Electic field geneated by an electic dipole ( x) 2 (22-7) We will detemine the electic field E geneated by the electic dipole shown in the figue using the pinciple of supeposition. The positive chage geneates at P an electic field whose magnitude 1+ ; 1 2x ( ) 2 4πε o 1 q ( + ) 2 4πε o + The negative chage ceates an electic field with magnitude E = E = 1 q
( x) 2 1+ ; 1 2x (22-7) The net electic field at P is: E 1 q = 4πε o q 2 2 + 1 q q = 4 πε o /2 /2 ( z d ) ( z+ d ) 2 2 q d d E = 1 1 2 + 4πε oz 2z 2z d We assume: = 1 2z 2 2 q d d E = 1 1 2 4πε oz + z z qd 1 p = = 3 3 2πε z 2πε z o o E = E E ( + ) ( )
Electic dipole : Two chages that ae equal in magnitude but of opposite signal. ( ) ( ) kq ( ) kq k q k q E = ˆ + ˆ= + 2 + 2 3 + 3 + + d d + = 2 = + 2 3 3 2 3 3 d d d ( ) 2 + = + = + + = 2 + 3 + +q d/2 ρ + ρ ρ E ρ ( ) =? >>d 2 4 3 3 o 2 2 2 2 d/2 ρ 2 d 2 d d = d + = 1 + 2 2 4 4 q 2 3 d d d 1>>Δx n 1, 2 >> ( 1+ Δx 2 2 ) 1+ nδx 2 4 3 3 3 3 3 1 ( ) 2 d kq d kq v 1 3 d 3( d ˆˆ = = = ) d ( ) 2 = 3 2 = 3 3 2 3 2 d d 3 3 d d E = kq 1+ kq 1 + 2 2 3 2 2 2
v Dipole moment : p = qd v kq ( ) kp 1. d E = d = 3 3 v v 2. / / d, d > 0 kq kq v kpˆ E = 3 [ 3dˆ dˆ] = 2d 2 3 3 = v v 3. / / d, d < 0 kq kq v kpˆ E = 3 [ 3( d ) ˆ dˆ] = 2d = 2, ( ˆ= dˆ ) 3 3
Effect of dielectic mateial E 0 = k 0 q 2 fee q bound q fee ε 0 ( k 0 : vacuum) q fee E ρ induced E kq 1 q kq E = E E = = = = 0 0 fee fee fee 0 induced 2 2 2 ε ε 4πε0ε
Foces and toques exeted on electic dipoles by a unifom electic field Conside the electic dipole shown in the figue in the pesence of a unifom (constant magnitude and diection) electic field E along the x-axis. The electic field exets a foce F + = qe on the positive chage and a foce F = qe on the negatice chage. The net foce on the dipole F = qe qe = 0 net (22-14)
The net toque geneated by F+ and F about the dipole cente is: d d τ = τ+ + τ = F+ sinθ F sinθ = qedsinθ = pesinθ 2 2 In vecto fom: τ = p E The electic dipole in a unifom electic field does not move but can otate about its cente. F 0 τ = p E net = (22-14)
Potential enegy of an electic dipole in a unifom electic field U U = pecosθ U = p E B 180 θ θ U = τdθ = pesinθdθ θ 90 90 θ U = pe sinθdθ = pecosθ = p E 90 p E At point A ( θ = 0) U has a minimum value U min = pe It is a position of stable equilibium At point B ( θ = 180 ) U has a maximum A (22-15) value U max =+ pe It is a position of p unstab E le equilibium
Chapte 24 Gauss s Law
Electic Flux Electic flux is the poduct of the magnitude of the electic field and the suface aea, A, pependicula to the field Φ E = EA
Electic Flux, Geneal Aea The electic flux is popotional to the numbe of electic field lines penetating some suface The field lines may make some angle θ with the pependicula to the suface Then Φ E = EA cos θ
Electic Flux, Intepeting the Equation The flux is a maximum when the suface is pependicula to the field The flux is zeo when the suface is paallel to the field If the field vaies ove the suface, Φ = EA cos θ is valid fo only a small element of the aea
Electic Flux, Geneal In the moe geneal case, look at a small aea element ΔΦ = EΔ A cosθ = E ΔA E i i i i i In geneal, this becomes Φ = lim E Δ A = d E i i ΔA 0 i E A suface
Electic Flux, final The suface integal means the integal must be evaluated ove the suface in question In geneal, the value of the flux will depend both on the field patten and on the suface The units of electic flux will be N. m 2 /C 2
Electic Flux, Closed Suface Assume a closed suface The vectos ΔA i point in diffeent diections At each point, they ae pependicula to the suface By convention, they point outwad
Active Figue 24.4 (SLIDESHOW MODE ONLY)
Flux Though Closed Suface, cont. At (1), the field lines ae cossing the suface fom the inside to the outside; θ < 90 o, Φ is positive At (2), the field lines gaze suface; θ = 90 o, Φ = 0 At (3), the field lines ae cossing the suface fom the outside to the inside;180 o > θ > 90 o, Φ is negative
Flux Though Closed Suface, final The net flux though the suface is popotional to the net numbe of lines leaving the suface This net numbe of lines is the numbe of lines leaving the suface minus the numbe enteing the suface If E n is the component of E pependicula to the suface, then Φ E = E da = EndA 旄
Gauss s Law, Intoduction Gauss s law is an expession of the geneal elationship between the net electic flux though a closed suface and the chage enclosed by the suface The closed suface is often called a gaussian suface Gauss s law is of fundamental impotance in the study of electic fields
PI Electostatics CT9 A cylindical piece of insulating mateial is placed in an extenal electic field, as shown. The net electic flux passing though the suface of the cylinde is 1 positive 2 negative 3 zeo