Lecture 2:electro-magnetism Dr. Anyuan Chen • Electric field , electric flux density, and electric potential • Conservative vector field: E in stationary electric field • Gauss’s law Integration and vector caculus dL direction Vector: Effective part of A is the the component along L direction Vector: Effective part of A the the component along S direction S direction is perpendicular to the tangent plane to that surface at S Scalar: no directon Divergence and Stoke’s theorem 휕풇 휕풇 휕풇 Gradient:Greatest rate of increase of field in spatial. 훻풇 = 풙ෝ+ 풚ෝ+ 풛ො 휕푥 휕푦 휕푧 휕퐸푥 휕퐸푦 휕퐸푧 Divergence: Flux out of a point 훻 ∙ 퐸 = + + 휕푥 휕푦 휕푧 휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 휕퐴 how much does a field circulate around a point 훻 × 퐴 = ( 푧- 푦)푥ො + ( 푥- 푧)푦ො + ( 푦- 푥)푧Ƹ 휕푦 휕푧 휕푧 휕푥 휕푥 휕푦 Electric field and flux density (vector) A static distribution of charges produces an electric field E 푸 Vector: 푬 = ퟐ 풓ො . q ퟒ흅휺ퟎ풓 Q Q 푬=푭/풒 풒푸 ′ Vector: 푭 = ퟐ 풓ො ∶ 푪풐풖풍풐풎풃 풔 풍풂풘 ퟒ흅휺ퟎ풓 휺ퟎ is the permittivity of vaccum. ε0 = 8.85e-12 F/m Electric flux density D and D= 휀0퐄. 푄 Vector: 푫 = 풓ො 4휋푟2 Electric flux density is independant of materials. Superposition: Vector E Electric potential 푊 ׬ 푭풅풍 Electric potential definition 푉 = = = ׬ 푬풅풍 푞 푞 풒푸 ′ Vector: 푭 = ퟐ 풓ො ∶ 푪풐풖풍풐풎풃 풔 풍풂풘 ퟒ흅휺ퟎ풓 푸 Vector: 푬 = ퟐ 풓ො ퟒ흅휺ퟎ풓 The potential difference between two point A and B: 푟퐵 푄 푄 1 1 − =׬ 2 푑푟푖= 푟퐴 4휋휀0푟푖 4휋휀0 푟퐵 푟퐴 Electric field and potential ( voltage) • Potential is a scalar. 휕풇 휕풇 휕풇 훻풇 = 풙ෝ+ 풚ෝ+ 풛ො 휕푥 휕푦 휕푧 푉 = ׬ 푬풅풍 • 풓 풓 푄 푄 =V=׬ 푬푑풍=׬ 2 푑풍 ∞ ∞ 4휋휀0푙 4휋휀0푟 Conservative vector field: stationary electric field Electric field is a conservative vector. Conservative vector fields have the property that the line integral is path independent. A conservative vector field is also Ir-rotational. In three dimensions , it has vanishing curl. 훻 × 푓=0. Example: find out the uknown potentials c c d d e e in a stationaly electric field, the potential at each point is listed in a) in the same stationary field and the potential at point d is 3V with another reference point. what are the potential that are missed in b). Example: field of a ring of charge Charge ring with radius a and line charge density: ρl Let calculate E at pont on the z axis. x 풏 풒 푬 = ෍ i 풓ෝ 4흅휺 풓2 풊 x 풊=1 ퟎ 풊 푛 2휋푎 푧휌푙.푐표푠휃 2휋푎 푧휌푙푑푙 2휋 휌푙푧푎푑ф 휌푙푎푧 퐸 = ׬ 푑푙 = ׬ 3 = ׬ 3 = 3 ෍ 푞푖 = න휌푙푑푙 = න휌푙푎푑ф 0 4휋휀푟2 0 4휋휀 푎2+푧2 ൗ2 0 4휋휀 푎2+푧2 ൗ2 2휀 푎2+푧2 ൗ2 푖=1 푟2 = 푎2+푧2 푑푙= 푎푑ф 푐표푠휃 = 푧 ൘ 1 푎2+푧2 2 Example: electric field around a charged ball What is the electric field outside a charged ball with total charge Q? ( r>>a ) Gauss’ law Electric flux flowing out of a closed surface = charge enclosed ׯ 푫푑풔=Q 푸 푸 Electric flux density D and D= ε퐄. Vector: 푬 = ퟐ , 푫 = ퟐ ퟒ흅풓 휺ퟎ ퟒ흅풓 D is independent of medium: 휌 • 훻. 퐸 = 휀 • 훻. 퐷 = 휌 -Q −푸 −푸 = ׯ 푬푑풔=ׯ ퟐ 푑풔 ퟒ흅풓 휺ퟎ 휺ퟎ Example:coaxial cable Assumming : infinite long cable Calculating electric field E between the out surface of inner cable and inner surface of the outer cable. Assuming Unit length ing: and the surface charge density is Circumference Poisson equation Point, line, surface and space charge Electric field in dielectric 풏 풒풒i : 푭 = ෍ 풓ෝ Force in vaccum 풕풐풕 4흅휺 풓2 풊 풊=1 0 풊 푸 Vector: 푬 = ퟐ 풓ො ퟒ흅휺ퟎ풓 In dielectric media, the force between charges depends upon the presence of media/ dielectric: 1: Once electric filed exists in dielectric media, the atoms are distorted or polarized, similar distortions can occur in molecules 2: Electric polarization produced by the electric field in a material depends upon material properties. 3: Polarization influence the force between charges. Electric polarization and material permittivity The influenc of electric polarization: 푫 = 휀0푬 + 푷 • 푫 = 휀0푬 + 푷 • 푷 = 휀0χ푒푬 • 푫 = 휀0푬 + 휀0χ푒푬 = 1 + χ푒 휀0푬 : χ푒 called electric susceptibility • 1 + χ푒 = 휀푟 relative permittivity: • 휀 = 휀푟휀0 electric permittivity .( dielectric material property) Example: coaxial cable with dielectric material and airgap a) D is material independant: b) D is material independant: Assumming : infinite long cable Calculating electric field E between the out surface of inner cable and inner surface of the outer cable. a) The permittivity is b) when.