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Necessary Derivations For P1

The document discusses the divergence of a vector function in spherical coordinates, detailing the mathematical definitions and integrals involved. It explains the gradient of scalar fields and derives Laplace's equation in spherical coordinates, emphasizing the relationship between electric potential and electric fields. The document also outlines the application of Gauss's law and provides equations relevant to charge distributions in spherical systems.

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aashbral7
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12 views4 pages

Necessary Derivations For P1

The document discusses the divergence of a vector function in spherical coordinates, detailing the mathematical definitions and integrals involved. It explains the gradient of scalar fields and derives Laplace's equation in spherical coordinates, emphasizing the relationship between electric potential and electric fields. The document also outlines the application of Gauss's law and provides equations relevant to charge distributions in spherical systems.

Uploaded by

aashbral7
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF or read online on Scribd
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Pivesgence in Spharreal Coordinates = > w Livergence of a vector functim F of the spherival eonrching tes Ce, 4,6) is define! o& a g. = bn a7 Fee) = Hm (f Pat) , yao v have the curfate integral 's cared out on the surface of an element yolume centertol at (2/98) Li Flo 3 bi date perpndisolar b Q, = 1d a etde\_F, ie Ker wea ilo teat age) Ket ory = vost (OFS a4 (i) Flan X gor poves perpenchroutar te 6, ao = rine) apiel (ore, Hag 0149] _ KY vain (9) 14°F (rtd, @ 424, 0) ] 1 =a" (2G) : Wii) Flex Ky for faces perpendicular to Ky = (9492)d6(1-434) anny Fores drag e64e)) — Paosiabod 6 (x, frig, nee) 2 iy = L0sin6I4 a(«'F.) Add at eve flaw and dLivicte bog tue volume (iimesbdods) we get PPL y Vt) + 2 DF) + 2 9(F4) ia a8 rsing 26 rsin@ o¢ The Grractfent ‘m Sphasical Coordin tes, In any coordina fe apse, toe procient F a calor function ix: Grts)= 987 + Wm + O84, 92 Dm dn a4 wher, fm and ” are. ru tually Petpenolicular vectors at tu point of observation ond IS 98 R D9 aw the reigecbive, partial obtriva ves ah” ow mn Z A A a s Ace In ths sphericcel coordina te gystem, £m, a woud be 6, © B teach el and differen trod fengths ang tam would be: dy xsinddd & rdO rape To pave tmit "yo do o *% r Radial Pirection Azimuntnal Pivec tion Polar. Divection Beli eee Azimuntnal Fivteni BOTA NCEE Hence, te gradient of scaler fred ot point Cr, b,6) rs grid. S = dee rt /Os|$ + 2 as)é ay sind Dp r (20 Loplaces Equation Vv zy ) By row Divergence iv olegined, Div. = bin fe ae vo or any volume element taken at a in space. Hence, By gauss (aw, ss fret = Je , Zo D Equation Cj) yids tue Ppferential form df Gawrs's Law. SO E = -... @® 5 = Eni qadien of wee potenting Jumtion VO,9, 9) af tte cordinater WR y, oe FJ) + 12 O(Fosing) + 2 9(F9) v rsing a8 ying O46 Se, the a the & field wou give ua = P= 49 vi)r & Caste) ta Xp). cm) r dr ren vind Je" Next, we know that ete electric Pretd we eye gradient ff potential Bs 6, fo + Egbt GO a 4 vie eta wS cam) oe 5dB rsind Ob with there three equationd, we obtain => polsson's EQUATION TN GPHERICAL COORPTMATES 1 D6 ar «7sing Og YEON) + 2 damit) + 4 y= 24 x’sin’O or ~ For pre je. space here charge distw'oetion is vere, Uris sare egu ation fa eotled Laplace s Equation in spherical coord ina ter

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