Úvod. bodových elektrických nábojů: «1 of Přiklad na použití Biot - Savartova zákona:
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1 Printed from the Mathematica Help Browser 1 «1 of 14 Úvod - Přiklad na použití Biot - Savartova zákona: Výpočet vektoru magnetické intenzity na ose závitu protékaného stacionárním proudem - Výpočet a znázornění intenzity ve Fraunhoferových ohybových jevech: Výpočet a grafické znázornění rozložení intenzity fe Fraunhoferově ohybovém jevu na obdélníkovém otvoru - Výpočet a znázornění elektrických polí elektrostatických multipólů složených z bodových elektrických nábojů: Ekvipotenciální hladiny a orientace vektoru elektrické intenzity v polích axiálních multipólů - dipól, axiální oktupól «2 of 14
2 2 Printed from the Mathematica Help Browser Příklad 1/1 Výpočet velikosti vektoru magnetické intenzity H na ose kruhového závitu, známe-li poloměr zavitu R a velikost protékajíciho stacionárního proudu I Vykreslete závislost H = f HzL, kde z je vzdálenost bodu na ose závitu od roviny závitu, pro hodnoty R = mm a I = 2 A. - Doporučení pro Mathematica CalcCenter : Vykreslit graf v mezích -.1 az.1 m - Počítáme pomocí Biot - Savartova zakona : přírůstek vektoru magnetické intenzity v bodě na ose závitu od elemntu závitu dl je dh = I dl sina 4 p r 2, kde r je vzdálenost mezi elementem závitu dl a bodem na ose závitu, a je uhel mezi dl a r. V uvažovanem příkladě je dl kolmé na r, tedy sina = 1, a dl = R df - Každý element dl závitu vyvolá magnetickou intenzitu v bodě na ose závitu jiného směru. Magnetická intenzita, vyvolaná dvěma protilehlými elementy dl dává výslednou intenzitu orientovanou kolmo na rovinu závitu, přičemž orientace vektoru magnetické intenzity je dána pravidlem pravé ruky. Za tohoto předpokladu je přírůstek velikosti vektoru magnetické intenzity na ose závitu od dvou protilehlých elementů dl dán výrazem 2 dh sinb, kde b je úhel mezi r a osou závitu - Potom celková velikost vektoru magnetické intenzity v bodě na ose závitu ve vzdálenosti z od roviny závitu je H = Ÿ p I R sinb f 2 p r 2 = I R sinb 2 r 2, kde r = R 2 + z 2 a sinb = R, tedy finalne r H = I R 2 2 IR 2 +z 2 M 3ê2 ;@AêmD «3 of 14 Příklad 1/2 hh = ii R RêH2 HR R + z zl ^H3ê2LL; R =.; R = ^ 2; ii =.; ii = 2; Plot@hh, 8z,.1,.1<D
3 Printed from the Mathematica Help Browser 3 Mathematica i Mathematica CalcCenter «4 of 14 Příklad 1/3 Vykresleme graf v širších mezích, např. 1, m, +1, m, Mathematica CalcCenter z Mathematica
4 4 Printed from the Mathematica Help Browser Problém : Mathematica CalcCenter nemá možnost volby PlotRange v příkazech Plot «of 14 Příklad 1/4 Pomocí 3D grafů je možno vykreslit závislost velkosti maghentické intenzity na parametrech úlohy, kterými jsou velikost proudu I a poloměr závitu. - Závislost na proudu je lineární, tedy nezajímavá - Vykreslíme závislost na poloměru závitu v rozsahu např. cm až 12 cm ii=.; ii = 2; Plot3D@ii* R* RêH2*HR*R+z*zL^H3ê2LL, 8R, * ^-2, 12* ^-2<, 8z, -.1,.1<, PlotPoints Æ D Mathematica
5 Printed from the Mathematica Help Browser Mathematica CalcCenter z.8 R Problém : Mathematica CalcCenter nemá možnost volby PlotPoints v příkazu Plot3D «6 of 14 Příklad 1/ Zavislost velikosti vektoru mg. intenzity na velikosti protékajícího proudu v mezích od 2 do A
6 6 Printed from the Mathematica Help Browser a=.; a = *^-2; Plot3D@ii*a*aêH2*Ha* a+z*zl^h3ê2ll, 8ii, 2, <, 8z, -.1,.1<, PlotPoints Æ D «7 of 14
7 Printed from the Mathematica Help Browser 7 Příklad 2/1 Fraunhoferovy ohybové jevy na obdélníkovém otvoru - osa otvoru je totožná se souřadnou osou x, strany otvoru jsou rovnoběžné se souřadnými osami y, z. - Intenzita ve Fraunhoferově ohybovém obrazci při ohybu na obdélnikovém otvoru je dána vztahem I=I *J sinhkbal kba N 2 *J sinhkgbl N 2, kgb kde k= 2 p je vlnovéčislo, 2a, 2b jsou rozměry obdélnikového otvoru a b, g jsou uhly, které svíra osa obdélníkového otvoru se spojnicí středu l otvoru s bodem, ve kterém je intenzita v ohybovém obrazci počítána, ve směrech souřadných os y, z - Vykresleme intenzitu I v ohybovém obrazci za nasledujících předpokladů: I = 1, kba se mění v mezích -6,+6, kbb se mění v intervalu -12, +12. To odpovida např. těmto realným parametrům: l = nm, 2a =.4 mm, 2b =.8 mm, b max = g max = ± 2.63 mrad. - Intenzitu vykreslíme nejprve v Mathematice postupně ve formě 3D grafu v různých stupních rozlišení - např. 7, 1 a 2 bodů/šířka grafu a ve formě "hustotniho" grafu, jehož výsledky se blíží fotografiím ohybových jevů - Následně se pokusíme získat obdobné grafy v Mathematice CalcCenter «8 of 14 Příklad 2/2 Mathematica
8 8 Printed from the Mathematica Help Browser ^2 ^ 2, 8x, 6, 6<, 8y, 12, 12<, PlotPoints 7, Axes > True, PlotRange > 8, 1<D
9 Printed from the Mathematica Help Browser 9 Plot3D@HSin@xDêxL ^2 HSin@yDêyL ^ 2, 8x, 6, 6<, 8y, 12, 12<, PlotPoints 1, Axes > True, PlotRange > 8, 1<D
10 Printed from the Mathematica Help Browser ^2 ^ 2, 8x, 6, 6<, 8y, 12, 12<, PlotPoints 2, Mesh False, Axes > True, PlotRange > 8, 1<D «9 of 14
11 Printed from the Mathematica Help Browser 11 Příklad 2/3 ^2 ^ 2, 8x, 6, 6<, 8y, 12, 12<, PlotPoints 2, Mesh False, Axes > FalseD «of 14 Příklad 2/4 Mathematica CalcCenter:
12 12 Printed from the Mathematica Help Browser Plot3DB x 2 Sin@ yd y 2, 8x, -6, 6<, 8y, -12, 12<, "Default", "None"F y x 2. - Opět narážíme na problém, že Mathematica CalcCenter nemá v příkazech Plot volby PlotPoints a PlotRange «11 of 14
13 Printed from the Mathematica Help Browser 13 Příklad 2/ DensityPlotB x 2 Sin@ yd y 2, 8x, -6, 6<, 8y, -12, 12<, "Default", "None"F «12 of 14 Příklad 3/1 Elektrické pole elektrostatických multipólů složených z bodových nábojů Uvažujeme multipól složený z N bodových nábojů o velikostech Q n ; n = 1, 2,..., N. Polohy nábojů jsou popsány polohovými vektory r n.
14 14 Printed from the Mathematica Help Browser Skalární potenciál elektrostatického pole takových multipólů v bodě P je ϕ HPL = N 1 4 π n=1 Q n» r P r n», intenzita elektrostatického pole ve stejném bodě je E HPL = grad ϕ HPL Znázornění elektrostatického pole : pomocí siločar. Rovnici siločar v kartéských souřadnicích v rovině lze získatřešením diferenciální rovnice dx E x = dy E y, obdobnou diferenciální rovnici lze definovat i v libovolných křivočarých souřadnicích. V řešení. pomocí znázornění ekvipotenciálních ploch a orientací vektoru elektrické intenzity ve zvolených bodech. Řešeno v publikaci R. L. Zimmerman, F. I. Olness, Mathematica for Physics. Addison Wesley Publishing Company, Inc., New York, 22. «13 of 14 Příklad 3/2 Vytvorim si vlastni funkciveplot@d, ktera mi v jednom grafu zobrazi ekvipotenciolove plochy HVL a silocary HEL. K tomu je třeba nahrát i potrebne knihovny. Needs@"Graphics`PlotField` "D VEPlot@potential_, xlim_, ylim_, opts D := Module@8plot1, plot2<, plot1 = PlotGradientField@ potential, xlim, ylim, PlotPoints 24, ScaleFunction H1 &L, DisplayFunction IdentityD; plot2 = ContourPlot@potential, xlim, ylim, ContourShading False, ContourSmoothing True, DisplayFunction Identity, PlotPoints D; Show@8plot1, plot2<, opts, DisplayFunction $DisplayFunction DD; VEPlot@potential_, opts D := VEPlot@potential, 8x, 2, 2<, 8y, 2, 2<, optsd Elementární dipól dipole = 1êHSqrt@x ^2 +Hy + 1ê2L^ 2DL 1êHSqrt@x ^2 +Hy 1ê2L^ 2DL x 2 +J yn2 x 2 +J yn2
15 Printed from the Mathematica Help Browser 1 VEPlot@dipole, Epilog 88Hue@.3D, Disk@8, 1 ê 2<,.1D, GrayLevel@D, Text@StyleForm@" ", FontSize 24, FontWeight > "Bold"D, 8, 1ê2<D<, 8Hue@.9D, Disk@8, 1ê2<,.1D, GrayLevel@D, Text@StyleForm@"+", FontSize 24, FontWeight > "Bold"D, 8, 1 ê 2<D<<D; + «14 of 14 Příklad 3/3 Elementární axiální oktupól octupole = 1êSqrt@x ^2 +Hy 3ê2L ^ 2D 3êSqrt@x^ 2 + Hy 1L ^ 2D + 3êSqrt@x ^2 +Hy 1ê2L^ 2D 1êSqrt@x ^ 2 + Hy L ^ 2D 1 x 2 + J yn2 3 x 2 +H 1 + yl x 2 +J yn2 1 x 2 + y 2
16 16 Printed from the Mathematica Help Browser 8x, 2.1, 2.1<, 8y, 1.1, 3.1<, Epilog 8 8Hue@.9D, Disk@8, 3 ê 2<,.1D, GrayLevel@D, Text@StyleForm@"+", FontSize 24, FontWeight > "Bold"D, 8, 3 ê 2<D<, 8Hue@.3D, Disk@8, 1<,.2D, GrayLevel@D, Text@StyleForm@" ", FontSize 24, FontWeight > "Bold"D, 8, 1<D<, 8Hue@.9D, Disk@8, 1 ê 2<,.2D, GrayLevel@D, Text@StyleForm@"+", FontSize 24, FontWeight > "Bold"D, 8, 1 ê 2<D<, 8Hue@.3D, Disk@8, <,.1D<, GrayLevel@D, Text@StyleForm@" ", FontSize 24, FontWeight > "Bold"D, 8, <D<D;
17 Printed from the Mathematica Help Browser
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