STiCM Select / Special Topics in Classical Mechanics P. C. Deshmukh Department of Physics Indian Institute of Technology Madras Chennai 600036 [email protected] STiCM Lecture 26 Unit 8 : Gauss’ Law; Equation of Continuity Essential tools to study Fluid Mechanics / Electrodynamics, etc. PCD_STiCM 1 Previous Unit: Unit 7 Physical examples of fields. Potential energy function. Gradient, Directional Derivative, …. PCD_STiCM 2 d uˆ Mathematical ds r dr Connections….. uˆ lim s0 s ds Differentiation Potentials Fields Integration / Constant of Integration Boundary Value problem SCALAR VECTOR POINT FUNCTIONS POINT FUNCTIONS PCD_STiCM 3 Johann Gauss’ Law; Carl Friedrich Equation of Continuity. Gauss Hydrodynamics & 1777 - 1855 Electrodynamics illustrations. Learning goals: WhenWhat there can is no cause source and a nochange sink, the density in the of matter indensity a volume element of the can fluid change inside if and only the of kettle?matter flows in, or out, of that region across the surface that boundsAt what that volume rate regioncan .that density The divergence theorem : an exact mathematical change? PCD_STiCM 4 expression of a conservation principle. The result is equally consequential with regard to fields just as well as for matter. We shall develop further handle on methods of vector calculus and apply the techniques to study fluid dynamics and electrodynamics. Johann Carl Friedrich Gauss and Wilhelm Weber http://www.gap-system.org/~history/PictDisplay/Gauss.html ˆ dV A()() r A r dSn volume surface regionTerrestrial Magnetism enclosing that region PCD_STiCM 5 Consolidated expressions for the GRADIENT Cartesian Coordinate System d uˆ ds =eˆx e ˆ y e ˆ z x y z r dr uˆ lim s0 Cylindrical Polar Coordinate System s ds 1 sr =eˆ e ˆ e ˆ z z Spherical Polar Coordinate System 11 =eˆ e ˆ e ˆ r r r r sin PCD_STiCM 6 d uˆ The ‘GRADIENT’ is a vector operator ds – it is of course not a vector. The operator would operate on an operand and generate new entities as a result of the operation. Operand: SCALAR POINT FUNCTION. RESULT : Other operations using GRADIENT OPERATOR Ar( ) : DIVERGENCE of a VECTOR POINT FUNCTION Ar( ) : CURL of a VECTOR POINT FUNCTION PCD_STiCM 7 GRADIENT of SCALAR POINT FUNCTION. RESULT : Other operations using GRADIENT OPERATOR A(r ) : DIVERGENCE of a VECTOR POINT FUNCTION This is NOT a scalar product of two vectors! Ar( ) : CURL of a VECTOR POINT FUNCTION This is NOT a vector product of two vectors! PCD_STiCM 8 Field strength: Field Lines 1 q VECTOR POINT E() r eˆ 2 r FUNCTION 4 0 r Field intensity fall like 1/r2 PCD_STiCM 9 Vector Fields: V V()(,,) r V x y z ‘Point’ V V(,) r t V(,,)(,,) r V z function PCD_STiCM 10 Vector Fields: V V()(,,) r V x y z ‘Point’ V V(,) r t function V(,,)(,,) r V z In the ‘continuum model’, the velocity field V V() r is a vector point function. PCD_STiCM 11 Ar( ): A vector point function. Define: “Flux” of a vector point function Flux crossing a surface A ( r ) dSnˆ surface Flux: additive property - obtained by integrating the quantity what is the direction/orientation of nˆ ? UNIT NORMAL TO THE SURFACE AT A GIVEN POINT .... but which surface? .. what is the directionPCD_STiCM of the unit normal?12 Ar( ) : A vector point function. Flux crossing a surface A ( r ) dSnˆ surface Elemental directed/oriented ‘Area’ PCD_STiCM 13 Flux crossing a surface The direction of the vector A() r dSnˆ surface element must be defined surface in a manner that is consistent with the forward-movement of a right-hand screw. C traversed one way right-hand-screw convention must be C traversed followed. PCD_STiCM the other way14 PCD_STiCM 15 Consider the sense/direction in … right which the rim of the hand net can be traversed screw What about some other point ? … and what if you ‘pinched’ the net and pulled it ‘above’ the rim? PCD_STiCM 16 Are there non-orientable surfaces? The surface under consideration, however, better be a ‘well-behaved’ surface! A cylinder open at both ends is not a ‘well-behaved’ surface! A cylinder open at only one end is ‘well-behaved’; isn’t it already PCD_STiCMlike the butterfly net? 17 Consider a rectangular strip of paper, spread flat at first, and given two colors on opposite sides. Now, flip it and paste the short edges on each other as shown. Is the resulting object three-dimensional? How many ‘edges’ does it have? How many ‘sides’ does it have? PCD_STiCM 18 How is the earring? It is MOBIUS !!! PCD_STiCM 19 http://mathssquad.questacon.edu.au/mobius_strip.html The Möbius strip used to be common in belt drives (eg. car fan belt). Modern belts are made from several layers of different materials, with a definite inside and outside,PCD_STiCM and do not have a twist. 20 Maurits Cornelius Escher 1898 - 1972 http://www.mcescher.com/ “The laws of mathematics are not merely human inventions or creations. They simply 'are'; they exist quite independently of the human intellect. The most that any(one) ... can do is to find that they are there, and to takePCD_STiCM cognizance of them. ” 21 Flux of a vector field Ar( ) : A vector nrˆ( ) : unit vector point function normal to the surface at the point r Flux crossing a surface A() r dSnˆ surface PCD_STiCM 22 Flux crossing a surface Is there any net accumulation of A() r dSnˆ the flux in a volume element? surface Source Sources and Sinks may be present in the region ! Sink What happens when the size of the volume element shrinks, becoming infinitesimallyPCD_STiCM small? 23 Consider a mass/charge density mc or crossing a certain cross-section of area at a certain rate. m (rr )v( ) has the dimensions -3 1 -2 1 ML LT ML T -3 1 -2 1 c (rr )v( ) has the dimensions QL LT QL T Amount of charge crossing unit area in unit time Amount of mass/charge crossing unit area in unit time PCD_STiCM 24 Consider a point P( x0 , y 0 , z 0 ) in a region of the vetor field Ar( ) How shall we determine the flux crossing a surface; A() r dSnˆ ? surface PCD_STiCM 25 Z Y δx (0,0,0) δz X δy PCD_STiCM 26 Z Y eˆz δx Two faces parallel to the xy plane δz (x0,y0,z0) The point P( x0 , y 0 , z 0 ) is in the δy X region of the vectore field Ar( ) eˆz NET Flux through the xy-face, (perpendicular to eˆz ), is zz Az x 0, y 0,, z 0 A z x 0, y 0 z 0 x y 22 Note! Only the z-component Flux crossing a surface contributes to the total flux ˆ through the faces parallel to A() r dSn the xy plane. PCD_STiCM surface 27 Flux crossing a surface Z Y A() r dSnˆ surface δx NET Flux through the xy-face, (perpendicular to eˆz ), is zz A x y,, z A x y z x y z 0, 0 0 z 0, 0 0 (x0,y0,z0) δz 22 δy X A Az z z x y V z z (,)x0, y 0 z 0 (,)x0, y 0 z 0 Flux crossing all the six surface elements Adding the flux through all faces, total flux that enclose the cell A Ax y Az A() r dSnˆ A( r ) dSnˆ dV surface x y z whole PCD_STiCM 28 cube Flux crossing a surface Adding the flux through Z Y A() r dSnˆ all faces, total flux surface A δx Ax y Az A( r ) dSnˆ dV x y z closed (x ,y ,z ) surface 0 0 0 δz X The integrand of the volume integral is δy called the divergence of the vector. A Ax y Az div AA x y z ˆ d A()() r A r dSn volume surface region enclosing that region PCD_STiCM 29 Gauss’ Divergence Theorem Divergence of a polar vector is a scalar Divergence of an axial-vector is a pseudo-scalar PCD_STiCM 30 We shall take a break here……. Questions ? Comments ? [email protected] http://www.physics.iitm.ac.in/~labs/amp/ Next: L27 Equation of Continuity PCD_STiCM 31 STiCM Select / Special Topics in Classical Mechanics P. C. Deshmukh Department of Physics Indian Institute of Technology Madras Chennai 600036 [email protected] STiCM Lecture 27 Unit 8 : Gauss’ Law; Equation of Continuity PCD_STiCM 32 Flux crossing a surface Adding the flux through A() r dSnˆ all faces, total flux surface ˆ d A()() r A r dSn volume surface region enclosing that region Gauss’ Divergence Theorem PCD_STiCM 33 ˆ Application: d E(r) Er( ) dSn volume surface electric intensity region enclosing field due to a that region point charge q d Er() eˆˆ r2 sin d d e 2 rr volume surface 4 0r region enclosing that region q 1 ddEr() d volume0 0 volume volume 0 region region region Er() Differential (or ‘point’) form of PCD_STiCM 34 0 the Gauss’s law Flux crossing a surface Z Y A() r dSnˆ surface A δx Ax y Az A( r ) dSnˆ dV closed x y z surface (x0,y0,z0) δz X The integrand of the volume integral is δy called the divergence of the vector. A A A dVdiv A AA()() r x y Az r dSn ˆ Gauss’ Divergence volumex surface y z region enclosing Theorem that region Cartesian expression of ‘divergence of the vector’ not its definition! PCD_STiCM 35 Physical meaning of ‘divergence’; definition free from coordinate system ˆ Take the limit of dV A()() r A r dSn volume surface the ratio of region enclosing that total flux over δs region to δV A(r) ndSˆ enclosing flux per unit volume, surface divAA lim at that point V 0 V remember: flux is defined through a surface, whereas divergence is defined at a point Flux is a scalar quantity.