Coordinate System

iCoordinate SystemPoint, , ? (1,2,3)Point is a sphere of radius 'r'whererOxx(3-constants)or"No variablesLine l2 constants x & Zy=oor(a,o,b)(a, 2, b)\ variable i.e "y.line is a cyllinder of radiuse0^3Surface:-Iconstanstant (x)Ad3orIgKdy2variables (y & 2)x(aN)#By definition surface is a Vector, ds = dydz axSurface vector defined as S = (Area) onfor abovesurfaceVolumeNo constantsorI3- - variables(x, y.2)aBy definition volume isScalaraydvdxdy dzai Unit Vector:-axlaxlIx-> i or as or 5'sair is in the direction of increasing^x Valvey j or ay or vig- -axk or ciz or U3= --ax is in the direction of decreasingx value#aIf a Vector A = Ax ax + Ay ay + Az az^thenUnitvectorinthedirectionofAisAAawhere A =AAxan+=Ayay+AzazIAI2Ax + Ay + AZwhere 1 A lmagnitude of vectorAsisthei.e = (Al =AX + Ay + A2sScalarorDot productofA.vectors is,^NIfA=Axax+Ayajy+A2azisxOAB32 = Bx an + By ay + Bz az^AThen Scalar Product :-A.B =TAllslCosQABor->=COS OAB =A.B=AxBx + Ay By + AzB2->A11B1AX+A Ajt + AZ BiAax. ax = 10x1/an/coso= Ias ay = I = ay . az = = 0ax I ay 1az. a3 = - ; 92 = o unit vector in different coordinate Systems:-Cartesiana: , ag ) azai 1 ay ^ 1 azCyllindricalai, as , azas 1 as 1 a3aAsphericalair, ao , asair 1 a 1 asVectorofanRoll your fingersfrom^azThumbA to Bwill pointIf A X BaNtoward theindir" of a.N.inaixayi.e=>ax a x ay = Sin 90° a - 3 = a3aazByayayx xaxany3as airaearaza zaaya'oasaocartesianco-ordinatecyllindricalSphericalco-ordinateco-ordinateScolar: It is a quantity having magnitude and signE.:- Q : charge (coulomb)Y : Electric flux(couloub)t : time ( second)I : current ( Amp)P : magnetic flux (wb)Vector It is a quantity having magnitude & directionEx:- J Current density (Amp/m )B : Magnetic flux density (wb/m2)D : Electric flux density ( c/m2)E: Electric field Intensity ( Volt/m)B = liole, HD = E. tr E# Peline charge density =Q=c/mllsSurface charge density = Q = c/m2Aiv = Volume charge density = Q = c / m 3V Note: To multiply with meter take Sde [ line IntegralEx:- If Amp/m is given then multiply to get Amp.I =SH.dlwhere Hnon-uniformAmp/m-s=H.dlIf Huniform^#H=2ax Amp/m it is uniform and staticIT = 2x ax Amp/in : n " non- uniform and Stat isH = 2 Sinwt ax Amp/m : u in uniform G2. To multiply with m2 take lldsEx:I=J.dsIf J : non - uniform Amp/m2- J.dsIfJ :uniformAmp/m23. To multiply with m3 take115dvQ =S Pv dv for Pv = c/m3 non- uniform=Pv(volumefor evuniform#Sedl=PeL = Q#Q = 11 P ds ls (area)==non-uniformuniformnon-uniuniformline chargedusity # Let A be the any vector which is the Sum of of its"->"tangential vector At and normalvector"AN"A=At + ANi.e1A1IANI01.At= = lAlCos OIf 1A1 & 01 At 1is known.An =IAICosO2. AA =1A,12 IANI2If Atl & IANIis Known.O = tan -1 AN1 AtlIRectangular or Cartesian co-ordinate.zyTx2Circular or cyllindrical Co-ordinate.2a:- Cifl. to Rest: x = Pcos 0y- PsingP(x,y,2)Z = 2Zyb. Rect to cyl - e. = jx'+y2xa = tan' Y/XZ = Z 3.Spherical Co-ordinate System:-aza. Rec from Sph.p(x,0,9)x = rsino CosP(x,y,2)y= r sino sin &Zoe(e,6,2)Z= r cosoIagb. Cyl. from Sph.eP= r sino&0 =2 = rccose OinC. Sph. from cyl.# O= (as"((z)r= Je2 z2& = &0 = cos' Zo = CosZe2+22p2+z20= cos' Zx2 +y2+ 22Rest. Sys.Cyt Co.or. syssph. co- Sys.1.General(x,y,2)(e, o, z )PaintP. radius of a circle","radius ofa sphereorcylinder"verticle anglehorizontal angle(b/w Z-axis & r-line)(b/w x-axis &0horizontal anglee- line)razairdo7dp 2. UnitGa; ay ; azVectorsae 1 as 1 azair 1 a'o 1. asaax1 aj 1 azapris radiallyis is radially outwardsoutwardfrom origin (or)^a,Spherical surface.Is horizontally1Circular inaois vertically circularanticlockwisein downwards directiondirection an aan a Spherical Surfacecylinderas is horizontalycircular in anticlockusedirection on a sphericalSurface.3 Shapesconst. surface isconstant is a cylin. stconstant is a spleaved Surfaceofarectangular11Rec. softhe9" " 4 conical "surfacesSurfaceZ-> >,circular sofyo "=)"semicircular Srf.P= constant2->"azL=Dazaz^^eayOZ=OconstantsurfaceSurfaceRonge- D < xose< 00os r e aofParame ters-D < y<00to cover all05 & < 2KO < 0 < R- D< 24.00regions00 < Z < 00osps. 2T Restain. Co-orcyl - co-sysph - co-Sys511xdxederdrDifferentialydyQedoOrdoLengthN->dzZdz0rsino doIDifferential length Vector:-ARec.dl = dx ax + dy aj + dz az^de =nCyp.de a, + e do as + dz a3Sph.dl = dr a" + r do ao + r sing do as2Differential Surface VectorsaRect. ds = dy dz ax + dx dz ay + dx dy az^cgl. ds = Pdd dz ap + de dz as + ede do az A^^^Sph. ds = r'sinodode a + r sino dr de ag + rdr do as3Differential Volume vector:-Rut . dv = dx dy dzcyl. . dr = rdr do dzsph. dv = r2 sino dr do do Conversion Tables0=0+90^Iaeapazx = ecosyy = esingaxCos- sing0Z = ZaaySin &COS &0a3OOIMethodAaax =AaCosp a - sing ao +oaza^ap =cose ax + sing ay + oaz^as =^- Sin $ax+(os 0ay+oazMethod QIf givenA= = Ap ap A+Apas+Azaz(Given)theninRec.Sys.AS=Axan +Ayag+AzazwhereAxcosd- Sing0ApAy=singcosqOAsAzOOIAZor->^IfA= Axax+ As ay t Az a3(Given)thenA,cosesin oAxAg- Sing(osso=AyAZOo1A2Knownunknown nA= 2a, a + 5 a g + 6 apXAXair ao aoa sin o cose 0\as O 0az coso -Sino 0Aa = Sino ae + cosa azao a = (coso ai - sino az )as ^ = a.sAepz = +5 (cosoal-sino az) -+ 6 00A= (2 sino + Scoso ) a e + Gap+ (2 coso - 5 sinQ) az or^A=2a. + 500 t 6 asredIconstants., ArAoApAAAp 02 = Apa + Ap ay + AzazAesin oCOSOoAr=)iAo=As0O=COS. OsinooofAQAZApsin oCOSOO2=OI5AsO6COSOsinoOAZL_Ap = 2 Sino + SCOSOAp = 6A2 = 2COSO - s sinoAAppz = (2 Sino + scoso)a" + 6 as a t (2coso - s sino) az-> 0=0+9010=90a0=0190A2 ndairaoas.x rsinocosesairSino CostCOSO Cos 0- Singy - r Sino SingAZ = r cosoaySino Singcoso SingCos &aazCos 0- SinoO3rda^.araoapP= r sinoaSinoCOSOOZ = r cosoAasOO!Q = 0 .asaCoso- sino0 TEAT Row st element is aszx= e cosInYE e Sinco2andRaw1st element 1/4 AtI2= ZLaeasazareCosp +90° -Singox+90}:- drs 'o'to yood notaySingCospoA42OOcostsinp= 1DeterminantSingcos $1stelementSimilarly :-x r sinocase=4thy =Sino singZ=r COS Q -> r element^a^as 0=0+90aoasOD1ax Sino cosecose cost,- -sinpus020192only 0+901&Sino Singcose singo costay0+909=0ttaA1927 Coso-sing -1 ONo relationP= r sino4 = &Z= r costas ^AA90+0aoapAapsinoCOSOOAasO0Aorio- sinoOazcoso Some operators:-1.Vi.cVector differentialoperatorAV= d dx ax + d ay + 2/2 d azoyozaGradiant (or) Grad((A)where A is any scalarslopei.egivesnormal tosurface.#A = a (xyz) ax + a (xyz) ag + d(ayz) 131If A = xyzdxdydzVA = yz as + xz aj + xyazIf A= 02 VA= d (es2) ae A + 1 a (exz) apt a dz (eoz) azdeP 00ATA = oz a + Z as + eqaz#It A = roo7A = (r 80) ar + \d (rop) ao a +-a (rop) an/rasxtorsino 00^TA = 00 ag^+0ao+OapsinoAIT:a, +1dCyp.=ap + dazP 00dz^Sph.V= d ai +\ddao +asorr00rsino00ExEDVwhere'-' indicates E is from higher=Pot". to lower Pot"Valtm14velt 2 - Divergence ( or) Div or V.(vecTor)Let A be any Vector A= Ax ax + Ay any + Az asV.A =Ve[Divergence of A&+veshowsAisgoingawayfrom Source]T.A =Ve[Convergencei.eAis coming towards sink]V.A =O[Divergence lessorSolenoidal ]->Divergence of A=) V.A = (2) dx az + dy a ag + dz a as) (AxaintAging + Azag)= d Aza+AytAzdxdydzCyl. A=A, a, + Ap as + Az azV.A = (6 2/6 de a + + e d 00 ag + 1/2 az) (PAP a + Ap ap t Azaz)T.A =Id (PAP) + 1 e d 00 Apd Az+P deJzA#A=Arao^Sph.+Aoag+Asap3.A =11/2 82 ord(r2Ar)+1d (sino Ao)1+d(Ap)r Sino 00rsino 00#VD=evscalar(c/m2)DotvectorVectorProductcharge(scalar(c/m2)density.( in )product) 3. Curl (vx A') i.e CurlRotationanayazV&A =aoaoxoyozAxAyA2^Acyl.sys.appapazVXA =adaPde00ozApPAsAz# Sph. Sys.^^arr aoV x A -/addr2 sinoordoasArrr Agrsino AsEx:-vectorV x H/(Amp/m2)VectorvectorVectorcurrent(1)Product(Amt/m)density.IfAsscalarVAScalar= 4 Laplacianoperator:=2If Aryectorvectorascuar vector } BothLaplacian22Ret V2 2 = ( (A) +à (Ad+0x22aydzcyl= T'A - IofA),1+22 (A) 2 (A)P dedee 2dz2V'A =Id(r2)a(A)a( sino(A)+82 orsinodosino 002 #Properties of Gradiant of Any VectorThe magnitude of v v equals The max rate of changein unit afrector per unit distanceGradiant of "v" points in the direction of the max in rate ofchange in VGradiant is normal (b ) to the constant value surface a givenpoint.It operates an a scale function and results in a vectorfunction.The projection or component of V in the direction of a unitvector 'a' is TV a ^ " and it is called DirectionalDerivativeProperties of CorlIt tells us the idea of Vector field Rotation about point.VxA = 0 i.e A has no - rotation or Irrotational VectororConservative= -ve i.e " " clock use rotation=+ve " " " Anti-"Curl of any Scalar has no Sense"#If A is scalar & if then A is harmonicI(VA) i.e CG=O2J. ( A ) =0 i.e DC=O3V2A =V (V.A) - Vx(DxA) i.e GD-CC -4v.(BxC) = C. (VxB) - B. (vxc) i.e Cab - bac5AxBxc = B (A.C) - C (A.B)6A.(BxC) = (.(AxB) = B. (AXC) Two Imp. theorems1. Stokes - "Circulation of vector field in a closed pathis equal to surface integral of corl of thatVector bounded by the closed loop."I A.dl=dswhere C isCclosed phathclosedSurface by the closedphath2. Divergence Theorem : Total outward flux through any closedof a vector is equal to volume integral of divergenceof that vector.$A.ds(V.A) dvVolume=Sclosed surfaceSource - a. Non moving charge or static chargeProduces Static Electric field. (I=0)i.e dq0dtb.DC current produces static magnetic fieldCTime varying current produces time varying electricand magnetic field.# Some variablesGCLIN/W&(F)(H)(Valt)AmpCwbvariablestheir unitsDEllEMTC(0% (F/m) (H/m) (v/m) AwbVariablesm2 Stokes LawoGiven E= - yesing ap find PE.di onthegivenclosedpathxAcc to stokes theo. PE.D =01 2Soly^I(0,1)0-line (0 to i) at P=1 i.e ap(10)dle-line ( 0 to 1) i.e al line (1-00) i.e ap$ E.II = S He sin o l a, . de a + S (He sinp) a P do ag AACBPEOTOL020 00+ J 4e a"BOPE (1 too)= O + O +4 sinP de = - 222II(EXE) ds VXE = 1 ap as az2/%ae00 0212/2upsing O 0= 1 - 40 COS 9) a3 = -4 coss a3 ^e= II (-4COS p) a3 . Ledidzait7/2= -4J close ededo = -200= Divergence theoremOilA= xyan + yzaj + zx az then find I A.ds for OEXS2SSal Acc. to Divergence theorem :- paids(V.A)dvS2D PAIDSSsA. (-dydz ai)= -fxydydzOx=0x=0&,-dydzaxx 2 S (A. dydzai) - x-2 jjxydydz = dz2H= 4x2=82.2dydzaxy=o=>IIA. (-dxdz aj) = = O. 2 =2youy=2 Surface = SA. (dydzaj)If2 2=y2dxdz =y=22=0 Surface(A.(-dxdyqz)=2=0Z=2 surfaceJA. (dxdyaz)=2x dx dy = 800Z=2SAIDS = 0+8+0870+8 = 242S(V.A) dv =) J.A = y+z+z222dxdy dz =00000058+8+8= 24 Positional VectorCartesian co-or.2 cylin co-or3 sph. co.or2zA(P,P,2)d A(x,Y,2)(8,0,0)dd)>(0,9,0)oY(010,0)(0,0,0)OA (p-o)aeOA=(r-o)a'OA=lap+203OA=rard=10A= 810A1d= xt+y2+z2d=p2+22d= x2fy4zzN/W forEMTI=GV - ResistanceJETEQ=CV- Capacitance , D=EE-=LI < Inductor->B = liftA+=>J.J + dev# And we knowEquation of Continuityotthat J.D=PvV.J=-dpot[[e] E -der itat t= ek=lv =V.OE = - derPr=eet -It ekItv it -dev- - deE evblu T E [V.EE] atPv= Proce- It tk= lupvEblu [[.V.D]= -dev at-everIt+K Helmholtz's Theorem1.Solenoidal and IrrotationalV.V = OandDXV=0Eg: A static electric field in charge - free region2. Solenoidal but not irrotationalD.F = 0 & JxF #0Eg . A steady magnetic field in a current carrying Cond.3. Irrotational but not SolenoidalTxF = 0 & J.F # OEg. A static electric field in a charged region4. Neither Solenoidal nor irrotationalJ.F IO & DxF # OEg: . An electric field in charged medium with a timevarying magnetic fieldXIEq" of Line . x-x, = Y-Y, = Z - Z, or x-x, = y-y, Z-2,x2-x1 Y2-Y, Z2 - 2, a 6Cx7/21x2g,22Where a, s &c are direction cosine of line2 = m + in where is n -vector 11 to given livem :- Line is passing through 131Eq" of Plane :ax + by +CZ +d=0Intersection of the & plane28. in =d in is I to the planeunit vector" in pleas by put