Collection of Fomuls Electomgnetic Fields EITF8 Deptment of Electicl nd Infomtion Technology Lund Univesity, Sweden August 8
/ ELECTOSTATICS field point '' ' Oigin ' Souce point Coulomb s Lw The foce F on point chge q t point, esulting fom point chge q t point ' qq F, whee '' ' nd 4 ' ' Electic Field Intensity E in Vcuum Fom point chge q t ' q E 4 4 Fom chge distibution E ' whee dq dq dv fo volume chge density ' s ds fo sufce chge density s dl fo line chge density p Fom point dipole p p E cos sin 3 4 l l Fom line chge (of density ) E l l Foce F on point chge q qe fo electosttics F qe vb fo genel cse
3/ Electic Potentil V Fom point chge q t ' V q 4 4 Fom chge distibution V dq ' Fom point dipole p p Fom line chge (of density l V ) V p pcos 3 4 4 l ln E V fo electosttics Electic Flu Density D Guss Lw D nds dv whee n is the outwd unit vecto noml to the sufce of the volume. Polition P eltionship between P, E nd D d P p dv D E P D E fo genel cse Boundy Conditions Et is continuous s D D n whee s is the sufce chge density of fee chges, nd egion towd egion. n is the noml unit vecto pointing fom
4/ STEADY ELECTIC CUENTS Cuent density J I J ds n Eqution of Continuity (fom pinciple of consevtion of chge) In diffeentil fom J t In integl fom Conductivity Totl Powe Dissiption P Boundy Conditions Time Constnt dq J n ds dt J E P J E dv Js n J J fo no sufce cuent Js E is continuous t C Anlogy between Electosttics nd Stedy Electic Cuents E, V E, V D J Q I C G
5/ Mgnetic Flu Density B in Vcuum MAGNETOSTATICS Fom point dipole m m m B cos sin 3 4 Fom cuent density J ' B J 4 ' dv' Fom cuent pth B 4 Idl' Fom cicul wie loop B, y, Fom long stight wie pth B Ib b I 3/ Vecto Mgnetic Potentil A in Vcuum Fom point dipole m A Fom cuent density J ' A m 4 3 4 J ' ' Fom cuent pth A 4 I Idl Fom long stight wie pth A ln dv ' Mgnetic Flu Flu Linkge Λ Λ N B n ds A d l Self-inductnce L nd Mutul-inductnce M Λ L I M I Λ L I M I
6/ Mgnetic Field Intensity H Ampee s Lw H J H dl J ds I n enclosed eltionship between mgnetition vecto M, B nd H B H M B H fo genel cse Boundy Conditions H H J n B n is continuous s whee J s is the cuent density of fee sufce cuent, nd egion towd egion. n is the noml unit vecto pointing fom eluctnce l S Mgnetic Foce df m I dl B Mgnetic Moment m fo Cuent Loop m I n ds Toque T m on Mgnetic Dipole m T m mb Foce on Mgnetic Dipole m F m B mb
7/ ELECTOMAGNETIC FIELD Fdy s Lw of Electomgnetic Induction d I dt Induced emf E v B d l d dt fo coils Fdy s Lw of Electomgnetic Induction In diffeentil fom B E t In integl fom E dl B t n ds Mwell s Equtions B E t D H J t D B Electomgnetic Constnts 9 7 8 c 4 H/m, F/m, 3 m/s 36,, 377 c Potentils eltionship between vecto mgnetic potentil A nd mgnetic flu density B B A eltionship between scl electic potentil V, vecto mgnetic potentil A nd electic field intensity E Poynting Vecto A E V t P, t E, th, t
8/ TIME-HAMONIC FIELD Plne Wve E Eˆ cos t k E Ee E E jk E ˆ j E e pek vlue Eˆ e j E instntneous vlue comple fom oot-men-sque (ms) vlue Goup Velocity,, k k k Chcteistic Impednce ight Hnd ule, E H,, E B k E H k E B VECTO IDENTITIES A BC B CA C AB ABC BA C C A B Divegence Theoem Volume A dv A ds Sufce n Stoke s Theoem Sufce A n ds A d l Cuve Null Identities V A
9/ COODINATE SYSTEM Ctesin Coodintes, y, Position vecto yy Line element dl d dy y d Volume element dv d dy d V V V Diffeentil opetos V y y A A y A A y A A y A A A y A A y y y V V V V y Cylindicl Coodintes,, Position vecto Unit vectos cos sin y sin cos Line element dl d d d Volume element dv d d d y Diffeentil opetos V V V V A A A A A A A A A A A V V V V
/ Spheicl Coodintes,, Position vecto Unit vectos sin cos sin sin cos y cos cos cos sin sin sin cos y y Line element Volume element Diffeentil opetos dl d d sin d dv sin d d d V V V V sin A A A sin A sin sin A A A A sin A sin sin A A V V V V sin sin sin
/ n n d, n n d ln d ln d ln d 3/ d csin d ctn d tn cos d ln tn sin ln d ln INTEGALS TIGONOMETIC IDENTITIES cos cos cos sin sin cos cos cos sin sin sin sin cos cos sin sin sin cos cos sin cos sin cos cos sin sin sin cos cos cos cos sin cos cos cos cos cos cos sin sin sin sin sin cos sin sin cos sin sin t b cost b sin t b sin,cos b b