lim = F F = F x x + F y y + F z

Transcription

1 Physics 361 Summary of Results from Lecture Physics 361 Derivatives of Scalar and Vector Fields The gradient of a scalar field f( r) is given by g = f. coordinates f g = ê x x + ê f y y + ê f z z Expressed in Cartesian The differential change in f produced by a displacement d l is df = g d l. The gradient is a vector field that points in the direction of the steepest increase in f with a value that gives the slope in that direction. Surfaces of constant f are perpendicular to the gradient at every point. The flux of a vector field is F is Φ = F d A. The ratio of the flux through a closed surface to the volume enclosed, in the limit that the surface shrinks to a point, gives the divergence of the vector field at that point Φ S lim S 0 V S = F In Cartesian coordinates the divergence is expressed as the sum of partial derivatives F = F x x + F y y + F z z The divergence measures the flow of F into or out of every point in space. The divergence theorem is Φ S = F da = F dv V S S The total flux through a closed (not necessarily small) surface is given by the sum of the fluxes through the surfaces around every enclosed volume element. The circulation of a vector field is C L = F d l. The ratio of the circulation on a closed loop to the area enclosed by the loop is the curl: ê L F C L = lim L 0 A L where ê L is a unit vector along the direction of the area vector obtained by integrating d a on a surface that terminates on the contour L. In Cartesian coordinates the components of the curl can be expressed as differences of cross derivatives of the Cartesian resolved components of F ( F Fz = y F ) y ê x + z ( Fx z F ) ( z Fy ê y + x x F ) x ê z y

2 The curl measures the flow of a vector field around a point. Stokes Theorem states L F d l = F da A L so that the total circulation in a closed loop L is the sum of the circulations around differential loops bounded by L. Derivatives of the central inverse square field (ê r /r 2 ) = 4πδ 3 ( r) where δ 3 ( r) is a quantity that is zero when its argument is nonzero, and infinite at r = 0 such that dv δ 3 ( r) = 1. Since ê r /r 2 = (1/r) we have ( ) 1 2 = 4πδ 3 ( r) r The one dimensional variant δ(x) is zero for nonzero argument and infinite for x = 0 such that dx δ(x) = 1. Helmholtz Theorem for Vector Fields A vector field F that goes to zero at infinity (faster than r 2 ) can be reconstructed knowing its divergence D( r) and curl C( r): F = U + W where U = 1 4π W = 1 4π d 3 r D( r ) r r d 3 r C( r ) r r and W = 0. Some Useful Electric Field Formulas Field exterior to a bounded, isotropic charge distribution with total charge Q E( r) = 1 4πϵ o Q r 2 êr Field in the interior of a bounded, isotropic charge distribution with total charge Q, radius R and uniform charge density E( r) = 1 4πϵ o Qr R 3 êr The electric field inside a uniform shell of charge is zero.

3 Field exterior to a cylindrically symmetric charge distribution with charge per unit length λ E( r) = 1 2πϵ o λ s ês Field exterior to a uniform sheet of charge with areal charge density σ E( r) = σ 2ϵ o ê n Fields for a polarized spherical shell of charge with radius R and surface charge density σ = σ o cos θ E > = σ or 3 3ϵ o r 3 (2 cos θê r + sin θê θ ) E < = σ o 3ϵ o ( cos θê r + sin θê θ ) This is a dipole shell with total dipole moment p = 4πσ o R 3 /3 ê z. Boundary Conditions on the Electrostatic Fields at a Surface E n,i E n,ii = σ ϵ o (discontinuous) E,I = E,II (continuous) Electric Potential E = V where V ( r) = 1 4πϵ o d 3 r ρ( r ) r r The difference in electric potential can be calculated by evaluating a line integral B V (B) V (A) = A E d l Electric potential exterior to a bounded isotropic charge distribution with total charge Q V ( r) = 1 Q 4πϵ o r Electric potential in the interior of a bounded isotropic charge distribution with total charge Q, radius R and uniform charge density V ( r) = 1 Q 4πϵ o R ( ) 3 2 r2 2R 2

4 The electric potential inside a uniform shell of charge is constant. Electric potential exterior to a cylindrically symmetric charge distribution with linear charge density λ and with V = 0 at s = a V = 1 ( ) a λ ln 2πϵ o s Electric potential for a polarized spherical shell of charge with radius R and surface charge density σ = σ o cos θ Electrostatic Energy V > = σ or 3 3ϵ o r 2 cos θ V < = σ or cos θ 3ϵ o The interaction energy of a collection of point charges q i at positions r i is U = <i,j> 1 4πϵ o q i q j r i r j where the sum extends over all pairs of charges, each charge counted once. This can be expressed as a sum over all pairs of charges U = 1 2 i j which can be equivalently expressed as U = πϵ o q i q j r i r j d 3 r ρ( r)v ( r) which can be equivalently expressed as U = d 3 r ϵ o 2 E( r) 2 The integrand in this expression u = ϵ o E 2 /2 is the energy density: energy per unit volume stored in the electric field. Energies of Some Simple Charge Distributions The energy of a uniform ball of charge with total charge Q and radius a is U = 1 3 4πϵ o 5 The energy of a uniform shell of charge with total charge Q and radius a is Q 2 a U = 1 Q 2 8πϵ o a

5 The energy of a polarized spherical shell of charge with radius R and surface charge density σ = σ o cos θ is Conductors in Equilibrium U = 2πσ2 or 3 9ϵ o In static equilibrium the electric field inside a conductor (in the conducting region) is zero. It follows that the volume charge density is zero everywhere in the conducting region, any uncompensated charge resides at the boundary of the conductor, the normal component of the external electric field is the areal charge density σ = ϵ o E > n and each point of the conductor has a common electric potential. Capacitively Coupled Conductors The interactions of N electrostatically coupled conductors can be described by an N N matrix of potential coefficients, and it s inverse, the capacitance matrix. Thus V i = q i = N P ij q j j=1 N C ij V j j=1 The electrostatic energy can be represented as an inner product using either of these matrices Laplace Equation U = 1 P ij q i q j = 1 2 i,j=1,n 2 i,j=1,n C ij V i V j In a charge free region of space 2 V = 0. If V is a function of only one spatial variable x, then the solution of this equation is V (x) = Ax + B In higher dimensions, in special coordinate systems, Laplace s equation admits separable (factorizable) solutions. Examples of these solutions are (2D Cartesian solutions) (2D polar solutions) V (x, y) = (A k e kx + B k e kx )(C k sin(ky) + D k cos(ky)) V (s, ϕ) = (A m s m + B m s m )(C m sin(mϕ) + D m cos(mϕ))

6 along with an isotropic solution (3D axially symmetry polar solutions) V (s) = A o + B o ln s V (r, θ) = (A l r l + B l r l+1 )P l(cos θ) where P l (u) are the Legendre polynomials, tabulated in many math and physics texts. A solution for V usually requires a coherent superposition of these separable solutions with coefficients chosen to satisfy the boundary conditions for V. Determination of the coefficients is possible using a projection method that isolates individual coefficients in a superposition using the following orthogonality relations: For a Fourier series in a channel of width a a ( ) ( ) mπy nπy sin sin dy 0 a a = 0 m n (and similar for for the cos(mπy/a) series.) For a Fourier series on a circle = a 2 m = n 2π 0 sin (mϕ) sin (nϕ) dϕ = 0 m n = π m = n (and similar for for the cos(mϕ) series.) For a Legendre expansion with u = cos θ 1 1 P l (u)p n (u) du = 0 l n = 2 2n + 1 l = n Some Image Solutions to Laplace s Equation Point charge q a distance d above a ground plane: The exterior potential from the induced surface charge is the potential of an image charge q = q a distance d below the ground plane. Point charge q a distance a from the center of a grounded conducting sphere of radius R (a > R): The exterior potential from the induced surface charge is the potential of an image charge q = qr/a a distance b = R 2 /a from the center of the sphere. There is an attractive force F = q 2 Ra/(4πϵ o (a 2 R 2 ) 2 ) on the exterior charge. Point charge q a distance a from the center of an isolated conducting sphere with total charge Q s : The exterior potential from the induced surface charge is

7 the superposition of the potential of a primary image charge q = qr/a a distance b = R 2 /a from the center and a secondary image charge q c = Q s +qr/a located at the center. For the special case of an uncharged isolated conducting sphere q c = qr/a. Point charge q at distance b from the center of a spherical cavity with radius R in a grounded conductor (b < R): The interior potential from the induced surface charge is the potential of an image point charge q = qr/b a distance a = R 2 /b from the center of the cavity. For the interior solution the image charge is not equal to the total charge ( q) actually induced on the walls of the cavity. Multipole Expansion The potential of a bounded axi-symmetric charge distribution ρ( r) can be represented by the multipole expansion V (r, θ) = 1 Q l 4πϵ o r P l(cos θ) l+1 l=0 where Q l = 2π ρ( r )(r ) l+2 P l (cos θ ) sin θ dθ dr If the charges are point charges along the z axis, this simplifies to Q l = i q i z l i Retaining the monopole l = 0 and dipole l = 1 terms this reads V (r, θ) = 1 ( Q0 4πϵ o r + Q ) 1 cos θ r 2 where Q O is the total charge and Q 1 is the total dipole moment. Torque, Force and Energy of a Dipole in an Electric Field The torque on a dipole p in a uniform electric field E is N = p E The force on a dipole in a uniform electric field is zero. In a nonuniform field the force is F = ( p ) E = ( p E) The energy of a dipole in an electric field is Fields and Interactions for Dipoles U = p E

8 The electric field of a dipole p 1 at the origin is E( r) = 1 4πϵ o [ ] 3( p1 ê r )ê r p 1 The interaction energy of a pair of dipoles is therefore U = 1 4πϵ o r 3 (3( p 1 ê r )( p 2 ê r ) p 1 p 2 ) The force law for two dipoles p 1 and p 2 separated by r is F 2 = r πϵ o r [ 15( p 4 1 ê r )( p 2 ê r )ê r + 3( p 1 p 2 )ê r + 3 p 1 ( p 2 ê r ) + 3 p 2 ( p 1 ê r )] = F 1 Polarized Matter The polarization of macroscopic matter that has a net dipole moment is P ( r): this is the dipole density or the dipole moment per unit volume. The polarization can be spontaneous or induced by an applied field. The electric potential from polarized matter is V ( r) = 1 4πϵ o S d 2 r σ b ( r ) r r + 1 4πϵ o V d 3 r ρ b ( r ) r r where the bound charge densities are σ b = P e n (surface bound charge areal density) and ρ b = P (volume bound charge density). In polarized media it is conventional to define the displacement field D = ϵ o E + P so that D = ρ f where ρ f is the free charge density, i.e. the volume density of charges that are not microscopically bound in dipoles. In general D is not simply a gradient since D 0. At an interface, in the absence of free charge and for static fields, the parallel component of E is continuous (E < = E> ) and the normal component of D is continuous (D < = D> ). Linear Dielectric Media In a linear homogeneous isotropic dielectric the local value of P is proportional to the local value of the electric field E: P = ϵo χ E, where χ is the (dimensionless) electric susceptibility of the medium In a linear medium D = ϵ o (1 + χ) E = ϵ o κ E where κ is the dielectric constant. The product ϵ = κϵ o is called the permittivity of the medium. In the body of a linear dielectric medium ( ρ b = 1 1 ) ρ f κ (1)

9 and therefore in the absence of a free charge density the volume bound charge density is always zero, and bound charges can occur only at boundaries of the medium. Dielectric Images (not in coverage for exam) A point charge a distance d above a semi-infinite dielectric slab with relative permittivity ϵ r induces a bound surface charge q = ((κ 1)/(κ + 1))q which produces the exterior potential of an image charge q located a distance d below the interface. Thus a free charge is attracted to a neutral semi-infinite dielectric medium with a force F = 1 16πϵ o ( ) κ 1 q 2 κ + 1 d 2 Similarly a point charge embedded within a semi-infinite dielectric, a distance d from the interface, is repelled from the interface (the force is directed into the bulk of the material) by a force F = 1 16πϵ o κ ( ) κ 1 q 2 κ + 1 d 2 This force is the interaction of the unscreened charge with the screened field. Work and Potential Energy in Dielectric Media In the presence of polarizable matter, the potential energy (i.e. the energy stored reversibly in a system of charges) is the work required to assemble the free charges by a quasistatic process. Using the charge-potential formulation U = 1 q i V ( r i ) 2 i where the sum over i extends over all the free charges, and the potential is the electric potential from all sources (free or bound). The potential energy is the volume integral of an energy density u = D E/2 integrated over all space. In vacuum this energy density reduces to the familiar u = ϵ o E 2 /2 result. Depolarizing Fields The induced polarization of a linear medium depends on χ but also on its shape. This occurs because the bound charge density can produce a shape dependent depolarizing field that acts to opposes the polarization of the medium. Examples are the polarization of a sphere for which P s = 3ϵ oχ 3 + χ E o or for a flat disk in a perpendicular field for which P d = ϵ oχ E 1 + χ o An object in the shape of a long thin needle has approximately no depolarizing field for an external field applied parallel to its long axis, so P n = ϵ o χ E o.

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