lectomagnetism Physics 15b Lectue #20 Dielectics lectic Dipoles Pucell 10.1 10.6 What We Did Last Time Plane wave solutions of Maxwell s equations = 0 sin(k ωt) B = B 0 sin(k ωt) ω = kc, 0 = B, 0 ˆk = Ê ˆB Popagates along k with speed c Two possible polaizations fo the same k negy flow given by the Poynting vecto Fo plane waves, aveage enegy flow is S = c 4π B S = c 4π c 2 ˆk = 8π 2 ˆk 0 1
Goals Today Intoduce dielectic mateial Synonym fo insulato ffects on electic field expessed by dielectic constant Univesally found in capacitos Look into the micoscopic oigin of dielectic Dipole moment of small chage distibution lectic field geneated by a dipole moment Foce on a dipole moment due to electic field Next lectue: constuct dielectic fom dipoles Mateial in a Unifom Field A slab of insulato is in a unifom field and chages feel the foce F = q They can t flow, but they do move slightly fom the natual (equilibium) positions xcess and chages appea on the sufaces Additional electic field appeas inside = 4πσ and opposite to It s easonable to assume σ Total electic field inside the insulato is F in = out = (1 k) out = 1 ε out ε = the dielectic constant of the mateial ε > 1, i.e., the electic field is weake inside σ σ 2
Dielectic Constant Dielectic constant ε depends on the density Gases vey close to 1 Liquids and solids 2 to 10 Liquids and gases of pola molecules have lage ε Molecules can otate to align with electic field Pime example: wate H O H H O H See textbook 10.6 and 10.12 Substance Condition ε Ai Gas, 0 C 1.0006 Methane Gas, 0 C 1.0009 HCl Gas, 0 C 1.0046 Wate Gas, 110 C 1.0126 Liquid, 20 C 80.4 Benzene Liquid, 20 C 2.28 Methanol Liquid, 20 C 33.6 Ammonia Liquid, 34 C 22.6 Mineal oil Liquid, 20 C 2.24 NaCl Solid, 20 C 6.12 Sulfu Solid, 20 C 4.0 Silicon Solid, 20 C 11.7 Polyethylene Solid, 20 C 2.25 2.3 Pocelain Solid, 20 C 6.0 8.0 Dielectic in Capacito Capacito is filled with a dielectic Chage ±Q on the plates ceate Dielectic polaizes and ceates suface chages Actual electic field inside is 0 = 4πQ A = 0 ε = 4πQ εa Potential diffeence is ight = ds = 4πQd C = Q left εa = εa Dielectic inceases the capacitance by a facto ε 4πd = ε C 0 aea A Real-wold capacitos use vaious dielectic mateials pape, ceamics, mica, oil, liquid electolyte, etc. Q Q Issues: field (voltage) toleance, fequency esponse, tempeatue dependence, polaity, long-tem stability d capacitance w/o dielectic 3
Half-Filled Capacito A ectangula capacito is patially filled with a dielectic If it wee empty, the capacitance a would be C 0 = ab 4πd Conside it as two capacitos: a(b x) mpty pat: C empty = b 4πd Filled pat: C filled = εax 4πd C total = d x ε a{b (ε 1)x} 4πd connected in paallel Suppose thee is chage Q in this capacito negy is U = Q2 2C = 2πdQ 2 du a{b (ε 1)x} dx = 2πdQ2 (ε 1) a{b (ε 1)x} < 0 2 Inceasing x deceases the potential enegy lectostatic foce pulls the dielectic into the capacito Small Chage Distibution So fa we had unifom field causing unifom polaization Moe geneal case (non-unifom ) equies a bette famewok Conside an abitay chage distibution of a small size, and the electic field fa away fom it lectic potential at point A is A ρ( )d v ϕ = Integal ove volume whee ρ 0 Since <<, we use Taylo expansion 1 = ( 2 2 cosθ 2 ) 1 2 = 1 1 cosθ 2 (3 cos 2 θ 1) 2 2 ρ 0 4
Moments We can now expess the potential at A as ϕ = 1 ρ( )d v 1 ρ( ) cosθ d v 1 (3 cos 2 θ 1) ρ( ) 2 2 d 3 2 v K 0 K 1 K 2 At lage distance, K 0 / >> K 1 / 2 >> K 2 / 3 >> We must conside highe-ode tems only if the peceding tems happen to be zeo K 0 is the net chage of the souce K 0 / is the familia Coulomb potential Fo a small chaged object, the Coulomb foce due to the net chage outweighs eveything else If the object (e.g. a molecule) is net neutal, the K 1 tem becomes impotant Dipole Moment We can ewite the K 1 tem as K 1 = 1 2 ρ( ) cosθ d v = ˆ 2 ρ( ) d v 2 Define the dipole moment of a chage distibution by p ρ( ) d v then the electic potential due to it is Q: Doesn t this definition depend on whee the oigin of the coodinate system is? A: It does. But that s OK ϕ = ˆ p 2 a vecto detemined by the chage distibution 5
Coodinate Oigin Let s move the oigin by Δ It s still inside the chage distibution Δ Δ The dipole moment becomes p ρ( )( Δ)d v = p QΔ Δ If Q = 0, no change to p If Q 0, we now have a diffeent p Re-calculate the fist two tems of the potential ϕ = Q ˆ p Q ( Δ) (p QΔ) 2 Δ Δ 3 Taylo expand by Δ/ and take the leading tems Q 1 ˆ Δ 2 ˆ (p QΔ) 2 xta tems cancel between the Coulomb and the dipole pats A Simple lectic Dipole Two chages, q and q, sepaated by a distance s p ρ( ) d v = qs q0 = qs This is a useful model fo any neutal object (e.g. a molecule) with a dipole moment p lectic field at lage distances will be identical We know how to daw field lines aound this q s q 6
Dipole lectic Field The electic field due to a dipole p is = ϕ = ˆ p 2 Use spheical coodinates, the pola axis paallel to p ϕ = p cosθ 2 = ϕ 2p cosθ = 3 θ = 1 ϕ θ = p sinθ 3 field deceases with 1/ 3 Faste than Coulomb, as expected p Foce on Dipole An electic dipole p is in an extenal field Foce q acts on each chage q Net foce is zeo if total chage Q is zeo Foces on q and q may be equal and opposite, but not on the same line Combined, they poduce a toque N = The toque N otates the dipole so that it will line up with Also: the net foce F might not be zeo if is not unifom This one is ticky, so let s look at a simple example q F = q ( q) = q( ) = qs = p F = q( ) q( ) = q(s ) = (p ) q s q q 7
Dipole and a Chage A dipole p is nea a chage Q Assume s << i.e., the dipole is attacted to the chage What if Q is negative? Toque will otate p so that it points towad the chage Net foce F will point towad the chage again Q q p is lined up with Net foce is Q F = q( s) q() = q ( s) Q 2 2 2qQ 3 s q lectic dipole is geneally pulled towad stonge field This is why neutal objects (e.g. dust paticles) ae attacted by static electicity q q Summay lectic field inside dielectic is educed ε = dielectic constant Capacitance is inceased by facto ε Fa field due to small chage distibution is detemined by: Fist, the net chage Q Coulomb field If Q = 0, then the dipole moment p ρ( ) d v Modeled by s p = qs q q A dipole in an electic field eceives: Toque N = p If is non-unifom, net foce F = (p ) Dipoles ae attacted to stonge field in = 1 ε out ϕ = ˆ p 2 8