ELECTRODYNAMICS—lecture notes second semester 2005 Ora Entin-Wohlman References: 1. J. D. Jackson, “Classical Electrodynamics”, Wiley. 2. G. B. Arfken, "Mathematical methods for Physicists", Academic Press. Summary of vector analysis 1. Gradient, F F F F xˆ yˆ zˆ x y z , a. F (r ) is perpendicular to the surface of constant F that includes the point r . r dF b. The gradient of F (r ) : F r dr 2. Divergence, a. V Vx Vy Vz , V x y z is the flux per unit volume through an infinitesimal closed surface, lim 1 V V nˆda v 0 v S where is the volume, S is the surface, and n̂ is a unit vector normal to the surface. 3. Curl (rotor), a. The n̂ xˆ V , x Vx yˆ y Vy zˆ z Vz component of the rotor is related to its circulation by lim 1 nˆ V V d S 0 S C where n̂ is normal to the plane where C lies. F (r ) V r F (r ) V 0 , V 0 . Exercise: Given the vector , find so that Solution: F (r ) const. / r 3 4. Div.grad=Laplacian, 5. Rot grad =0, , 2 2F 2F 2F F 2 2 2 x y z 2 F 0 2 V 0 6. Div.rot=0, 7. Rot 2 rot, ( V ) ( V ) V 8. Gauss’s theorem ˆ V n da d r V S V Important consequence: 1 r 4 dr r dr r 3 0 Exercise: Show that when V A P, where A is a constant vector, Gauss’s theorem gives ˆ n P da d r P . Hint: Use S V ˆn A P nˆ P A nˆ P A and A P P A A P . Solution: When A is a constant vector, then A P A P . Gauss's theorem gives ˆ ˆ d r A P n A P A n P A d r P. S S V The last equality gives the desired identity. 9. Stoke’s theorem V d V nˆda S Exercise: Show that when V AF , where A is a constant vector, Stoke’s theorem gives ˆ n Fda Fd . Hint: use S 3 ( AF ) F A (F ) A . Solution: When A is a constant vector, then ( AF ) (F ) A . Stoke's theorem gives ˆ ˆ ˆ FA d AF nda F A n A n F A Fd . S S S The last equality gives the desired identity. The delta-function Dirac delta-function obeys the following in one-dimension: ( x a) 0, x a a dx ( x a) 1 a and has the properties dxf ( x) ( x a) f (a) f ( x) i 1 df dx dxf ( x) ' ( x) f ' (a) ( x xi ) xi Delta functions of trigonometric functions (n and m are integers): 2 d cos m cos n n ,m 0 2 d sin m sin n n ,m 0 4 2 d cos m sin n 0 0 We also use: n n' a x sin x nn' a a 2 a dx sin 0 a/2 1 1 dx cos(x / a) cos(' x / a) ' a a / 2 2 Exercise: Calculate dxf ( x) (ax b) 1 b f ( ) . |a| a Solution: When a>0, we write y=ax+b and obtain 1 1 b y b dyf ( y ) f ( ) a a a a When a<0, we write y=-|a|x+b and obtain b y 1 1 b 1 b dyf ( y ) f ( ) f |a| | a | |a| |a| |a| a. Both results are obtained by using 5 (ax b) 1 b (x ) , |a| a with no need for the change of variables. In more than one dimension, the delta-function becomes a product. In Cartesian coordinates (r R) ( x X ) ( y Y ) ( z Z ) In spherical coordinates (r r ' ) Exercise: Calculate 1 (r r ' ) (cos cos ' ) ( ' ) r2 dxdyF ( r ) ( A r B), where A is a two by two matrix, B and r are two-dimensional vectors. Solution: Written explicitly, we have dxdyF ( r ) ( A11 x A12 y Bx ) ( A21 x A22 y B y ) . We now change the variables of the integration: x' A11 x A12 y y ' A21 x A22 y , 1 ( A22 x' A12 y ' ) det A . 1 y ( A21 x' A11 y ' ) det A x and This can be put in matrix notations as It remains to note that that Ar r ' and A1r ' r . x x' dxdy y x' x dx' dy ' y ' dx' dy ' y | det A | , y ' 6 to obtain the integral in the form 1 1 1 1 dx' dy ' F ( A r ' ) ( x' Bx ) ( y ' By ) F ( A B). | det A | | det A | Examples: 1. Gaussian representation of a one-dimensional delta-function, the limit 0 of x 2 ( x) exp 1 2. Gaussian representation of a three-dimensional delta-function, the limit 0 of 1 (r ) 2 3/ 2 1 2 2 2 exp x y z 2 2 3 1 3. Other one-dimensional delta-functions: (a) Lorentzian--the limit 0 of ( x) (b) of 1 1 ; 1 ( x / ) 2 sin( x / ) 1 ( x) x 2 1/ ixt dte 1 / (c) and of ( x) 1 e ( x / ) , x 0 4. From Gauss’s theorem, (r ) 1 2 1 4 r 7 8