Semester II, 17-18 Department of Physics, IIT Kanpur PHY13A: Lecture # 3 (Text Book: Intro to Electrodynamics by Griffiths, 3 rd Ed.) Anand Kumar Jha 8-Jan-18
Notes The first tutorial is tomorrow (Tuesday). Updated lecture notes will be uploaded right after the class. Office Hour Friday :3-3:3 pm Phone: 714(Off); 96-14-3993(Mobile) akjha@iitk.ac.in; akjha9@gmail.com Tutorial Sections have been finalized and put up on the webpage. Course Webpage: http://home.iitk.ac.in/~akjha/phy13.htm
Summary of Lecture # : Gradient of a scalar TT TT xx + yy + zz 3 1 1 3 Divergence of a vector VV 3 1 1 3 VV = vv xx + vv yy + vv zz 1 1 Curl of a vector VV VV = xx yy zz yy zz 1 1 1 vv xx vv yy vv zz 1 = vv zz vv yy xx + vv xx vv zz yy + vv yy vv xx zz 3 1 1
Integral Calculus: The ordinary integral that we know of is of the form: ff xx dddd aa In vector calculus we encounter many other types of integrals. Line Integral: VV ddll aa Vector field Infinitesimal Displacement vector Or Line element ddll = dddd xx + ddddyy + ddddzz If the path is a closed loop then the line integral is written as VV ddll When do we need line integrals? Work done by a force along a given path involves line integral. 4
Example (G: Ex. 1.6) Q: VV = yy xx + xx(yy + 1)yy? What is the line integral from A to B along path (1) and ()? Along path (1) We have dddd = dddd xx + ddddyy. (i) dddd = dddd xx ; yy=1; VV dddd= yy dddd = 1 (ii) dddd = dddd yy ; xx=; VV dddd= 4(yy + 1)dddd=1 1 Along path (): dddd = dddd xx + ddddyy ; xx = yy; dddd = dddd 1 VV dddd= (xx + xx + xx)dddd=1 VV dddd=11-1=1 This means that if VV represented the force vector, it would be a non-conservative force 5
Surface Integral: VV ddaa or VV ddaa Vector field Infinitesimal area vector Closed loop For a closed surface, the area vector points outwards. For open surfaces, the direction of the area vector is decided based on a given problem. When do we need area integrals? Flux through a given area involves surface integral. 6
Example (Griffiths: Ex. 1.7) Q: VV = xxxx xx + xx + yy + yy zz 3 zz? Calculate the Surface integral. upward and outward is the positive direction (i) x=, ddaa = dddddddd xx ; VV ddaa = xxxxxxxxxxxx = 4zzzzzzzzzz VV ddaa = 4zzzzzzzzzz=16 (ii) VV ddaa= (iii) VV ddaa=1 (iv) VV ddaa=-1 (v) VV ddaa=4 (vi) VV ddaa=-1 VV ddaa = 16 + + 1 1 + 4 1 = 8 7
Volume Integral: TT(xx, yy, zz)dddd In Cartesian coordinate system the volume element is given by dddd = dddddddddddd. One can have the volume integral of a vector function which V as VVdddd = VV xx dddd xx + VV yy ddττyy + VV zz dddd zz Example (Griffiths: Ex. 1.8) Q: Calculate the volume integral of TT = xxxxzz over the volume of the prism We find that zz integral runs from to 3. The yy integral runs from to 1, but the xx integral runs from to 1 yy only. Therefore, the volume integral is given by 1 yy 1 3 xxxxzz dddddddddddd = 3 8 8
The fundamental Theorem of Calculus: FF xx dddd = aa ddff dddd = ff ff(aa) dddd aa The integral of a derivative over a region is given by the value of the function at the boundaries Example xxdddd aa = aa dd xx dddd dddd = xx aa = aa In vector calculus, we have three different types of derivative (gradient, divergence, and curl) and correspondingly three different types of regions and end points. 9
The fundamental Theorem of Calculus: dddd dddd = ff ff(aa) dddd aa The integral of a derivative over a region is given by the value of the function at the boundaries The fundamental Theorem for Gradient: aa PPPPPPP TT ddll = TT TT(aa) The fundamental Theorem for Divergence VV dddd = VV ddaa VVoooo SSuuuuuu The integral of a derivative (gradient) over a region (path) is given by the value of the function at the boundaries (end-points) (Gauss s theorem): The integral of a derivative (divergence) over a region (volume) is given by the value of the function at the boundaries (bounding surface) The fundamental Theorem for Curl VV ddaa = VV ddll SSuuuuuu PPaaaaa (Stokes theorem): The integral of a derivative (curl) over a region (surface) is given by the value of the function at the boundaries (closedpath) 1
Spherical Polar Coordinates: rr ; θθ ππ; φφ ππ xx = rr sinθθ cos φφ, yy = rrsinθθ sinφφ, zz = rr cosθθ A vector in the spherical polar coordinate is given by AA = A rr rr + A θθ θθ + A φφ φφ rr = sinθθ cos φφ xx + sinθθ sin φφ yy + cosθθzz θθ = cosθθ cos φφ xx + cosθθ sin φφ yy sinθθzz φφ = sinφφxx + cosφφyy Griffiths: Prob 1.37 The infinitesimal displacement vector in the spherical polar coordinate is given by dl= dddd rr rr + dddd θθ θθ + dddd φφ φφ (In Cartesian system we have ddll = dddd xx + ddddyy + ddddzz ) 11
Spherical Polar Coordinates: xx = rr sinθθ cos φφ, yy = rrsinθθ sinφφ, zz = rr cosθθ A vector in the spherical polar coordinate is given by AA = A rr rr + A θθ θθ + A φφ φφ rr = sinθθ cos φφ xx + sinθθ sin φφ yy + cosθθzz θθ = cosθθ cos φφ xx + cosθθ sin φφ yy sinθθzz φφ = sinφφxx + cosφφyy Griffiths: Prob 1.37 The infinitesimal displacement vector in the spherical polar coordinate is given by dl= dddd rr rr + dddd θθ θθ + dddd φφ φφ (In Cartesian system we have ddll = dddd xx + ddddyy + ddddzz ) dl= ddddrr + rrrrrrθθ + rrsinθθθθθθφφ The infinitesimal volume element: dddd = dddd rr dddd θθ dddd φφ = rr sinθθddddddθθdddd The infinitesimal area element (it depends): ddaa = dddd θθ dddd φφ rr = rr sinθθddθθddddrr (over the surface of a sphere) 1
Cylindrical Coordinates: ss ; φφ ππ; zz AA xx = s cosφφ, yy = ss sinφφ, zz = zz A vector in the cylindrical coordinates is given by AA = A ss ss + A φφ φφ + A zz zz ss = cos φφ xx + sin φφ yy φφ = sinφφxx + cosφφyy zz = zz Griffiths: Prob 1.41 The infinitesimal displacement vector in the cylindrical coordinates is given by dl= dddd ss ss + dddd φφ φφ + dddd zz zz (In Cartesian system we have ddll = dddd xx + ddddyy + ddddzz ) 13
Cylindrical Coordinates: xx = s cosφφ, yy = ss sinφφ, zz = zz AA A vector in the cylindrical coordinates is given by AA = A ss ss + A φφ φφ + A zz zz ss = cos φφ xx + sin φφ yy φφ = sinφφxx + cosφφyy zz = zz Griffiths: Prob 1.41 The infinitesimal displacement vector in the cylindrical coordinates is given by dl= dddd ss ss + dddd φφ φφ + dddd zz zz (In Cartesian system we have ddll = dddd xx + ddddyy + ddddzz ) dl= ddddss + ssddddφφ + ddddzz The infinitesimal volume element: dddd = dddd ss dddd φφ dddd zz = ss ddddddφφdddd The infinitesimal area element (it depends): ddaa = dddd φφ dddd zz ss = ss ddddddddss (over the surface of a cylinder) 14
Gradient, Divergence and Curl in Cartesian, Spherical-polar and Cylindrical Coordinate systems: See the formulas listed inside the front cover of Griffiths 15