Magnetism from Conductors, and Enhanced NonLinear Phenomena


 Donald Goodwin
 3 years ago
 Views:
Transcription
1 Mnetism from Conductors, nd Enhnced NonLiner Phenomen JB Pendry, AJ Holden, DJ Roins, nd WJ Stewrt Astrct  We show tht microstructures uilt from nonmnetic conductin sheets exhiit n effective mnetic permeility, µ eff, which cn e tuned to vlues not ccessile in nturlly occurrin mterils, includin lre iminry components of µ eff. The microstructure is on scle much less thn the wvelenth of rdition, is not resolved y incident microwves, nd uses very low density of metl so tht structures cn e extremely lihtweiht. Most of the structures re resonnt due to internl cpcitnce nd inductnce, nd resonnt enhncement comined with compression of electricl enery into very smll volume retly enhnces the enery density t criticl loctions in the structure, esily y fctors of million nd possily y much more. Wekly nonliner mterils plced t these criticl loctions will show retly enhnced effects risin the possiility of mnufcturin ctive structures whose properties cn e switched t will etween mny sttes. Index Terms  effective permeility, nonlinerity, photonic crystls JB Pendry is with The Blckett Lortory, Imperil Collee, London, SW7 BZ, UK. AJ Holden, DJ Roins, nd WJ Stewrt re with GECMrconi Mterils Technoloy Ltd, Cswell, Towcester, Northmptonshire, NN 8EQ, UK. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
2 I. INTRODUCTION In sense every mteril is composite, even if the individul inredients consist of toms nd molecules. The oriinl ojective in definin permittivity, ε, nd permeility, µ, ws to present n homoeneous view of the electromnetic properties of medium. Therefore it is only smll step to replce the toms of the oriinl concept with structure on lrer scle. We shll consider periodic structures defined y unit cell of chrcteristic dimensions. The contents of the cell will define the effective response of the system s whole. Clerly there must e some restrictions on the dimensions of the cell. If we re concerned out the response of the system to electromnetic rdition of frequency ω the conditions re esy to define: << λ= πc ω () If this condition were not oeyed there would e the possiility tht internl structure of the medium could diffrct s well s refrct rdition ivin the me wy immeditely. Lon wvelenth rdition is too myopic to detect internl structure nd in this limit n effective permittivity nd permeility is vlid concept. In the next section we shll discuss how the microstructure cn e relted to ε, µ. eff eff In n erlier pper [] we showed how structure consistin of very thin infinitely lon metl wires rrned in 3D cuic lttice could model the response of dilute plsm, ivin netive ε eff elow plsm frequency somewhere in the ihertz rne. Theoreticl nlysis of this structure hs een confirmed y experiment []. Sievenpiper et l hve lso investited plsmlike effects in metllic structures [3,4]. Idelly we should like to proceed in the mnetic cse y findin the mnetic nloue of ood electricl conductor: unfortuntely there isn t one! Nevertheless we cn find some lterntives which we elieve do ive rise to interestin mnetic effects. Why should we o to the troule of microstructurin mteril simply to enerte prticulr µ eff? The nswer is tht toms nd molecules prove to e rther restrictive set of elements from which to uild mnetic mteril. This is prticulrly true t frequencies in the ihertz rne where the mnetic response of most mterils is einnin to til off. Those mterils, such s the ferrites, tht remin modertely ctive re often hevy, nd my not hve very desirle mechnicl properties. In contrst, we shll show, microstructured mterils cn e desined with considerle mnetic ctivity, oth dimnetic nd prmnetic, nd cn if desired e mde extremely liht. There is nother quite different motivtion. We shll see tht stron mnetic ctivity implies stronly inhomoeneous fields inside the mteril. In some instnces this my result in locl field strenths mny orders of mnitude lrer thn in free spce. Dopin the composite with non liner mteril t the criticl loctions of field concentrtion ives enhnced nonlinerity, reducin power requirements y the field enhncement fctor. This is not n option ville in conventionl mnetic mteril. We show first how to clculte µ eff for system, then we propose some model structures which hve mnetic ctivity nd ive some numers for these systems. Finlly we show how electrosttic enery cn e stronly concentrted in these structures nd hence the demonstrte potentil for enhncin non liner effects. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
3 II. DEFINING AN EFFECTIVE PERMEABILITY We re seekin to uild structures with effective epsilon nd mu, B D = µ µ H ve eff ve = ε ε E ve eff ve () where we ssume tht the structure is on scle much shorter thn the wvelenth of ny rdition so tht we cn sensily spek of n vere vlue for ll the fields. A key question is how do the veres differ? Clerly if the structure is mde of thin wires or sheets of metl then if the veres were tken over the sme reions of spce, ε, µ would lwys e unity. However we oserve tht Mxwell s equtions, eff eff H = + D / t E = B / t my e pplied in the interl form, zc zc H dl = + E dl = t t zs zs D. ds B. ds (3) (4) where the line interl is tken over loop c which encloses n re s. This form of the equtions immeditely suests prescription for verin the fields. For simplicity we shll ssume tht the periodic structure is descried y unit cell whose xes re orthoonl s shown in fiure elow. Some of the ruments used in this section re similr to those we used in derivin finite difference model of Mxwell s equtions [5]. Fiure. Unit cell of periodic structure. We ssume tht the unit cell dimensions re much smller tht the wvelenth of rdition, nd vere over locl vritions of the fields. In the cse of the B  field we vere over the fces of the cell nd in the cse of the H  field, over one of the edes. We choose to define the components of H ve y verin the H field lon ech of the three xes of the unit cell. If we ssume simple cuic system, /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 3
4 H H H ve x ve y ve z z zz r=,, = H dr r=,, r=,, = H dr r=,, r=,, = H dr r=,, (5) There is only one cvet concernin the definition of the unit cell: its edes must not intersect with ny of the structures contined within the unit cell. This leves us free to cut the structure into whole numer of unit cells when we come to crete surfce nd ensures tht the prllel component of H ve is continuous cross the surfce s required in consistent theory of n effective medium. To define B ve we vere the B field over ech of the three fces of the unit cell defined s follows: S x is the surfce defined y the vectors y, z S y is the surfce defined y the vectors x, z S z is the surfce defined y the vectors x, y Hence we define, B B B ve x ve y ve z = B ds S = B ds z zz S = B ds S x z y The rtio defines the effective epsilon nd mu from (), d i d i d i µ eff = B µ x ve x Hve x µ eff = B µ y ve y Hve y µ eff = B µ z ve z Hve z (6) (7) Thus if we seek lre effect we must try to crete fields tht re s inhomoeneous s possile. We shll explore vrious confiurtions of thin sheets of metl, derive µ eff, nd discuss the results with view to mkin the effect s lre s possile. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 4
5 III. EXAMPLES OF MAGNETIC MICROSTRUCTURES A. An Arry of Cylinders Fiure. Model A consists of squre rry of metllic cylinders desined to hve mnetic properties in the direction prllel to the xes of the cylinders. We strt with very simple structure for the purposes of illustrtion. Let us pply n externl field, H, which we shll tke to e prllel to the cylinders. We ssume tht the cylinders hve conductin surfce so tht current, j, per unit lenth, flows. The field inside the cylinders is, H = H + j j (8) where the second term on the riht hnd side is the field cused directly y the current, nd the third term is the result of the depolrisin fields with sources t the remote ends of the cylinders. If the cylinders re very lon the depolrisin field will e uniformly spred over the unit cell, ut will hve the sme numer of lines of force in it s the direct field inside the cylinders. We now clculte the totl emf round the circumference of cylinder: L M N M L N O P Q O P Q r emf = π r µ π H + j j t M P = + iωπ r µ H + j j M P σj σj where σ is the resistnce of the cylinder surfce per unit re. The net emf must lnce nd therefore, (9) j = iωπ r µ H = L r i r O r M L π ωπ µ P π σ M N Q N H O r i P + σ ω r µ Q () We re now in position to clculte the relevnt veres. The vere of the Bfield over the entire unit cell is, Bve = µ H () /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 5
6 However if we vere the Hfield over line lyin entirely outside the cylinders, Hve = H j = H L NM H O r i P + σ ω r µ Q = H L NM σ + i ω rµ O i P + σ ω rµ Q () Hence we define, µ eff σ + i B ve ω rµ = = µ Hve σ + i ω rµ L NM = + σ i ω rµ O QP (3) For n infinitely conductin cylinder, or in the hih frequency limit, µ eff is reduced y the rtio of the cylinder volume to the cell volume. This rtio of volumes will turn out to e the key fctor in determinin the strenth of the effect in ll our models. Evidently in the present model µ eff cn never e less thn zero, or reter thn unity. It should lso e mentioned tht to mximise the effect we could hve replced the metllic cylinders with prisms of squre cross section to mximise the volume enclosed within the prism. If the resistivity of the sheets is hih then the dditionl contriution to µ eff is iminry ut lwys less thn unity, µ eff 3 ω µ + i, σ >> ω rµ (4) σ A further point tht should e noted is tht ll the structures we discuss hve electricl s well s mnetic properties. In this prticulr cse we cn crudely estimte for electric fields perpendiculr to the cylinders, ε eff F HG = F = I KJ where F is the frction of the structure not internl to cylinder. In derivin (5) we ssume tht the cylinder is perfect conductor, nd nelect depolrisin fields risin from interction etween cylinders. Inclusion of ε eff in our clcultions removes one difficulty y ensurin tht, (5) lim ω d i (6) liht eff eff ω c = lim c ε µ = c Evidently without ε eff the velocity of liht in the effective medium would hve exceeded tht in free spce. Most of the structures discussed in this pper hve similr ε eff. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 6
7 B. A Cpcittive Arry of Sheets Wound on Cylinders The previous structure showed limited mnetic effect. Now we show how to extend the rne of mnetic properties ville to us y introducin cpcittive elements into the structure. We tke the sme structure of cylinders s efore except tht the cylinders re now uilt in split rin confiurtion shown elow in fiure 3. Fiure 3. Model B consists of squre rry of cylinders s for model A ut with the difference tht the cylinders now hve internl structure The sheets re divided into split rin structure nd seprted from ech other y distnce d. In ny one sheet there is p which prevents current from flowin round tht rin. The importnt point is tht there is p which prevents current from flowin round ny one rin. However there is considerle cpcitnce etween the two rins which enles current to flow, Fiure 4. When mnetic field prllel to the cylinder is switched on it induces currents in the split rins s shown in the fiure. The reter the cpcitnce etween the sheets, the reter the current. Detiled clcultions ive, µ eff = F σi 3 + ω rµ 3 π µ ω Cr (7) where F is the frctionl volume of the cell side occupied y the interior of the cylinder, r F = π (8) nd C is the cpcitnce per unit re etween the two sheets, /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 7
8 ε C = = d dc µ (9) Hence, µ eff = σi 3dc + ω rµ 3 π ω r () Becuse we now hve cpcitnce in the system which cn lnce the inductnce present, µ eff hs resonnt form which we sketch elow in fiure 5. µ eff µ eff = ω ω mp ω Fiure 5. The effective mnetic permeility for model B shows resonnt structure dictted y the cpcitnce etween the sheets nd the mnetic inductnce of the cylinder. We sketch the typicl form for hihly conductin smple, σ. Below the resonnt frequency µ eff is enhnced, ut ove resonnce µ eff is less thn unity nd my e netive close to the resonnce. Fiure 5 illustrtes the eneric form of µ eff for ll the structures we present here. We define ω to e the frequency t which µ eff diveres, 3 3dc ω = = 3 3 π µ Cr π r () nd ω mp to e the mnetic plsm frequency ω mp = π µ 3 3 Cr F = π 3 r 3dc F HG I KJ () Note tht the seprtion etween ω ndω mp, which is mesure of the rne of frequencies over which we see stron effect, is determined y /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 8
9 F = (3) the frction of the structure not internl to cylinder. As for the cse A, the simple cylinder, the hih frequency limit is iven y, lim µ eff ω ω = (4) We mention in pssin tht the system sustins lonitudinl mnetic modes t the mnetic plsm frequency, the nloue of the plsm modes of s of free electricl chres [6,7]. Of course, we hve no free mnetic poles, only the ppernce of such s currents round the cylinders mke the cylinder ends pper to support free mnetic poles in the fshion of r mnet. Toether with ε eff iven in eqution (5), which is lso pplicle here, we cn illustrte eneric dispersion reltionship shown elow in fiure 6. Fiure 6. Generic dispersion reltionship for resonnt structures with µ eff. The solid lines represent twofold deenerte trnsverse modes, the dotted line sinle lonitudinl mnetic plsmon mode. The relevnt points to note re: (i) wherever µ eff is netive there is p in the dispersion reltionship, i.e. for, ω ω ω < < mp (5) (ii) lonitudinl mnetic plsm mode, dispersionless in this pproximtion, is seen t ω = ω mp. (iii) The dispersion reltion converes symptoticlly to the free spce liht cone. s discussed ove. In fct metllic structures in enerl represent fresh pproch to the photonic insultor concept introduced independently y Ylonovitch [8,9] nd John []. If we tke the followin vlues, 3 r =. m 3 = 5. m 4 d =. m (6) /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 9
10 we et,. Hz (7) f mp = π ω mp = 47 9 f F HG I 9 = fmp 94 KJ =. Hz (8) Note tht the frequency t which the structure is ctive corresponds to free spce wvelenth of cm, much reter tht the.5cm seprtion etween cylinders. This will e typicl of these cpcittive structures nd implies tht the effective medium pproximtion will e excellent. C Swiss Roll Cpcitor We tke the sme rrnement of cylinders on squre lttice s efore except tht the cylinders re now uild s follows: Fiure 7. In model C metllic sheet is wound round ech cylinder in coil. Ech turn of the coil is spced y distnce d from the previous sheet. The importnt point is in tht no current cn flow round the coil except y virtue of the self cpcitnce, Fiure 8. When mnetic field prllel to the cylinder is switched on it induces currents in the coiled sheets s shown in the fiure. Cpcitnce etween the first nd lst turns of the coil enles current to flow. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
11 In this instnce we find for the effective permeility, µ eff = F σi + ω rµ N 3 π r µ N ω C (9) = σi dc + ω rµ N 3 π r N ω where F is s efore the frction of the structure not internl to cylinder, nd the cpcitnce per unit re etween the first nd the lst of the coils is, ε C = = d N µ dc N (3) The criticl frequencies re, ω = dc = 3 π r µ C N π r N 3 (3) ω mp = dc = 3 F π r µ C N F I 3 π r N HG KJ (3) If we tke the vlues we used efore in (6), 3 r =. m 3 = 5. m 4 d =. m N = (33) we et, π ω. Hz (34) 9 f = = 38. (35) f mp = π ω mp = i.e. there is much more cpcitnce in this model nd the rne of ctive frequencies is n order of mnitude lower thn it ws in model C which used only two overlppin sheets. Choosin n even smller scle nd reducin the numer of turns in order to drive up the frequencies to our rne of interest, /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
12 4 r =. m 4 = 5. m 5 d =. m N = 3 (36) we et, 9 f = 85. Hz (37) f mp =. 5 9 Hz (38) Note tht the free spce wvelenth t the plsm frequency is round 3cm nd compre this to the very much smller spcin etween cylinders of.5cm. We shll now clculte the dispersion of µ eff for vrious prmeters. First let us tke the prmeters iven in eqution (36). The resultin dispersion of µ eff is shown elow in fiure 9 Fiure 9. Dispersion with frequency of µ eff for Swiss roll structure, clculted for the prmeters shown in eqution (36), ssumin tht the metl hs zero resistivity. Next we enquire wht is the effect of mkin the sheets resistive? Below we present series of clcultions for vrious vlues of the resistivity, σ, iven in Ω. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
13 Fiure. Dispersion with frequency of µ eff for Swiss roll structure, clculted for the prmeters shown in eqution (36), for vrious vlues of the resistivity of the sheets:.ω,.ω, 5.Ω,.Ω. In fiure we increse the resistivity from. Ω to. Ω. Note the rodenin of the resonnce, the complementry ehviour of µ rel nd µ im, dictted y Krmers Kroni, nd how resistivity limits the mximum effect chieved. Next we explore the dependence on the rdius of the cylinders. In fiure the rdius of the cylinders is decresed, reducin the volume frction occupied y the cylinders, nd risin the resonnt frequency y fctor of two. We lso decrese d, the spcin etween the sheets, incresin the cpcitnce in the system nd rinin the resonnt frequency ck down to its oriinl vlue. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 3
14 Fiure. Dispersion with frequency of µ eff for Swiss roll structure. Top: clculted for the prmeters shown in eqution (36), except tht the resistivity of the sheets is now.ω, nd the rdius of the cylinders hs een reduced from. 4 m to 6. 4 m, thus risin the resonnt frequency y fctor of two. Bottom: d, the spcin etween the sheets, hs een reduced to. 5 5 m rinin the resonnt frequency ck to the oriinl vlue. Usin cpcittive cylindricl structures such s the Swiss roll structure we cn djust the mnetic permeility typiclly y fctor of two nd, in ddition if we desire, introduce n iminry component of the order of unity. The ltter implies tht n electromnetic wve movin in such mteril would decy to hlf its intensity within sinle wvelenth. This presumes tht we re seekin rodnd effects tht persist over the reter prt of the GHz reion. However if we re prepred to settle for n effect over nrrow rne of frequencies spectculr enhncements of the mnetic permeility cn e chieved, limited only e the resistivity of the sheets nd y how nrrow nd we re willin to tolerte. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 4
15 III. AN ISOTROPIC MAGNETIC MATERIAL The structures shown ove ive mnetic properties when the field is lined lon the xes of the cylinders, ut hve essentilly zero mnetic response in other directions. They suffer from nother potentil prolem: if the lternte polristion is considered where the electric field is not prllel to the cylinders, the system responds like n effective metl ecuse current is free to flow lon the lenth of the cylinders. For some pplictions this hihly nisotropic ehviour my e undesirle. Therefore we redesin the system with view to restorin isotropy, nd minimisin purely electricl effects. To this end we need sic unit tht is more esily pcked into rrys thn is cylinder, nd which voids the continuous electricl pth provided y metl cylinder. We propose n dpttion of the split rin structure in which the cylinder is replced y series of flt disks ech of which retins the split rin confiurtion ut in slihtly modified form: see fiure. First we shll clculte the properties of disks stcked in squre rry s shown if fiure 3. This structure is still nisotropic, prolem we shll ddress in moment, ut y elimintin the continuous conductin pth which the cylinders provided, it elimintes most of the electricl ctivity lon this direction. Fiure. Left: pln view of split rin showin definitions of distnces. Riht sequence of split rins shown in their stckin sequence. Ech split rin comprises two thin sheets of metl. The rin shown is scled up version defined y the prmeters shown elow in fiure 3. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 5
16 Fiure 3. Pln view of split rin structure in squre rry, lttice spcin. The two dimensionl squre rry shown in fiure 3 cn e mde y printin with metllic inks. If ech printed sheet is then fixed to solid lock of inert mteril, thickness, the locks cn e stcked to ive columns of rins. This would estlish mnetic ctivity lon the direction of stckin, the z xis. The unit cell of this structure is shown in fiure 4 on the left. How do we mke symmetricl structure? Strt from the structure just descried comprisin successive lyers of rins stcked lon the z xis. Next cut up the structure into series of sls thickness, mkin incisions in the y z plne nd ein creful to void slicin throuh ny of the rins. Ech of the new sls contins lyer of rins ut now ech rin is perpendiculr to the plne of the sl nd is emedded within. Print onto the surfce of ech sl nother lyer of rins nd stck the sls ck toether in. The unit cell of this second structure is shown in the middle of fiure 4. In the next step third set of sls is produced y cuttin in the x z plne, printin on the surfce of the sls, nd ressemlin. Finlly we now hve structure with cuic symmetry whose unit cell is shown on the riht of fiure 4. Fiure 4. Buildin 3D symmetry: ech successive restckin of the structure dds rin to nother side of the unit cell. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 6
17 Of course n lternte method of mnufcturin this structure would e to strt from set of cues of the inert mteril nd loriously stick rins to their sides efore ssemlin the cues into lttice. The cut nd pste method we suest ove is much more efficient. Now let us clculte the effective permeility. First we need to clculte the cpcitnce etween the two elements of the split rin. We shll ssume: r >> c, r >> d (39) l < r (4) ln c d >> π (4) Under these conditions we cn clculte the cpcitnce etween unit lenth of two prllel sections of the metllic strips: ε c C c = ln = π d πµ c d ln (4) The effective mnetic permeility we clculte on the ssumption tht the rins re sufficiently close toether tht the mnetic lines of force due to currents in the stcked rins, re essentilly the sme s those in continuous cylinder. This cn only e true if the rdius of the rins is of the sme order s the unit cell side. We rrive t: µ eff = = σ + i 3 l l σ c ω rµ + l i 3 l 3 π µ ω C r ω rµ c πω d r 3 ln where σ is the resistnce of unit lenth of the sheets mesured round the circumference. (43) To ive some exmples let us choose convenient set of prmeters:  =. m 3 c =. m 4 d =. m 3 l =. m 3 r =. m (44) Fiures, 3 show the rins drwn to scle. These prmeters do not quite stisfy ll the inequlities, which is difficult to do with resonle numers, ut note tht the inequlities re only importnt to the ccurcy of our formule, not to the functionin of the structure. The resonnt frequency t which µ eff diveres is iven y, 3lc ω = = 7. c 3 π ln d r (45) /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 7
18 or, ω = π 35. GHz (46) If we choose to mnufcture the split rins from lyer of copper, it is esily possile to chieve σ.. Evidently from fiure 5, this produces hihly resonnt structure. Fiure 5. Plot of µ eff for the cuic split rin structure clculted usin the chosen prmeters. Left: for copper rins, σ =. ; riht: for more resistive rins, σ =.. In order to see sustntil effect we hve to increse the resistnce either y incresin the resistivity of the mteril of which the rins re mde, or y mkin them thinner. The sclin of frequency with size cn e deduced from (45) we see tht the resonnt frequency scles uniformly with size: if we doule the size of ll elements in iven structure, the resonnt frequency hlves. Nerly ll the criticl properties re determined y this frequency. IV. ENHANCED NONLINEAR EFFECTS We hve seen how the ddition of cpcitnce to the structure ives fr richer vriety of mnetic ehviour. Typiclly this hppens throuh resonnt interction etween the nturl inductnce of the structure nd the cpcittive elements, nd t the resonnt frequency electromnetic enery is shred etween the mnetic fields nd the electrosttic fields within the cpcittive structure. To put this more explicitly: tke the split rin structure descried in fiures, 3: most of the electrosttic enery of the cpcitor is locted in the tiny p etween the rins. Concentrtin most of the electromnetic enery in this very smll volume will results in n enormously enhnced enery density. If we wish to enhnce the nonliner ehviour of iven compound, we locte smll mount of the sustnce in the p where the stron electrosttic fields re locted. Since the response scles s the cue of the field mplitude, we cn expect enhncements of the order of the enery density enhncement squred. Furthermore not only does the structure enhnce the nonlinerity, it does so in mnner tht is very economicl with the mteril: less tht % of the structure need e filled with the nonliner sustnce. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 8
19 Note tht there is symmetry etween, on the one hnd the present structures desined to enerte mnetic permeility nd within which we find enhnced electrosttic fields, nd on the other hnd the erlier thin wire structures [,] desined to enerte netive electricl permittivity, nd within which we find enhnced mnetic fields. We shll now clculte the enery density in the cpcitnce etween the two split rins in fiures,3. First we clculte the volte etween the two rins s function of the incident mnetic field, H. V in V out s= s= s= V out V in Fiure 6. The emf ctin round one of the sheets of the split rin in fiure s function of the distnce, s, round the rin. V in denotes the emf on the inner rin, nd V out tht on the outer rin. Note tht this rin is cut t s = so tht the emf is discontinuous. The electric field etween the two hlves of the rin is then of the order, E rin V d (47) We clculte tht, 3li H V = ωcπ r L i 3 O r 3 M P + lσ l ω µ π µ ω C r N Q (48) Hence on sustitutin from (4) nd (47) into (48): E rin 3lµ c i H c dr L ω ln i c d O r c NM QP + lσ 3l ω µ 3 πω r ln d (49) Now we rue tht the electrosttic enery density in the incident electromnetic field is equl to the mnetic enery density, which in turn cn e relted to the electrosttic enery density in the rin. Hence, 3l ε E rin ε π r d ωcπ r = µ µ H lσ i 3 l ve + ω rµ 3 π µ ω C r If we evlute this formul on resonnce we et much simplified formul, (5) /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 9
20 ε Erin ω resonnt enhncement= Q = = µ H ω 3 πω r µ 4lσ dc (5) Let us tke s n exmple the prmeters used to clculte fiure 5, Hence, d =. m σ ω l =. m r =. m Q = R = =. ct = 7. L NM πω r µ 4lσdc O QP (5) = (53) A more detil picture of enhncement s function of frequency is shown in fiure 7. Fiure 7. Enhncement of the enery density of the electric field within the p etween the split rins (see fiures nd 3) for two different vlues of the resistivity of the metl sheet. The correspondin vlues of µ eff re shown in fiure 5. For exmple: em of microwves t 3.4GHz with power flux of 4 wm hs n electric field strenth of the order of 3 Vm in vcuo. If this em were incident on, nd entirely trnsmitted into, our mnetic structure it would enerte field strenth of the order of /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
21 4 Vm in the spce etween the split rins, or of the order of 6 V etween the edes of the two rins: more thn enouh to cuse electricl rekdown in ir! It is evident tht these structures hve considerle potentil for enhncin nonliner phenomen. Furthermore the nonliner medium need only e present in the smll volume within which the enery is concentrted, openin the possiility of usin smll quntities of expensive mteril, nd reducin ny requirements of mechnicl interity tht lrer structure would impose. In pssin we drw n nloy with surfce enhnced Rmn sctterin (SERS), oserved on rouh metllic surfces, typiclly silver surfces. The Rmn sinl from molecules dsored on these surfces my e enhnced y fctors of the order of 6 over tht seen on insultin surfces. The Rmn effect is proportionl to the second power of the electromnetic mode density t the surfce, 3 4 nd it is known tht rouhness cn enhnce the locl mode density y fctors of up to, hence the spectculr Rmn enhncement (see [] for further detils nd references). A very similr locl enhncement tkes plce in our system nd, we expect, cn e exploited in n nloous fshion. In conclusion: we hve shown how to desin structures mde from nonmnetic thin sheets of metl, which respond to microwve rdition s if they hd n effective mnetic permeility. A wide rne of permeilties cn e chieved y vryin the prmeters of the structures. Since the ctive inredient in the structure, the this metl film, comprises very smll frction of the volume, typiclly : 4, the structures my e very liht, nd reinforced with stron insultin mteril to ensure mechnicl strenth, without dversely ffectin their mnetic properties. It is likely tht the structures will e exploited for their ility to concentrte the electromnetic enery in very smll volume, incresin its density y hue fctor, nd retly enhncin ny nonliner effects present. REFERENCES [] J.B. Pendry, A.J. Holden, W.J. Stewrt, I. Youns, Extremely Low Frequency Plsmons in Metllic Meso Structures, Phys. Rev. Lett. vol. 76, pp , 996. [] J.B. Pendry, A.J. Holden, D.J. Roins, nd W.J. Stewrt, Low Frequency Plsmons in Thin Wire Structures, J. Phys. [Condensed Mtter], vol., pp , 998. [3] D.F. Sievenpiper, M.E. Sickmiller nd E. Ylonovitch, 3D Wire mesh photonic crystls Phys. Rev. Lett., vol. 76 pp , 996. [4] D.F. Sievenpiper, E. Ylonovitch, J.N. Winn, S. Fn, P.R. Villeneuve, nd J.D. Jonnopoulos, 3D MetlloDielectric Photonic Crystls with Stron Cpcitive Couplin etween Metllic Islnds, Phys. Rev. Lett., vol. 8, pp 8983, 998. [5] J.B. Pendry Clcultin Photonic Bnd Structure J. Phys. [Condensed Mtter], vol. 8 pp 858, 996. [6] D. Pines nd D. Bohm, A Collective Description of Electron Interctions: II Collective vs Individul Prticle Aspects of the Interctions, Phys. Rev. vol. 85 pp , 95. [7] D. Bohm nd D. Pines, A Collective Description of Electron Interctions: III Coulom Interctions in Deenerte Electron Gs, Phys. Rev. vol. 9. pp 6965, 953. [8] E. Ylonovitch, Inhiited Spontneous Emission in Solid Stte Physics nd Electronics, Phys. Rev. Lett., vol. 58, pp 596, 987. [9] E. Ylonovitch, Photonic Bnd Gp Crystls, J. Phys.: [Condensed Mtter], vol. 5, pp , 993. [] S. John, Stron Loclistion of Photons in Certin Disordered Lttices, Phys. Rev. Lett., vol. 58, pp , 987. [] F.J. Grci Vidl nd J.B. Pendry, Collective Theory for Surfce Enhnced Rmn Sctterin, Phys. Rev. Lett., vol. 77, pp 6366, 996. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
Week 11  Inductance
Week  Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationAn OffCenter Coaxial Cable
1 Problem An OffCenter Coxil Cble Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Nov. 21, 1999 A coxil trnsmission line hs inner conductor of rdius nd outer conductor
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationr 2 F ds W = r 1 qe ds = q
Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationSolar and Lunar Tides
Solr nd Lunr Tides evised, Corrected, Simplified Copyriht Steve Olh, Irvine, CA, 009 solh@cox.net Keywords: Tide, Lunr Tide, Solr Tide, Tidl Force Abstrct Solr nd lunr tides were prt of life since there
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This printout should hve 22 questions, check tht it is complete. Multiplechoice questions my continue on the next column or pge: find ll choices efore mking your
More informationVersion 001 CIRCUITS holland (1290) 1
Version CRCUTS hollnd (9) This printout should hve questions Multiplechoice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire
More information4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A
Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More information, and the number of electrons is 19. e e 1.60 10 C. The negatively charged electrons move in the direction opposite to the conventional current flow.
Prolem 1. f current of 80.0 ma exists in metl wire, how mny electrons flow pst given cross section of the wire in 10.0 min? Sketch the directions of the current nd the electrons motion. Solution: The chrge
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationAnswer, Key Homework 4 David McIntyre Mar 25,
Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his printout should hve 18 questions. Multiplechoice questions my continue on the next column or pe find ll choices before mkin your selection.
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationSirindhorn International Institute of Technology Thammasat University at Rangsit
Sirindhorn Interntionl Institute of Technology Thmmst University t Rngsit School of Informtion, Computer nd Communiction Technology COURSE : ECS 204 Bsic Electricl Engineering L INSTRUCTOR : Asst. Prof.
More informationHomework 10. Problems: 19.29, 19.63, 20.9, 20.68
Homework 0 Prolems: 9.29, 9.63, 20.9, 20.68 Prolem 9.29 An utomoile tire is inlted with ir originlly t 0 º nd norml tmospheric pressure. During the process, the ir is compressed to 28% o its originl volume
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More information** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand
Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationIn the following there are presented four different kinds of simulation games for a given Büchi automaton A = :
Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile
More information14.2. The Mean Value and the RootMeanSquare Value. Introduction. Prerequisites. Learning Outcomes
he Men Vlue nd the RootMenSqure Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationChapter G  Problems
Chpter G  Problems Blinn College  Physics 2426  Terry Honn Problem G.1 A plne flies horizonlly t speed of 280 mês in position where the erth's mgnetic field hs mgnitude 6.0µ105 T nd is directed t n
More informationTheories of light and Interference S BALASUBRAMANYA SGL IN PHYSICS SARVODAYA PU COLLEGE, TUMKUR Important paints
Theories of light nd nterference S BALASUBRAMANYA SGL N PHYSCS SARVODAYA PU COLLEGE, TUMKUR mportnt pints Theories of Light Newton s Corpusculr theory (1675) Christin Huygen s Wve theory (1678) Mxwell
More informationContent Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem
Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationElectric Circuits. Simple Electric Cell. Electric Current
Electric Circuits Count Alessndro olt (74587) Georg Simon Ohm (787854) Chrles Augustin de Coulomb (736 806) André Mrie AMPÈRE (775836) Crbon Electrode () Simple Electric Cell wire Zn Zn Zn Zn Sulfuric
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationSOLUTIONS TO CONCEPTS CHAPTER 5
1. m k S 10m Let, ccelertion, Initil velocity u 0. S ut + 1/ t 10 ½ ( ) 10 5 m/s orce: m 5 10N (ns) 40000. u 40 km/hr 11.11 m/s. 3600 m 000 k ; v 0 ; s 4m v u ccelertion s SOLUIONS O CONCEPS CHPE 5 0 11.11
More informationVector differentiation. Chapters 6, 7
Chpter 2 Vectors Courtesy NASA/JPLCltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higherdimensionl counterprts
More informationPHYS1231 Higher Physics 1B Solutions Tutorial 2
PHYS3 Higher Physics Solutions Tutoril sic info: lthough the term voltge is use every y, in physics it is mesure of firly bstrct quntity clle Electric Potentil. It s importnt to istinguish electric potentil
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationTheory of Forces. Forces and Motion
his eek extbook  Red Chpter 4, 5 Competent roblem Solver  Chpter 4 relb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics  Everything
More information1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?
Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationIncreasing Q of Waveguide PulseCompression Cavities
Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide PulseCompression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque
More informationDispersion in Coaxial Cables
Dispersion in Coxil Cbles Steve Ellingson June 1, 2008 Contents 1 Summry 2 2 Theory 2 3 Comprison to Welch s Result 4 4 Findings for RG58 t LWA Frequencies 5 Brdley Dept. of Electricl & Computer Engineering,
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationTHE PARAMETERS OF TRAPS IN KFELDSPARS AND THE TL BLEACHING EFFICIENCY
GEOCHRONOMETRIA Vol. 2, pp 152, 21 Journl on Methods nd Applictions of Asolute Chronology THE PARAMETERS OF TRAPS IN KFELDSPARS AND THE TL BLEACHING EFFICIENCY ALICJA CHRUŒCIÑSKA 1, HUBERT L. OCZKOWSKI
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationPHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS
PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationStrong acids and bases
Monoprotic AcidBse Equiliri (CH ) ϒ Chpter monoprotic cids A monoprotic cid cn donte one proton. This chpter includes uffers; wy to fi the ph. ϒ Chpter 11 polyprotic cids A polyprotic cid cn donte multiple
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationSymmetry in crystals National Workshop on Crystal Structure Determination Using Powder XRD
Symmetry in crystls Ntionl Workshop on Crystl Structure Determintion Using Powder XRD Muhmmd Sbieh Anwr School of Science nd Engineering Lhore University of Mngement & Sciences (LUMS) Pkistn. (Dted: August
More informationPhysics 2102 Lecture 2. Physics 2102
Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields ChrlesAugustin de Coulomb (17361806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other
More information4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.
4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.
More informationPlotting and Graphing
Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More informationm, where m = m 1 + m m n.
Lecture 7 : Moments nd Centers of Mss If we hve msses m, m 2,..., m n t points x, x 2,..., x n long the xxis, the moment of the system round the origin is M 0 = m x + m 2 x 2 + + m n x n. The center of
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More informationSolving Linear Equations  Formulas
1. Solving Liner Equtions  Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationLECTURE #05. Learning Objectives. How does atomic packing factor change with different atom types? How do you calculate the density of a material?
LECTURE #05 Chpter : Pcking Densities nd Coordintion Lerning Objectives es How does tomic pcking fctor chnge with different tom types? How do you clculte the density of mteril? 2 Relevnt Reding for this
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationA5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s R 2 s(t).
4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The
More information