Static Electric Fields. Coulomb s Law Ε = 4πε. Gauss s Law. Electric Potential. Electrical Properties of Materials. Dielectrics. Capacitance E.
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1 Coulomb Law Ε Gau Law Electic Potential E Electical Popetie of Mateial Conducto J σe ielectic Capacitance Rˆ V q 4πε R ρ v 2 Static Electic Field εe E.1
2 Intoduction Example: Electic field due to a chage Fom the field ditibution, we can gue that the divegence of the electic field i non-zeo and the cul of the electic field i zeo. ρ v E 0 E.2
3 Coulomb Law Coulomb law 1. An iolated chage q induce an electic field E at evey point in pace and i given by Ε Rˆ q 4πε R 2 ε i the electical pemittivity of the medium V / m E.3
4 Coulomb Law 2. In the peence of an electic field E at a give point in pace, the foce acting on a tet chage q i F q ' Ε N Fo a mateial with electical pemittivity ε, we have ε E ε ε E o 12 ε o F / m E.4
5 Electic Field due to Chage The expeion fo the electic field due to a ingle chage can be extended to find the field due to multiple point chage a well a continuou chage ditibution. The electic field at poition vecto R caued by chage q 1, q 2,, q N located at point with poition vecto R 1, R 2,, R N i given by Ε ( R 1 4πε R ) N q i ( R R i ) V / 2 R R R i 1 R i R Note: i the unit vecto fom q i to the i R i obevation point. emontation: 4.1, 4.2, M , M4.2 1, M i m E.5
6 Electic Field due to Chage Electic field due to a volume chage ditibution dε Rˆ Ε dq ' 4πε R dε v / ˆ ρ v dv ' R ' 2 2 ' 4πε R ' 1 ˆ ρ v dv R ' v 2 4πε R ' V ' / m E.6
7 Electic Field due to Chage The electic field due to chage on a uface i 1 Ε 4πε R ˆ ' ρ d R ' S ' 2 ' The electic field due to chage on a line i 1 Ε 4πε R ˆ ' ρ dl R ' l l ' 2 ' E.7
8 Gau Law Gau Law tate that the outwad flux of though a uface i popotional to the encloed chaged. ρ iffeential fom ds S v Integal fom E.8
9 E.9 Example: point chage Contuct a pheical uface with cente at q and adiu R. Electic field i the ame eveywhee on the uface. Applying integal fom of Gau law give R E R ˆ 4 / 4 ˆ 2 2 R q q R d d d R R R πε ε π
10 Example: infinite long line of chage Since the line of chage i infinite in extent and i along the z- axi, mut be in the adial -diection and mut not depend on o z. h 2π d z 0 φ 0 2πh ˆ ˆ dφdz ρ h ρ l E / ε ˆ 2πε l E.10
11 E.11 Example: Spheical chage ditibution Find the electic field due to a pheical chage ditibution of adiu a with unifom volume chage denity ρ v in fee pace. Gauian uface of adiu a Cae (i): Fo a Gauian uface of adiu le than the adiu a of the chage ditibution, the total chage encloed by the Gauian uface i: v ρ π Applying Gau Law to the Gauian uface, E ˆ 3 / 4 ˆ 2 o v o R d d ε ρ ε π
12 E.12 Example: Spheical chage ditibution Cae (ii): Fo a Gauian uface of adiu geate than a : v a ρ π Applying Gau law, E ˆ 3 / 4 ˆ a d d o v o R ε ρ ε π emontation: 4.9, 4.10, M
13 Electic Potential The electotatic field i iotational o conevative, i.e. E 0 E. dl 0 C Accoding to vecto identity, fo any cala V ( V ) 0 So we can define a cala electic potential V E V V P E dl The integal i independent of the path taken E.13
14 Electic Potential Electic potential due to point chage: N 1 q V i 4πε R R i 1 Electic potential due to volume ditibution V 1 4πε v' ρ v dv' R' Electic potential due to uface ditibution i V 1 4πε S ' ρ d' R' Electic potential due to line ditibution 1 ρ l V dl ' 4πε l ' R' emontation: M4.1.3 M4.2.2, M4.6 E.14.1
15 Electical Popetie of Mateial The electomagnetic contitutive paamete of a mateial medium ae Electical pemittivity ε Electical pemeability µ Conductivity σ Thoughout thi ubject, we aume the mateial ae Homogenou Contitutive paamete do not vay fom point to point Iotopic Contitutive paamete ae independent of diection E.15
16 Conducto A conducto ha a lage numbe of looely attached electon in the outemot hell of the atom. Upon applying an extenal electic field, the electon migate fom one atom to the next along a diection oppoite that of the extenal field. Thei movement give ie to a conduction cuent J σe Point fom of Ohm law E.16
17 Conductivity of ome common mateial Conducto Pefect dielectic: σ0 and then J 0 egadle of E Pefect conducto: σ and then E 0 egadle of J E.17
18 ielectic ielectic ae inulating mateial, and contain bound chage which cannot move feely to geneate cuent. Bound chage can move hot ditance unde an electic field to fom electic dipole. E.18
19 ielectic A dielectic medium polaized by an extenal electic field E.19
20 ielectic The polaization (o bound) chage can alo contibute to the electic field E a the fee chage. It can be hown that i elated to E and P accoding to the following equation: o ε E + P In mot mateial, P i in the ame diection and popotional to E o that ε o E + ε oχe ε ε E εe o whee ε o i the pemittivity of vacuum, ε i the elative pemittivity (o dielectic contant) of the mateial, ε i the pemittivity of the mateial, and χ i the uceptibility of the mateial. E.20
21 ielectic Stength If the applied electic field exceed the dielectic tength of the mateial, it will fee the electonic completely fom the molecule in the fom of a conduction cuent. Spak may occu ielectic beakdown E.21
22 Capacitance Capacitance between two conducto i defined a: C V (F ) whee i the chage on the conducto and V i the potential diffeence between the conducto. E.22
23 Example 4-11: Paallel-Plate Capacito Becaue of the applied voltage diffeence, chage + accumulate unifomly on the top plate and accumulate unifomly on the lowe plate. In the dielectic medium between the plate, the chage induce a unifom electic field (finging field i ignoed). E.23
24 E.24 Example 4-11: Paallel-Plate Capacito Applying Gau law, the electic field can be detemined: A z z ze ε ε ρ / ˆ / ˆ ˆ E ( ) d A d A Ed d V C d ε ε / 0 l E
25 Example: 4-12: Coaxial Line Contuct a Gauian cylindical uface of unit length and adiu. The E-field at adiu i: d E 2πl / ε ˆ 2πεl E.25
26 Example: 4-12: Coaxial Line V b a E dl b ˆ a ln 2πεl 2πεl ( b / a) ˆ d C V 2πεl ln ( b / a) E.26
27 Execie E.27
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