Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier transform Any periodic signa g( x), of period a, can aways be expressed with the so-caed Fourier series, where with G n the Fourier coefficient G n e g x, (.) n π nx i a G n π nx a i a g( x)e dx a. (.) a Aternativey, the Fourier series can be expressed using sine and cosine functions instead of an exponentia function. Equations (.) and (.) are then repaced by g( x) G + A n cos πnx a + B n sin πnx a n A n a g( x)cos πnx a a dx a B n a g( x)sin πnx a a dx. a (.) Note that G A, if n is aowed in the second of equations (.). If we set δ ( ma) g x, for x < a m (.4) then it seen from equations (.) and (.) that This is the cosure reation. δ ( ) a π n( x x ) e i a, for x < a n. (.5) 9
If we et a, we have the foowing transformations πn a n k dn a π dk (.6) G n π a G( k). The pair of equations (.) and (.) defining the Fourier series are repaced by a corresponding set G k g x G( k) e ikx dk g( x)e ikx dx π. (.7) These equations are usuay rendered symmetric by, respectivey, dividing and mutipying them by π. We then get the Fourier transform pair π g x π G k G( k)e ikx dk g( x)e ikx dx. (.8) We can evauate the cosure reation corresponding to the Fourier transform by setting g x x δ ( x ) in equations (.8), we thus obtain δ ( ) ik e ( x x ) π dk. (.9) The generaization of both the Fourier series and Fourier transform to functions of higher dimensions is straightforward... Spherica harmonics One possibe soution to the Lapace equation is a potentia function of poynomias of rank. If we use Cartesian coordinates we can write this potentia as Φ ( x) C n n n x n yn zn. (.) n +n +n
We now evauate the maximum number of terms in the summation. If we start by assigning a vaue m (one out of the + possibe vaues) to the exponent n, we can easiy determine that there are + m different combinations with which we can arrange the other two indices. If we denote by N ( ) the maximum number of terms composing the potentia, we have ( + m) N ( + m). (.) m But since this equation can aso simpy be written as N we can transform equation (.) as foows + m m + m, (.) + N ( ) ( + ) ( + ) m + + m N. (.) Then, N ( ) ( + + ). (.4) However, these N ( ) terms cannot a be independent since the potentia must satisfy the Lapace equation, which states that Φ. (.5) Because this ast equation has a rank of, the number of independent terms is given by N ( ) N ( ) + + ( ) + +. + (.6) If we make a change of coordinates from Cartesian to spherica with
x r sin( θ)cos ϕ y r sin( θ)sin( ϕ) z r cos( θ), (.7) and further x + iy r sin( θ)e iϕ x iy r sin( θ)e iϕ, (.8) we can write the potentia as Φ ( r,θ,ϕ) C m r Y m ( θ,ϕ), (.9) where the functions Y m ( θ,ϕ), with < m <, are the spherica harmonics. The Lapacian operator in spherica coordinates is given by m r r r + r sin θ r r r + Ω r, θ sin( θ) θ + sin θ ϕ (.) with Ω sin θ θ sin( θ) θ + sin θ ϕ. (.) Appying the Lapace equation to the term (, m) of the potentia using the ast of equations (.) yieds r r r + Ω r r Y m ( r + Y r r m ) + Ωr Y m ( + )r Y m + r ΩY m. (.) Thus, the differentia equation satisfied by the spherica harmonics is ΩY m ( θ,ϕ) + ( + )Y m θ,ϕ. (.) Equation (.) is commony known as the generaized Legendre differentia equation. Because the coordinates can be separated from each other in the Lapace equation, we
can better estabish a more genera functiona form for the soution to the Lapace equation in spherica coordinates. For exampe, if we act on U ( r) Φ r,θ,ϕ r P( θ)q( ϕ), (.4) with equation (.) (whie setting the resut of the operation equa to zero), mutipy by r sin ( θ), and divide by equation (.4), we find that ϕ is separated from r and θ and consequenty with m a constant. The soution to equation (.5) is d Q Q dϕ m. (.5) Q( ϕ) e imϕ. (.6) If we insert equations (.4) and (.6) into equation (.), mutipy by r, and divide by equation (.4), r is now separated from θ and ϕ with with a constant, and therefore r U U ( r) r d U dr ( + ), (.7) A r + B (.8) + r in genera. Finay, if we insert equations (.4), (.6), and (.8) in equation (.), mutipy by r, and divide by equation (.4), then θ is separated from r and ϕ with sin θ d dθ sin θ and a corresponding soution P m θ spherica coordinates is therefore with Φ r,θ,ϕ dθ + + dp m sin P, (.9) ( θ). The genera soution to the Lapace equation in r + Y m ( θ,ϕ), (.) A m r + B m m
Y m ( θ,ϕ) P m ( θ)e imϕ. (.) The spherica harmonics aso satisfy the foowing reations: a) The orthonormaity reation b) If we write π π * dϕ Y m ( θ,ϕ)y m ( θ,ϕ)sin( θ)dθ δ Y m δ m m. (.) r ( θ,ϕ) Y m r Y m ( n), (.) where n has the orientation π θ and π + ϕ, then the foowing symmetry reation appies Y m ( n) Y m n. (.4) c) If we substitute m m Y, m ( θ,ϕ) m Y m * ( θ,ϕ). (.5) d) The cosure reation * Y m ( θ, ϕ )Y m ( θ,ϕ) δ ϕ ϕ m δ ( cos( θ) cos ( θ )). (.6) Since the spherica harmonics form a compete set of orthonorma functions, any arbitrary function can be expanded as a series of spherica harmonics with A m Y m ( θ,ϕ) g θ,ϕ m π π * A m dϕ g( θ,ϕ)y m ( θ,ϕ)sin( θ)dθ. (.7) Here are some exampes of spherica harmonics 4
Y 4π Y Y,± Y 4π cos θ 8π sin( θ)e±iϕ 5 6π cos θ Y,± 5 8π sin( θ)cos( θ)e±iϕ Y,± 5 ( π sin θ)e ±iϕ. (.8) The Legendre poynomias are the soution to the generaized Legendre differentia equation (i.e., equation (.)) when m. They can be defined using the spherica harmonics as P cos( θ) 4π + Y ( θ). (.9) The Legendre poynomias aso form a compete set of orthogona functions with an orthogonaity reation and a cosure reation ( x)p ( x) dx P + δ (.4) where x cos( θ). Any arbitrary function of x cos θ series of Legendre Poynomias * P ( x )P ( x) + δ ( ), (.4) can, therefore, be expanded in a A P ( x) g x A + g( x)p ( x)dx. (.4) The first few Legendre poynomias are 5
P P P P ( x) ( x) x ( x) x ( x) 5x x (.4) P 4 ( x) ( 8 5x4 x + ). There exists a usefu reation between the Legendre poynomias and the spherica harmonics. The so-caed addition theorem for spherica harmonics is expressed as foows P cos( γ ) 4π * Y m ( θ, + ϕ )Y m ( θ,ϕ) (.44) m and where γ is the ange made between two vectors x and x of coordinates r,θ,ϕ ( r, θ, ϕ ), respectivey. These definitions for x and x aso impy that cos( γ ) cos θ cos θ + sin θ We can prove the addition theorem by noting that since cos γ sin( θ )cos( ϕ ϕ ). (.45) is a function of θ and ϕ through equation (.45), then, P cos γ can be expanded in a series of spherica harmonics Y m ( θ,ϕ). We therefore write P cos( γ ) c m Y m ( θ,ϕ). (.46) m Evidenty, since equation (.45) is symmetric in θ,ϕ and ( θ, ϕ ), we coud just as we express P cos γ with a simiar series containing θ and ϕ instead P cos( γ ) c m Y m ( θ, ϕ ). (.47) m Furthermore, because of the particuar form of equation (.45) we can aso write P cos( γ ) α m m Y m ( θ, ϕ )Y m ( θ,ϕ). (.48) m m 6
can be expressed by a [Note: Equation (.48) is justified because each term in cos γ sum of products of functions of θ and ϕ with functions of θ and ϕ through cos γ sin θ cos( ϕ)sin ( θ )cos ϕ + cos( θ)cos ( θ ), + sin θ sin( ϕ)sin ( θ )sin ϕ (.49) ( γ ) (which can simiary be expressed as a sum of products of functions of θ and ϕ ). For a given, an arbitrary function f ( θ,ϕ) (Y m ( θ, ϕ ) ); and therefore the same can be said of P cos γ, since it contains a sum of terms of the type cos n θ and ϕ with functions of ( f ( θ, ϕ )) can be expanded as a series of spherica harmonics Y m θ,ϕ ergo equation (.48).] However, because of equations (.) and (.45) the dependence on ϕ must be such that e i m ϕ e imϕ e i ( m ϕ + mϕ ) e im( ϕ ϕ ). (.5) That is, we must have m m, and P cos( γ ) α m Y, m ( θ, ϕ )Y m ( θ,ϕ). (.5) m Substituting equation (.5) for the first spherica harmonic, we get P cos( γ ) ( ) m * α m Y m ( θ, ϕ )Y m ( θ,ϕ). (.5) m If we now set γ so that cos( γ ), θ θ, and ϕ ϕ, we have from equation (.5) ( ) m α m Y m ( θ,ϕ) P, (.5) m and integrating both sides over a anges we find, because the normaization of the spherica harmonics (see equation (.)), Let us now square equation (.5) 4π ( ) m α m. (.54) m 7
P cos( γ ) m m m α m Y m * m m * α m Y m m α m Y m * ( θ, ϕ )Y m θ,ϕ ( θ, ϕ )Y m θ,ϕ ( θ, ϕ )Y m θ,ϕ ( ) m α m Y m θ, ϕ * ( )Y m ( θ,ϕ), m (.55) where we have used equation (.5) twice in the ast term, and made the transformations m m and as we. We integrate over the anges θ and ϕ to obtain m m P cos( γ ) dω m m m+ m α m α m Y m * * Y m ( θ,ϕ)y m ( θ,ϕ) dω, ( θ, ϕ )Y m θ, ϕ (.56) which according to equations (.) and (.4) gives 4π + α m m Integrating over the anges θ and ϕ this time, we get Y m ( θ, ϕ ). (.57) 4π 4π + α m. (.58) m Combining equations (.54) and (.58) we can determine the coefficient α m to be α m 4π ( )m +, (.59) which upon insertion in equation (.5) yied equation (.44), i.e., the addition theorem for spherica harmonics. Finay, another usefu equation is that for the expansion of the potentia of point charge, in a voume excuding the charge (i.e., x x ), as a function of Legendre poynomias x x r < + r P cos( γ ), (.6) > 8
where r < and r > are, respectivey, the smaer and arger of x and x, and γ is the ange between x and x (see equation (.45)). Upon substituting the addition theorem of equation (.44) for the Legendre poynomias in equation (.6), we find x x 4π r < + + r Y * m ( θ, ϕ )Y m ( θ,ϕ). (.6) > m. Soution of Eectrostatic Boundary-vaue Probems As we saw earier, the expression for the potentia obtained through Green s theorem is a soution of the Poisson equation Φ ρ ε. (.6) It is aways possibe to write Φ as the superposition of two potentias: the particuar soution Φ p, and the characteristic soution Φ c Φ Φ p + Φ c. (.6) The characteristic potentia is actuay the soution to the homogeneous Lapace equation Φ (.64) due to the boundary conditions on the surface S that deimitates the voume V where the potentia is evauated (i.e., Φ c is the resut of the surface integra in equation (.6) after the desired type of boundary conditions are determined). Consequenty, Φ p is the response to the presence of the charges present within V. More precisey, Φ p ( x) ρ x 4πε R d x. (.65) V This suggests that we can aways break down a boundary-vaue probem into two parts. To the particuar soution that is formay evauated with equation (.65), we add the characteristic potentia, which is often more easiy soved using the Lapace equation (instead of the aforementioned surface integra)... Separation of variabes in Cartesian coordinates The Lapace equation in Cartesian coordinates is given by 9
Φ x + Φ y + Φ, (.66) z where we dropped the subscript c for the potentia as it is understood that we are soving for the characteristic potentia. A genera form for the soution to this equation can be attempted by assuming that the potentia can be expressed as the product of three functions, each one being dependent of one variabe ony. More precisey, Φ( x, y, z) X ( x)y ( y)z ( z). (.67) Inserting this reation into equation (.66), and dividing the resut by equation (.67) we get d X ( x) X ( x) + dx Y y d Y ( y) + dy Z z d Z ( z). (.68) dz If this equation is to hod for any vaues of x, y, and z, then each term must be constant. We, therefore, write d X ( x) α X ( x) dx d Y y Y ( y) d Z z Z ( z) dy β dz γ, (.69) with α + β γ. (.7) The soution to equations (.69), when α and β are both positive, is of the type Exampe Φ( x, y,z) e ±iα x e ±iβy e ± α +β z. (.7) Let s consider a hoow rectanguar box of dimension a,b and c in the x, y, and z directions, respectivey, with five of the six sides kept at zero potentia (grounded) and the top side at a votage V ( x, y). The box has one of its corners ocated at the origin (see Figure.). We want to evauate the potentia everywhere inside the box. Since we have Φ for x, y, and z, it must be that 4
Figure. A hoow, rectanguar box with five sides at zero potentia. The topside has the specified votage Φ V x, y. X sin αx Y sin βy Z sinh( α + β z). (.7) Moreover, since Φ at x a, and y b, we must further have that α n nπ a β m mπ b (.7) γ nm π n a + m b, where the subscripts n and m were added to identify the different modes aowed to exist in the box. To each mode nm corresponds a potentia Φ nm sin( α n x)sin( β m y)sinh( γ nm z). (.74) The tota potentia wi consists of a inear superposition of the potentias beonging to the different modes A nm sin α n x Φ x, y, z sin( β m y)sinh( γ nm z), (.75) n,m where the sti undetermined coefficient A nm is the ampitude of the potentia associated with mode nm. Finay, we must aso take into account the vaue of the potentia at z c with 4
A nm sin α n x V x, y sin( β m y)sinh( γ nm c). (.76) n,m We see from equation (.76) that A nm sinh( γ nm c) is just a (two-dimensiona) Fourier coefficient, and is given by A nm sinh( γ nm c) 4 ab a b dx dy V ( x, y)sin( α n x)sin( β m y). (.77) If, for exampe, the potentia is kept constant at V over the topside, then the coefficient is given by A nm sinh( γ nm c) V 4 π nm n m. (.78) We see that no even modes are aowed, and that the ampitude of a given mode is inversey proportiona to its order.. Mutipoe Expansion Given a charge distribution ρ ( x ) contained within a sphere of radius R, we want to evauate the potentia Φ( x) at any point exterior to the sphere. Since the potentia is evauated at points where there are no charges, it must satisfy the Lapace equation and, therefore, can be expanded as a series of spherica harmonics Φ( x) Y c m θ,ϕ m, (.79) 4πε r + m where the coefficients c m are to be determined. The radia functiona form chosen for the potentia in equation (.79) is the ony physica possibiity avaiabe from the genera soution to the Lapace equation (see equation (.)), as the potentia must be finite when r. In order to evauate the coefficients c m, we use the we-known voume integra for the potentia Φ( x) ρ x 4πε d x, (.8) V with equation (.6) for the expansion of the denominator. We, then, find (with r < r x and r > r x, since x > x ) ε Φ x * Y m θ, + ( ϕ ) r ρ ( x )d x Y m θ,ϕ. (.8) r + m 4
Comparing equations (.79) and (.8), we determine the coefficients of the former equation to be c m 4π + Y * m ( θ, ϕ ) r ρ ( x )d x. (.8) The mutipoe moments, denoted by q m ( c m ( + ) 4π ), are given by * q m Y m ( θ, ϕ ) r ρ ( x )d x (.8) Using some of equations (.8), with r sin ( θ )e ±iϕ x ± i y r cos θ z, (.84) we can cacuate some of the moments, with for exampe q 4π ρ ( x )d x q 4π q q,± 4π cos( θ) r ρ ( x ) d x 4π p z 8π sin( θ)e iϕ r ρ ( x ) d x ( 8π p ip x y ) z ρ 4π ( x ) d x x i 8π ( y )ρ( x ) d x (.85) for, and the foowing for 4
q 5 cos ( θ) 6π r ρ ( x )d x 5 6π Q 5 6π q,± 5 8π sin( θ)cos( θ)e iϕ r ρ ( x )d x 5 8π q,± 5 ( 8π Q iq ) 5 π ( sin θ)e iϕ r ρ ( x )d x 5 ( π Q iq Q ). 5 π z r ρ ( x )d x z ( x i y )ρ( x )d x (.86) ( x i y ) ρ ( x )d x In equations (.85) and (.86), q is the tota charge or monopoe moment, p is the eectric dipoe moment p x ρ ( x )d x, (.87) and Q ij is a component of the quadrupoe moment tensor (here, no summation is impied when i j ) Q ij ( x i x r δ )ρ( x )d x. (.88) j ij Using a Tayor expansion (see equation (.84)) for x x, we can aso express the potentia in rectanguar coordinate (the proof wi be found with the soution to the first probem ist) with Φ( x) 4πε q r + p x + r Q ij x i x j + r 5. (.89) The eectric fied corresponding to a given mutipoe is given by and from equations (.79), (.8), and (.8) we find E Φ m, (.9) 44
q m E r Φ r + ε + E θ Φ r θ ε + E ϕ Φ r sin( θ) q m ( θ,ϕ) Y m r + r + q m ϕ ε + θ Y m θ,ϕ r + im sin( θ) Y m ( θ,ϕ). (.9) For exampe, for the eectric monopoe term ( ) we find E r q 4πε r and E E θ ϕ, (.9) which is equivaent to the fied generated by a point charge q, as expected. For the eectric dipoe term ( ), we have E r ε r q, Y, + q Y + q Y ( p ε r x + ip y )sin( θ)e iϕ + p z cos θ 8π 4πε r sin( θ)e iϕ + p x ip y p x sin( θ)cos( ϕ) + p y sin( θ)sin( ϕ) + p z cos( θ) E θ Y q, ε r, θ + q Y θ + q Y θ ( p ε r x + ip y )cos( θ)e iϕ p z sin θ 8π 4πε r cos( θ)e iϕ + p x ip y p x cos( θ)cos( ϕ) + p y cos( θ)sin( ϕ) p z sin( θ), (.9) and E ϕ i ε r sin θ i ε r sin θ 4πε r q, Y, + q Y 8π ( p x + ip y )sin( θ)e iϕ + ( p x ip y )sin θ p x sin( ϕ) + p y cos( ϕ). e iϕ (.94) Because the unit basis vectors of the spherica and Cartesian coordinates are reated by the foowing reations 45
e r sin( θ)cos( ϕ)e x + sin( θ)sin( ϕ)e y + cos( θ)e z e θ cos( θ)cos( ϕ)e x + cos( θ)sin( ϕ)e y sin( θ)e z e ϕ sin( ϕ)e x + cos( ϕ)e y, (.95) then we can write the dipoe eectric fied vector as E 4πε r ( p r e r p). (.96) If the dipoe is ocated at x (which we now consider to be the origin of the coordinate system), with r x x and n the unit vector inking x to x (we coud use e r instead of n ), we can write the dipoe eectric fied vector as foows n( n p) p E x 4πε x x. (.97) It is important to note that in genera the mutipoe moments q m depend on the choice of the origin. Coming back to the evauation of the eectric fied due to the dipoe moment term, we note that if we cacuate the average vaue of the eectric fied inside a sphere of radius R centered at x, we find that E( x)d x E r e r + E θ e θ + E ϕ e ϕ r sin( θ)drdθdϕ, (.98) r < R r < R for which, according to equations (.9)-(.94), each term wi contain ony terms proportiona to one of the foowing integras π π π sin( ϕ)dϕ, or cos( ϕ)dϕ, or sin( θ)cos( θ)dθ. (.99) Since a of these integras vanish, we find that the contribution of the dipoe term to the voume-averaged eectric fied aso vanishes. That is, E( x)d x. (.) r < R This resut may not be surprising, since this woud be expected if one considers a simpe dipoe configuration (i.e., two point charges of opposite poarity dispaced equay far on either side of x ) and integrate over the sphere centered on the dipoe. Things are different, however, if we consider an arbitrary charge distribution ρ x. We again 46
consider a sphere of radius R centered on the origin of the coordinate system, and which contains the whoe charge distribution. We start by writing E( x)d x Φd x r < R r < R Φ( x)n R dω, r R (.) where equation (.) was used, and n x R is the outward unit vector norma to the surface of the sphere. Repacing the potentia Φ with its integra form, we have E( x)d x R r < R d x ρ 4πε x n ( ) dω. (.) r R Since the unit vector can be written as a function of the anges with n sin( θ)cos( ϕ)e x + sin( θ)sin( ϕ)e y + cos( θ)e z, (.) each component n i of n can be expanded in a series of spherica harmonics where n i c m Y m ( θ,ϕ). (.4) m Because of this, each anguar integra on the right-hand side of equation (.) can be transformed to n i dω Y c m θ,ϕ m r R r R m dω r c < m + r P cos( γ ) Y m θ,ϕ > m r R dω r c < m Y + ( θ,ϕ)y m ( θ,ϕ) dω r >, r R m (.5) where we used equations (.6) and (.9), and cos( γ ) cos θ cos θ + sin θ sin( θ )cos( ϕ ϕ ). (.6) Upon using the orthogonaity property of spherica harmonics (i.e., equation (.)), we find that the ony term aowed is that where in the expansion for x x. Equation (.) can therefore be written as 47
E( x)d x R r < R r < The anguar integra can now be cacuated as foows dω dω d x ρ 4πε x ncos γ r >. (.7) r R π π ncos γ dϕ sin( θ)cos( ϕ)e x + sin( θ)sin( ϕ)e y + cos( θ)e z r R cos( θ)cos ( θ ) + sin( θ)sin ( θ )cos( ϕ ϕ ) sin( θ)dθ π sin θ cos ϕ π e x + sin ( ϕ )e y sin ( θ)dθ π +π cos ( θ )e z sin( θ)cos ( θ)dθ 4π sin ( θ )cos( ϕ )e + sin ( θ )sin( ϕ )e + cos ( θ )e x y z (.8) 4π n, where the ast two of equations (.8) serve as an expicit definition of the unit vector n x r. Equation (.7) is now simpified to E( x)d x R r < R r < ρ x ε n d x. (.9) r > With the origina assumption that the charge distribution is entirey contained within the sphere, we can write r < r and r > R, and equation (.9) reduces to E( x)d x R r < R ρ x ε r n d x R x ρ ( x )d x ε p ε, (.) from equation (.87). This resut states that the average vaue of the eectric fied over a voume containing the charge distribution responsibe for the fied is proportiona to the dipoe moment of the charge distribution with respect to the center of the sphere. It can be combined to the previous resut expressed in equation (.97) to yied E( x) 4πε n( n p) p x x 4π pδ x x. (.) 48
Equation (.) does not change the vaue of the eectric fied as cacuated with equation (.97) when x x, but it correcty takes into account the required voume integra (.). Interestingy, if we consider a sphere to which the charge distribution is externa, we have r > r and r < R in equation (.9), and E( x)d x R r < R ρ x ε R ρ x ε R ρ x n r d x x r d x x d x x x (.) 4π R E( ), where we used Couomb s Law (i.e., equation (.47)) for the ast step. In other words, the average vaue of the eectric fied over a sphere containing no charge is the vaue of the fied at the centre of the sphere..4 Eectrostatic Potentia Energy We know from the cacuations that ed to equation (.6), on page, that the product of the potentia and the charge can be interpreted as the potentia energy of the charge as it is brought from infinity (where the potentia is assumed to vanish) to its fina position. More precisey, we define the potentia energy W i of a charge q i with W i q i Φ( x i ). (.) If the potentia is due to an ensembe of n charges q j, then the potentia energy of the charge q i becomes and the tota potentia energy W is W i q i W 4πε n q j, (.4) 4πε j x i x j n q i q j. (.5) i j <i x i x j Aternativey, we can write a more symmetric equation for the tota energy by summing over a charges and dividing by two 49
W 8πε n q i q j. (.6) i j i x i x j Equation (.6) can easiy be generaized to continuous charge distributions with W 8πε or, aternativey, if we use equation (.57) for the potentia ρ( x)ρ ( x ) d x d x (.7) W ρ( x)φ( x)d x (.8) Furthermore, we can express the potentia energy using the eectric fied instead of the charge distribution and the potentia. To so, we repace the potentia with the Poisson equation in equation (.8) and proceed as foows W ε ε ε Φ Φd x ( Φ) Φ Φ d x ( Φ Φ) nda E d x S, (.9) where we used equation (.6) and the divergence theorem for the second and ast ines, respectivey. Because the integration is done over a of space, the surface in integra wi vanish since im R ( Φ Φ) nda S R S im im R R R R dω 4π R. (.) The potentia energy then becomes W ε E d x (.) It foows from this resut that the potentia energy density is defined as 5
w ε E. (.) It is important to note that the energy cacuated with equation (.) wi be greater than that evauated using equation (.7) as it contains (that is, equation (.)) the sef-energy of the charge distribution. We are now interested in expressing the potentia energy of a charge distribution subjected to an externa fied as consisting of the contributions from the different terms in the mutipoe expansion. In this case, we know from equation (.) that W ρ( x)φ( x) d x. (.) We start by expanding the potentia function with a Tayor series (see equation (.84)) around some predefined origin Φ( x) Φ( ) + x Φ x + x x i j Φ + x i x j x (.4) Φ( ) x E( ) x E j ix j +, x i x where we used E Φ, and summations over repeated indices were impied. Since E for the externa fied, we can subtract the foowing reation from the ast term of equation (.4) without effectivey changing anything 6 r E x 6 r δ ij E j x i x (.5) to get Φ( x) Φ( ) x E( ) ( 6 x x r i j δ ij ) E j + (.6) x i x If we insert equation (.6) into equation (.8) for the potentia energy, and using equation (.88) for the components of the quadrupoe moment tensor, we get W qφ( ) p E( ) 6 Q E j ij + (.7) x i x 5
Using equation (.) for the eectric fied generated by a dipoe, and equation (.7), we can cacuate the energy of interaction between two dipoes p and p ocated at x and x, respectivey, when x x. Thus, W p E x p E ( x ) ( p n) p p p n, 4πε x x (.8) where n ( x x ) x x. The dipoes are attracted to each other when the energy is negative, and vice-versa. For exampe, when the dipoes are parae in their orientation, and to the ine joining them, then, W < and they wi attract each other..5 Eectrostatic fieds in Matter So far in deaing with the equations of eectrostatic, we were concerned with, and derived, the microscopic equations of eectrostatic. That is, we considered probems invoving mainy charge distributions without the presence of any ponderabe media. When anayzing eectrostatic fieds in matter, we need to make averages over macroscopicay sma, but microscopicay arge, voumes to obtain the macroscopic equations of eectrostatics. In the first pace, if we think of the macroscopic eectric fied E at a given point x as some average of the microscopic eectric fied E µ over some surrounding voume ΔV, we can write E( x) ΔV If we cacuate the cur of the macroscopic fied, we have E µ ( x + x )d x. (.9) ΔV E( x) ΔV ΔV ΔV E µ ( x + x )d x E µ ( x + x )d x, ΔV (.) and since from the microscopic equation E µ, then E (.) for the macroscopic fied. So the same reation between the eectric fied and the potentia exists when deaing with ponderabe media. That is, E( x) Φ. (.) 5
When a medium made up atoms or moecues is subjected to an externa eectric fied, the charges making up the moecues wi react to its presence by individuay producing a, or enhancing an aready existing, dipoe moment. The materia as a whoe wi, therefore, become eectricay poarized, the resuting dipoe moment being the dominant mutipoe term. That is, an eectric poarization P (dipoe moment per unit voume) given by N i x P x p i (.) i is induced in the medium, where p i is the average dipoe moment of the ith type of moecues in the medium (cacuated in the same manner as the eectric fied was in equation (.9)), and N i is the average number density of the same type of moecues at point x. The average charge density ρ x is evauated using the same process. Simiary, we can use equation (.89) to cacuate the contribution to the potentia at the position x from the macroscopicay sma voume eement ΔV ocated at x ΔΦ( x, x ) ρ ( x )ΔV 4πε + ( ) P ( x )ΔV, (.4) where ρ ( x )ΔV and P ( x )ΔV are, respectivey, the average monopoe (or charge) and dipoe moments contained in the voume. Taking the imit ΔV d x, and integrate over a space, we get the potentia Φ( x) ρ x d x 4πε x x + P ( x ), (.5) where we used equation (.55). Transforming the second term, we find P x ( ) d P x x P x d x P ( x ) n d a P ( x ) d x P x d x, (.6) since the first integra on the right-hand side is over an infinite surface. The potentia now becomes Φ( x) d x 4πε ρ ( x ) P x x ( x ). (.7) 5
Equation (.7) has the same form as equation (.57) for the potentia, as ong as we define a new effective charge density ( ρ P). By effective charge density, we mean that if the poarization is non-uniform in a given region (i.e., P ), then there wi be a net change in the amount of charge within that region. Upon using equation (.), we can write E( x) Φ 4πε ρ ( x ) P ( x ) d x ρ ( ) δ ε x P x d x (.8) P( x) ε ρ x. We now define the eectric dispacement D as D ε E + P (.9) and equation (.8) becomes D ρ (.4) In most media P is proportiona to E (i.e., the media are inear and isotropic), and we write P ε χ e E, (.4) where χ e is the eectric susceptibiity of the medium. In such cases, D is aso proportiona to E and with the eectric permittivity ε defined by D εe, (.4) ε ε ( + χ e ). (.4) The quantity ε ε is caed the dieectric constant. If the medium is aso homogeneous, then the divergence of the eectric fied becomes 54
E ρ ε. (.44) Consequenty, any eectrostatic probem in a inear, isotropic, and homogeneous medium is equivaent to one set in vacuum (soved using the microscopic equations) as ong as the eectric fied is scae by a factor ε ε. Finay, the boundary conditions derived for the microscopic eectric fied (see equations (.79)) can easiy be extended to such medium ( D D ) n σ n, E E (.45) where n is a unit vector extending from medium to medium, and norma to the boundary surface, and σ is the surface charge..6 Eectrostatic Energy in Dieectric Media In evauating the amount of energy contained in a dieectric, we must be cautious to carefuy incude not ony the energy needed to assembe the charge distribution (by bringing it from infinity), as was done in section.4, but aso the energy spent in poarizing the medium. So, if we consider a change δρ in the macroscopic charge distribution, which extends over a space, then the work done to accommodate this change is (see equation (.)) δw δρ( x)φ( x)d x, (.46) where Φ is the potentia due to the aready existing charge density ρ( x). Using equation (.4), we aso write the change in charge density as Inserting this resut in equation (.46) we get δρ ( δd). (.47) ( δd) δw Φ d x δdφ E δd d x, ( δd) Φ d x (.48) since the integra of the divergence on the second ine vanishes when transformed into a surface integra (over a space) using the divergence theorem. Initiay, before the dieectric medium is assembed and the eectric density attains its fina vaue D, we must have D. Therefore, 55
If the medium is inear, then D εe and W d x E δd. (.49) D δ ( E D) εδ ( E E) ε ( E δe) ( E δd), (.5) and the tota eectrostatic energy in the medium is W E D d x (.5) Finay, et s consider a dieectric ε immersed in another (with permittivity ε ) where a fixed eectric fied E exists. We assume that there are no free charge distributions. That is D D. (.5) If the first dieectric ε were absent, the energy contained within the equivaent voume of space V it occupies woud be whereas it is W E D d x, (.5) W with it in pace. The difference between the two energies is E D d x (.54) W W W ( E D E D ) d x V d x ( D D ) d x E D E D + E + E V. V (.55) and, therefore, E + E Φ, and ( D D ) However, since E + E from equation (.5), the second integra on the right-hand side of equation (.55) is shown to vanish from 56
( E + E ) ( D D ) d x V Φ ( D D ) d x V V { ( D D )Φ Φ ( D D )} d x V ( D D )Φ d x (.56), Φ( D D ) n da S where the integra of the divergence term was transformed into a surface integra over S, the surface deimiting the voume V, which vanishes because D D at the surface (or just beyond it). Hence, d x W E D E D. (.57) V Aternativey, if we use D ε E and D ε E, and the region occupied by the object beongs to free space (i.e., the permittivity ε is that of vacuum), then W ( ε ε )E E d x. V P E d x. V (.58) We can, therefore, define the energy density of a dieectric in a fixed eectric fied E as w P E. (.59) If we aow for an increase δe in the eectric fied reative to some coordinate x i, then equation (.58) tes us that there wi be a corresponding decrease δw in the potentia energy. We, therefore, find that the force on the dieectric is δw F im >, (.6) δ E δ x i and that the dieectric wi tend to acceerate toward regions of increasing eectric fied intensity provided that ε > ε. 57