Fields. Coulomb s Law

Transcription

1 Coulomb s Law q t -q q 2

2 Electic Field Vecto valued function ligned with foce F = q E -q q 2 Supeposition of Electic Field q t -q q 2

3 Potential Enegy U = U() U() = q du = F d = qe d U = F = qe E d E = E F = q E q 2 -q Electic Potential Scala field Genealization of potential enegy as E is a genealization of F lways elative measue Note sign convention Potential Enegy U = U() U() = q Electic Potential u = E = u E d nothe definition of Electic Field E d

4 Electic Potential Field Scala valued field Related to enegy and wok Independent of path q 2 -q Ohm s Law in a Volume Conducto Continuous fom of discete vesion Epessed in tems of field paametes j = i = v() v() R j = σ v j = σ E

5 Scala and Isocontous Each line epesents a egion of constant scala value y Scala Diffeence φ φ 2 φ φ/ d dφ φ d l φ 2 dy â y d aˆ φ 2 = φ φ d φ dy (...) y

6 We wote fo the scala diffeence: φ 2 φ = dφ The Gadient = φ/ d φ/ y dy Now with d l = d a dy a y we can genealize this as a scala poduct by defining the vecto opeato: φ = φ/ a φ/ y a y So that the poduct becomes dφ = φ d l We efe to!" as the gadient of the scala field ". (It is anothe vecto field, i.e., associated with all points in space. Featues of the gadient dφ = φ d l y C φ d l long which dl is d" a maimum? - a minimum? What is the diection of!" elative to the iso-value contou? If " is electic potential, what is -!"?

7 Computing the Gadient φ = φ a φ y a y Only in Catesian coodinates! Gadient Eample

8 dy y j 2 j c d 4 3 Div j = lim y 0 Divegence Flu =(j c j dy...) 2 Flu 2 = (j c j dy...) Flu 3 =(j cy 2 j y d...) 2 y Flu 4 = (j cy j y d...) 2 y ΣFlu = ( j / j y / y) ddy ut the value of flu depends on ddy, so we define a nomalized quantity: ΣFlu ddy = j j y y = j Divegence Eample Div = 0 Div > 0 Div < 0

9 Gadient to Laplacian Gadient Divegence Divegence of the gadient = Laplacian φ = φ a φ y a y j = j j y y φ = 2 φ 2 2 φ y 2 = 2 φ Eample of Laplacian Lap = 0 Lap > 0 Lap < 0

10 ack to (ioelecticity) asics pply the vecto calculus to cuents, electic potentials etend to thee dimensions note: fomulas apply only to Catesian coodinates ioelectic volume conductos fo thee to be cuent, thee must be potential diffeence quasi static assumption (impedance = esistance) may be unifom, piecewise unifom, o anisotopic egula cicuits ae discete/lumped vesion of volume conductos Electic Potential Scala field Genealization of potential enegy as E is a genealization of F lways elative measue Note sign convention Potential Enegy U = U() U() = q Electic Potential u = E = u E d nothe definition of Electic Field E d

11 Potentials,, Cuents in Volume Conductos E = φ E = φ J = σ E = σ φ Definition of E Ohm s Law J = σ φ = σ 2 φ = I v 2 φ = I v σ 2 φ =0 # v is souce density Poisson s Equation Laplace s Equation y Cuent Monopole Field J = σ E = σ φ I v = J d = J 4π 2 z d I 0 " "2 " 3 σ φ = I v 4π 2 a φ = I v 4πσ 2 a φ m = φ m = I v 4πσ 2 a d I v 4πσ C

12 y d -I 0 z = 0 d I 0 φ m = 0 Cuent Dipole Field I v 4πσ C d d... = φ d a d I v I v φ d = 4πσ 0 4πσ I v φ d = I v 4πσ 0 4πσ φ d = I 0 4πσ I 0 a d d = p φ d = I 0 4πσ φ d = 4πσ ssumes that d --> 0, mathematical dipole 0 d d d... a d d p d y Eample φ d = I 0 4πσ p p $ φ d = p cos θ 4πσ 2 z Potential fom a dipole in an infinite homogeneous medium

13 Why Dipoles? Repesent bioelectic souces Membane cuents Coupled cells ctivation wavefont Whole heat