be obtained from dapat T 1 as follows : + higher order and negligible And therefore T = T 2 - T 1 can be approximated as
If change of coordinate is very small ( x 0, y 0, z 0) then Position differential . d. T = grad T. dr = T. dr where (del) is a differential operator which in Cartesian
Gradient formula
Gradient of a scalar function is conservative i. e. : From the figure :
Example
DIVERGENCE OF VECTOR FIELD Vector F and flux
Consider F as a vector in Cartesian Fluks F out from closed S covered v = x y z is as follows. Since v is a cubic and therefore total flux
To make simple the can be divided for each direction. Flux in X direction Ffron and Fback can be related as Constribution for front and back fluxes is
Using similar way for flux in Y direction An for flux in Z direction
For all surfaces If the results is divided with volume differential v If v 0 then
Divergence of a vector F If operator nabla is is defined as then
Divergences for each coordinate systems
DIVERGENCE THEOREM
EXAMPLE
DIVERGENCE AND MAXWELL EQUATIONS From Gauss Law for electric field From divergence theorem Elimination of dv . ( o. E) = v Using the same way for magnetic field Maxwell equations in diff. form or . (D) = v . B = 0
Gradient and divergence in electrostatics Gradient Potential V Divergence Field E Integration Charge density r
Example In free space potential is expressed as V=2 x 2 y+3 z +100 V Determine E and r at A(1, 1, 1) m
Potential in free space is independent on y and z and only a function of x and expressed graphically below V (volt) 10 0 2 4 6 8 -10 Determine a. Electric field between x=0 and x = 8 b. Charge density c. Total charge in a sphere with radius of 1 m centered at (6, 6, 6) x (m)
CURL OF A VECTOR FIELD Curl indicates net circulation of a vectior field around closed contour( or about a point). For example water flow in a canal with non uniform velocity If F is a vector as function of x, y and z then curl of F can be expressed as Curl F = Curl F x ax + Curl F y ay + Curl F z az
Curl meter for water flow in a canal Z Y Curl V = 0 Curl V > 0 Curl V < 0
Z Y
Curl releases a vertical force
For the case of water flow in a canal at Y-Z plane For F in cartesian
For small contour c, using Taylor expansion Then
Using same way for y component And for z component Complete expression for curl is
Curl in different coordinate systems Cartesian Cylindrical Spherical
Illustration of divergence and Curl
Example
From chapter I for a conductor with radius of a and uniform current I
STOKES THEOREM Surface Integral relates with line integral as AMPERE’s AND DIFFERENTIAL Ampere’s Law FARADAY’S LAWS IN THE FORM OF
For obtaining differential form take a small contour c enclosed a surface of s , then With taking surface element of ds in the direction of ax ; ds=ds ax then Jx and Ex are J and E in the direction of x. Using similar way for y direction
For z direction. Then for complete Curl And can be written as Ampere’s Law in diferential form
Another approach is using Stokes theorem Taking surface element of s And then
For Faraday’s law Using Stokes theorem And for surface element s Or
For static field then (dt) = 0 And therefore
Summary of Maxwell Equations in differential forms
CONTINUITY EQUATION AND DISPLACEMENT CURRENT Electric charge is conservative i. e. can not be created and can not be destroyed. The possibility is displacement of charges to form electric current. For a volume enclosed by a surface S with a charge density of v and a current density J J S ρv Net flux outward from closed surface S is equal with the decrease of charge in the volume enclosed by S
Using divergence theorem Taking volume element of v then The last equation is continuity equation representing charge conservation. Initially Ampere’s law is expressed without correlation between electric and magnetic fields i. e.
Divergence of J In vector mathematics the divergence of a curl should be zero then . J = 0 This is inconsistent with continuity equation. Maxwell introduced JD as
Then continuity equation is satisfied or This is displacement current introduced by Maxwell. Thus Ampere‘s law can be expressed as
For steady current div J=0 because d /dt=0 then
Example In a material with s= 5 S/m and relative permittivity of 1 electric field is expressed as E=250 sin 1010 t (V/m). Determine conduction current and displacement current. At which frequency the 2 currents are the same ?
WAVE EQUATION IN SOURCE FREE REGION For source free region =J=0 then
There is a coupling between electric and magnetic fields in an electromagnetic field. Taking curl from curl of E then. Substitute curl B with its E coupling In vector mathematics there is a vector identity as follows x x E = (. E) - 2 E The last term is Laplacian operator
If Then each component Since . D = o . E = 0 then This is E wave equation in source free region.
Using similar way magnetic field wave equation is.
MEDAN HARMONIK DAN PHASOR Untuk membahas gelombang elektromagnetik kita gunakan fungsi sinusoidal. Rapat arus dan muatan merupakan fungsi dan posisi dan waktu Dengan fungsi kompleks waktu dapat ditulis sebagai Persamaan Maxwell dalam fungsi kompleks dapat ditulis sebagai . o ( . ( (r))ej t = ej t (r) ej t) = 0 x ( (r) ej t ) = -j ( (r) ej t) dan
Eliminasi menghasilkan persamaan Maxwell dalam bentuk phasor . o ( (r)) = . ( (r) = 0 x ( (r) = -j (r)
Bila rapat arus dan muatan = 0 yaitu daerah bebas sumber maka persamaan Maxwell menjadi . o . x =0 =0 = -j Untuk mendapatkan informasi medan listrik dan magnet dari bentuk phasor dapat dilakukan dengan mengalikan dengan dan mengambil bagian riil. E (r, t) = Re( (r) ej t) B (r, t) = Re((r) ej t)
Untuk fungsi kosinuisoidal maka bila komponen riil medan pada arah x x (r) = Eoej maka fungsi Ex sebenarnya dapat ditulis sebagai Ex (r, t) = Re(Eoej ej t) = Eo cos ( t + ) Bentuk terakhir ini sangat mudah untuk dimengerti.
Dari persamaan Maxwell bentuk phasor untuk daerah bebas sumber, dapat diturunkan persamaan gelombang listrik dan magnet dalam bentuk phasor dengan subtitusi dan yaitu 2 + 2 oeo =0 =0
PROPAGASI GELOMBANG DATAR DI RUANG BEBAS Pandang suatu gelombang datar yang merambat pada arah z. Dengan demikian baik E maupun B adalah fungsi dari z bukan dari x dan y.
Karakteristik gelombang demikian dapat diringkas sebagai ; 1. Turunan parsial terhadap x atau y sama dengan nol 2. Daerah yang diselidiki bebas sumber artinya J = = 0 sehingga persamaan Maxwell ditulis sebagai
Untuk gelombang datar menghasilkan dimana turunan parsial terhadap x dan y = 0. Penyelesaian persamaan ini menghasilkan
Hal yang sama berlaku untuk atau dan menghasilkan
Diferensiasi dari Ex (z) terhadap z menghasilkan Mengingat Ex hanya fungsi dari z bukan x dan y maka dapat dipakai ekspresi differensial eksak. Solusi umum Ex untuk persamaan gelombang ini adalah
dimana dan C 1 dan C 2 konstanta. Dengan mengganti konstanta C 1 dan C 2 dengan amplitudo Ex diperoleh Bila amplitudo dibuat riil maka Fungsi waktu dari Ex didapat dengan mengambil komponen riil yaitu
Em+ cos ( t - o z) adalah gelombang yang merambat ke z positif Em+ cos ( t + o z) adalah gelombang yang merambat ke z negatif
untuk Ex pada arah z positif maka pada t = 0 dan untuk t > 0 untuk medan magnetik diperoleh dimana yang merupakan cepat rambat cahaya di ruang bebas.
Untuk gelombang yang berpropagasi pada z negatif rasio Ex dan By adalah Jadi rasio Ex dan By juga c. Seringkali dipakai rasio antara E dan H yang akan menghasilkan
o merupakan impedansi intrinsik ruang bebas. Secara umum impedansi intrinsik bahan adalah Beberapa parameter gelombang yang perlu diketahui adalah
1. Panjang gelombang ( ) dimana o =2 atau 2. Kecepatan fasa yaitu kecepatan rambat gelombang yang dalam kasus ini ke arah z. Dengan mengambil suatu t - oz = constant maka Dengan media ruang bebas maka yaitu kecepatan cahaya sehingga
Contoh – contoh Soal Dalam ruang bebas medan listrik adalah, E(z, t)=103 sin ( t - z)ay (V/m). Tentukan persamaan H(z, t). Dilihat dari fasa E yaitu, t - z, menunjukan arah perambatan E adalah +z Karena E x H juga harus dalam arah + z, maka H mesti dalam arah –ax. Sehingga atau dan
Medan listrik pada arah X dengan f=1 MHz dari suatu medan datar merambat pada arah +Z di ruang bebas. Harga puncak E adalah 1, 2 m. V/m pada t=0 dan z=50 m. Tentukan E(z, t) dan H(z, t) serta plot pada t=0.
Pergeseran p/3
Medan magnet di ruang bebas ke arah Z positif dinyatakan sebagai H(z, t)= 4 x 10 -6 cos (2 x 107 t- oz) ay A/m. tentukan o dan panjang gelombang. Tentukan pula E(z, t).
Suatu gel. datar dengan f=1 GHz merambat pada arah X positif pada bahan dielektrik sempurna dengan konstanta 2, 1 dan permeabilitas relatif 1. a. Tentukan v, impedansi intrinsik, dan b. Bila E= 100 sin (at – bx) az V/m. Tentukan persamaan lengkap E dan H
Tetapkanlah konstanta dari gelombang pada soal sebelumnya jika diberikan frekuensi f = 95, 5 MHz. Secara umum , = Dalam ruang bebas = 0, sehingga Hasil ini menunjukan bahwa faktor atenuasi = 0 dan konstanta penggeseran fasa = 2, 0 rad/m.
Tetapkan konstanta propagasi bagi suatu bahan dengan dan = 0, 25 p. S/m, Jika frekuensinya 1, 6 MHz. Dalam hal ini, sehingga Jadi = + j j 9, 48 x 10 -2 m-1. Bahan bersifat seperti dielektrik sempurna pada frekuensi yang diberikan. Konduktivitas dalam orde 1 p. S/m menunjukan sifat bahan yang lebih sebagai isolator daripada konduktor.
Pada frekuensi-frekuensi berapa tanah dapat dianggap dielektrik sempurna jika = 5 x 10 -3 S/m, r = 1, dan er = 8 ? Dapatkah dianggap nol pada frekuensi itu? Ambil saja Maka Untuk / kecil, Jadi, bagaimanapun tingginya frekuensi, sekitar 0, 333 Np/m, atau hampir 3 d. B/m sehingga tak dapat dianggap nol.
Dalam ruang bebas E(z, t) = 50 cos ( t - z)az (V/m). Tetapkan daya rata-rata yang melewati suatu permukaan lingkaran berjari-jari 2, 5 m dalam bidang z = konstan. Dalam bentuk kompleks, E = 50 ef( t - z)ax (V/m) Karena = 120 dan perambatan dalam arah +z, maka Arus tersebut normal terhadap permukaan tadi, hingga
Diketahui dielektrik dengan r = 1, r = 10, = 20 n S/m. medan listrik a. Tentukan f dimana b. Tentukan c. Tentukan Jawab a. agar jadi agar Maka f = 36 Hz
b. f = 36 Hz c. Selalu terpisah 90 o. Arus kapasitif IC Arus resistif IR
Suatu gelombang datar dengan f = 1 GHz merambat pada teflon ( r=2, 1, r=1, =0). Tentukan , , dan Jawab
Arus dakam medium : Untuk frekuensi sudut Rasio menentukan apakah medium konduktor, isolator atau diantaranya. Contoh 1. Tembaga = = 58 MS/m , = o pada f = 1 MHz Konduktor sangat baik 2. Plastik = = 3. 10 -8 S/m , = 2, 1 o pada f = 1 MHz isolator/dielektrik baik Sifat konduktor/isolator tergantung frekuensi
POLARIZATION OF PLANE WAVES =( ax + ay) e-j z = eja = ejb
= ( ax + ay) e-j( z-a) E = ( ax + ay) cos ( t - z + a)
Ex = cos ( t + a - z) Ey = cos ( t + b - z)
Ex(z, t) = cos ( t - z) Ey(z, t) = - sin ( t - z) E (z, t) = Ex(z, t) ax + Ey(z, t) ay
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