Review: Electrostatics and Magnetostatics

Transcription

1 Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion in time. Theefoe, ρ, E and D ae constant and thee is no magnetic field H, since thee is no cuent density J. MAGNETOSTATICS The chage cossing a given cosssection (cuent) does not vay in time. Theefoe, J, H and B ae constant. Although chages ae moving, the steady cuent maintains a constant chage density ρ in space and the electic field E is static. Amanogawa, 2006 Digital Maesto Seies 5

2 The equations of electostatics ae obtained diectly fom Maxwell s equations, by assuming that / t, J, H and B ae all zeo: E = 0 D = 0 D=εE The electostatic foce is simply F = qe We also define the electostatic potential φ by the elationship E = φ Amanogawa, 2006 Digital Maesto Seies 6

3 The electostatic potential φ is a scala function of space. Fom vecto calculus we know that This potential is vey convenient fo pactical applications because it is a scala quantity. The potential automatically satisfies Maxwell s cul equation fo the electic field, since E φ=0 = φ= Fom a physical point of view, the electostatic potential povides an immediate way to expess the wok W pefomed by moving a chage fom location a to location b: b W = F dl a Hee, l is the coodinate along the path. The negative sign indicates that the wok is done against the electical foce. 0 Amanogawa, 2006 Digital Maesto Seies 7

4 By intoducing the electostatic potential, we obtain W q E dl q dl q b a q b a b a ( ) ( ) = = φ = [ φ φ ] = δφ NOTE: The line integal of a gadient does not depend on the path of integation but only on the potential at the end points of the path. The electostatic potential is measued with espect to an abitay efeence value. We can assume fo most poblems that a convenient efeence is a zeo potential at an infinite distance. In the esult above fo the electostatic wok, we could set a zeo potential at the initial point of the path, so that φ(a)=0. Eithe choice of potential efeence would give the same potential diffeence δφ. Amanogawa, 2006 Digital Maesto Seies 8

5 It is quite convenient to expess also the divegence equation in tems of the electostatic potential. If we assume a unifom mateial medium: 2 D=ε E= ε φ= ε φ=ρ This esult yields the well known Poisson equation 2 ρ φ= ε In the case of ρ = 0, we have the classic Laplace equation 2 φ=0 Amanogawa, 2006 Digital Maesto Seies 9

6 If the poblem involves a non-unifom medium with vaying dielectic pemittivity, a moe geneal fom of Poisson equation must be used ( ε φ ) = ρ Anothe impotant equation is obtained by integating the divegence ove a cetain volume V V DdV= ρ Gauss theoem allows us to tansfom the volume integal of the divegence into a suface integal of the flux V V S dv DdV = D ds Component nomal to the suface Amanogawa, 2006 Digital Maesto Seies 10

7 The volume integal of the chage density is simply the total chage Q contained inside the volume V ρ dv = Q The final esult is the integal fom of Poisson equation, known as Gauss law: S D ds = Q Most electostatic poblems can be solved by diect application of Poisson equation o of Gauss law. Analytical solutions ae usually possible only fo simplified geometies and chage distibutions, and numeical solutions ae necessay fo most geneal poblems. Amanogawa, 2006 Digital Maesto Seies 11

8 The Gauss law povides a diect way to detemine the foce between chages. Let's conside a sphee with adius suounding a chage Q 1 located at the cente. The displacement vecto will be unifom and adially diected, anywhee on the sphee suface, so that 2 4 S D ds D = π = Q Assuming a unifom isotopic medium, we have a adial electic field with stength E D ( ) = = 1 ε If a second chage Q 2 is placed at distance fom Q 1 the mutual foce has stength Q ( ) 1 Q F = Q 2 2 E = π ε 2 4 Q π ε Amanogawa, 2006 Digital Maesto Seies 12

9 The electostatic potential due to a chage Q can be obtained using the pevious esult fo the electostatic wok: b b Q φ ( b) =φ( a) E d=φ( a) d a a 2 4 πε Q 1 1 =φ ( a) + 4 πε b a whee indicates the distance of the obsevation point b fom the chage location, and a is a efeence point. If we chose the efeence point a, with a efeence potential φ(a) = 0, we can expess the potential at distance fom the chage Q as Q φ ( ) = 4 π ε Amanogawa, 2006 Digital Maesto Seies 13

10 The potential φ indicates then the wok necessay to move an infinitesimal positive pobe chage fom distance (point b) to infinity (point a) fo negative Q, o convesely to move the pobe fom infinity to distance fo positive Q (emembe that the wok is done against the field). The pobe chage should be infinitesimal, not to petub the potential established by the chage Q. The wok pe unit chage done by the fields to move a pobe chage between two points, is usually called Electomotive Foce ( emf ). Dimensionally, the emf eally epesents wok athe than an actual foce. The wok pe unit chage done against the fields epesents the voltage V ba between the two points, so that: emf = E d l = V b a ba Amanogawa, 2006 Digital Maesto Seies 14

11 In electostatics, thee is no diffeence between voltage and potential. To summaize once again, using fomulas, we have Potential at point a Potential at point b φ a b a = E dl φ b = E dl Potential diffeence o Voltage between a and b φ φ = = = b b a ba a ( ) V E dl e. m. f. Amanogawa, 2006 Digital Maesto Seies 15

12 In the case of moe than one point chage, the sepaate potentials due to each chage can be added to obtain the total potential 1 φ ( xyz,, ) = 4 πε i Q i i If the chage is distibuted in space with a density ρ(x,y,z), one needs to integate ove the volume as 1 φ ( xyz,, ) = πε 4 V ρ ( x', y', z' ) dv Amanogawa, 2006 Digital Maesto Seies 16

13 In the magnetostatic egime, thee ae steady cuents in the system unde consideation, which geneate magnetic fields (we ignoe at this point the case of feomagnetic media). The full set of Maxwell's equations is consideed (setting / t = 0 ) with the complete Loentz foce E = 0 H = J D =ρ B = 0 D=εE B=µ H F q E = + v B ( ) Amanogawa, 2006 Digital Maesto Seies 17

14 It is desiable to find also fo the magnetic field a potential function. Howeve, note that such a potential cannot be a scala, as we found fo the electostatic field, since H = J Cuent density is a vecto 0 We define a magnetic vecto potential A though the elation A = B = µ H This definition automatically satisfies the condition of zeo divegence fo the induction field, since B = A = ( ) 0 The divegence of a cul is always = 0 Amanogawa, 2006 Digital Maesto Seies 18

15 The vecto potential can be intoduced in the cul equation fo the magnetic field ( ) ( ) 2 H = B= A = A A= J µ µ µ Howeve, in ode to completely specify the magnetic vecto potential, we need to specify also its divegence. Fist, we obseve that the definition of the vecto potential is not unique since: A ' = A + ψ = A + ψ = A ( ) ( ) ψ = Scala function Always = 0 Amanogawa, 2006 Digital Maesto Seies 19

16 In the magnetostatic case it is sufficient to specify (in physics teminology: to choose the gauge) A = 0 so that A= A+ ψ = A+ ψ= A+ ψ= ( ) 2 0 We simply need to make sue that the abitay function ψ satisfies 2 ψ = 0 We can then simplify the pevious esult fo the cul equation to 2 A= µ J the magnetic equivalent of the electostatic Poisson equation. Amanogawa, 2006 Digital Maesto Seies 20

17 The geneal solution of this vecto Laplacian equation is given by µ A( xyz,, ) = π 4 V J x ( ', y', z' ) which is simila to the fomal solution obtained befoe fo the electostatic potential fo a distibuted chage. If the cuent is confined to a wie with coss-sectional aea S and descibed by a cuvilinea coodinate l, we can wite I = I = J S dv = S dl dv with a final esult µ I A( x, y, z) = π 4 l dl (note that the total cuent I is constant at any wie location). Amanogawa, 2006 Digital Maesto Seies 21

18 The solution fo the magnetic field obtained fom the vecto potential leads to the famous Biot-Savat law: B 1 I dl I dl H = = A= 4 l = µ µ π 4π l I 1 I dl iˆ = dl 4 l = π 4π l 2 dl I i (x,y,z) Amanogawa, 2006 Digital Maesto Seies 22

19 The magnetic field can also be detemined by diect integation of the cul equation ove a suface S H ds= J ds= I Stoke's theoem can be used to tansfom the left hand side of the equation, to obtain the integal fom of Ampee's law l H dl = In many applications it is useful to detemine the magnetic flux though a given suface. The vecto potential can be used to modify a suface integal into a line integal, using again Stoke's theoem Magnetic Flux S I = B ds= A ds= A dl S S l Amanogawa, 2006 Digital Maesto Seies 23