Magnetism from Conductors, and Enhanced Non-Linear Phenomena

Size: px
Start display at page:

Download "Magnetism from Conductors, and Enhanced Non-Linear Phenomena"

Transcription

1 Mnetism from Conductors, nd Enhnced Non-Liner Phenomen JB Pendry, AJ Holden, DJ Roins, nd WJ Stewrt Astrct - We show tht microstructures uilt from non-mnetic 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 non-liner 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, non-linerity, photonic crystls JB Pendry is with The Blckett Lortory, Imperil Collee, London, SW7 BZ, UK. AJ Holden, DJ Roins, nd WJ Stewrt re with GEC-Mrconi 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 plsm-like 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 non-linerity, 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 B-field 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 H-field 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 two-fold 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 rod-nd 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 re-stckin 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 NON-LINEAR 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 non-liner 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 non-linerity, it does so in mnner tht is very economicl with the mteril: less tht % of the structure need e filled with the non-liner 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 non-liner 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 non-mnetic 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 non-liner 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 Metllo-Dielectric Photonic Crystls with Stron Cpcitive Couplin etween Metllic Islnds, Phys. Rev. Lett., vol. 8, pp 89-83, 998. [5] J.B. Pendry Clcultin Photonic Bnd Structure J. Phys. [Condensed Mtter], vol. 8 pp 85-8, 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 69-65, 953. [8] E. Ylonovitch, Inhiited Spontneous Emission in Solid Stte Physics nd Electronics, Phys. Rev. Lett., vol. 58, pp 59-6, 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 63-66, 996. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe

Week 11 - Inductance

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 information

EQUATIONS OF LINES AND PLANES

EQUATIONS 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 point-direction nd twopoint

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire 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 information

Reasoning to Solve Equations and Inequalities

Reasoning 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 information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix 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 information

COMPONENTS: COMBINED LOADING

COMPONENTS: 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 information

Answer, Key Homework 10 David McIntyre 1

Answer, Key Homework 10 David McIntyre 1 Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your

More information

Experiment 6: Friction

Experiment 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 information

Version 001 CIRCUITS holland (1290) 1

Version 001 CIRCUITS holland (1290) 1 Version CRCUTS hollnd (9) This print-out should hve questions Multiple-choice 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 information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial 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 information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 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 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.

, 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 information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. 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 so-clled becuse when the sclr product of two vectors

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. 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 information

Factoring Trinomials of the Form. x 2 b x c. Example 1 Factoring Trinomials. The product of 4 and 2 is 8. The sum of 3 and 2 is 5.

Factoring Trinomials of the Form. x 2 b x c. Example 1 Factoring Trinomials. The product of 4 and 2 is 8. The sum of 3 and 2 is 5. Section P.6 Fctoring Trinomils 6 P.6 Fctoring Trinomils Wht you should lern: Fctor trinomils of the form 2 c Fctor trinomils of the form 2 c Fctor trinomils y grouping Fctor perfect squre trinomils Select

More information

Answer, Key Homework 4 David McIntyre Mar 25,

Answer, Key Homework 4 David McIntyre Mar 25, Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his print-out should hve 18 questions. Multiple-choice questions my continue on the next column or pe find ll choices before mkin your selection.

More information

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand

** 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 information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 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 information

Section 5-4 Trigonometric Functions

Section 5-4 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 information

Square Roots Teacher Notes

Square 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 information

Treatment 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.

Treatment 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 t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

PHY 140A: Solid State Physics. Solution to Homework #2

PHY 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 information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 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. nm-wide 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

. 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 information

6.2 Volumes of Revolution: The Disk Method

6.2 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 so-clled volumes of

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs 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 information

Lecture 5. Inner Product

Lecture 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 information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use 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 information

SOLUTIONS TO CONCEPTS CHAPTER 5

SOLUTIONS 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 information

Vector differentiation. Chapters 6, 7

Vector differentiation. Chapters 6, 7 Chpter 2 Vectors Courtesy NASA/JPL-Cltech 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 higher-dimensionl counterprts

More information

Pure C4. Revision Notes

Pure 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 information

Vectors 2. 1. Recap of vectors

Vectors 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 information

5.2. 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. 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 rel-vlued

More information

Factoring Polynomials

Factoring 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 information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.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 information

THE PARAMETERS OF TRAPS IN K-FELDSPARS AND THE TL BLEACHING EFFICIENCY

THE PARAMETERS OF TRAPS IN K-FELDSPARS AND THE TL BLEACHING EFFICIENCY GEOCHRONOMETRIA Vol. 2, pp 15-2, 21 Journl on Methods nd Applictions of Asolute Chronology THE PARAMETERS OF TRAPS IN K-FELDSPARS AND THE TL BLEACHING EFFICIENCY ALICJA CHRUŒCIÑSKA 1, HUBERT L. OCZKOWSKI

More information

Operations with Polynomials

Operations 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 information

1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?

1. 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 information

Increasing Q of Waveguide Pulse-Compression Cavities

Increasing Q of Waveguide Pulse-Compression Cavities Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide Pulse-Compression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque

More information

Dispersion in Coaxial Cables

Dispersion 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 information

Homework 3 Solutions

Homework 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 information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or 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 information

Regular Sets and Expressions

Regular 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 information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up 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 information

Rotating DC Motors Part II

Rotating 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 information

Helicopter Theme and Variations

Helicopter 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 information

Strong acids and bases

Strong acids and bases Monoprotic Acid-Bse 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 information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.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 information

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

PHY 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 information

4. 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. 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 information

AREA OF A SURFACE OF REVOLUTION

AREA 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 information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 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 information

Physics 2102 Lecture 2. Physics 2102

Physics 2102 Lecture 2. Physics 2102 Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields Chrles-Augustin de Coulomb (1736-1806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 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 information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR 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 information

Plotting and Graphing

Plotting 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 information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 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 information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 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 information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. 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 information

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00

Two 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 information

Math 135 Circles and Completing the Square Examples

Math 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 information

Applications to Physics and Engineering

Applications 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 information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL 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 information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL 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 information

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.

addition, 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 information

Lesson 13 Inductance, Magnetic energy /force /torque

Lesson 13 Inductance, Magnetic energy /force /torque Lesson 3 nductnce, Mgnetic energy /force /torque 楊 尚 達 Shng-D Yng nstitute of Photonics Technologies Deprtment of Electricl Engineering Ntionl Tsing Hu Uniersity, Tiwn Outline nductnce Mgnetic energy Mgnetic

More information

LECTURE #05. Learning Objectives. How does atomic packing factor change with different atom types? How do you calculate the density of a material?

LECTURE #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 information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. 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 information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 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 information

The Acoustic Design of Soundproofing Doors and Windows

The Acoustic Design of Soundproofing Doors and Windows 3 The Open Acoustics Journl, 1, 3, 3-37 The Acoustic Design of Soundproofing Doors nd Windows Open Access Nishimur Yuy,1, Nguyen Huy Qung, Nishimur Sohei 1, Nishimur Tsuyoshi 3 nd Yno Tkshi 1 Kummoto Ntionl

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes. LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

NQF Level: 2 US No: 7480

NQF Level: 2 US No: 7480 NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................

More information

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility

More information

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING Sung Joon Kim*, Dong-Chul Che Kore Aerospce Reserch Institute, 45 Eoeun-Dong, Youseong-Gu, Dejeon, 35-333, Kore Phone : 82-42-86-231 FAX

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Project 6 Aircraft static stability and control

Project 6 Aircraft static stability and control Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The

More information

Simulation of operation modes of isochronous cyclotron by a new interative method

Simulation of operation modes of isochronous cyclotron by a new interative method NUKLEONIKA 27;52(1):29 34 ORIGINAL PAPER Simultion of opertion modes of isochronous cyclotron y new intertive method Ryszrd Trszkiewicz, Mrek Tlch, Jcek Sulikowski, Henryk Doruch, Tdeusz Norys, Artur Srok,

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

Morgan Stanley Ad Hoc Reporting Guide

Morgan Stanley Ad Hoc Reporting Guide spphire user guide Ferury 2015 Morgn Stnley Ad Hoc Reporting Guide An Overview For Spphire Users 1 Introduction The Ad Hoc Reporting tool is ville for your reporting needs outside of the Spphire stndrd

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

Section A-4 Rational Expressions: Basic Operations

Section A-4 Rational Expressions: Basic Operations A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

More information

Straight pipe model. Orifice model. * Please refer to Page 3~7 Reference Material for the theoretical formulas used here. Qa/Qw=(ηw/ηa) (P1+P2)/(2 P2)

Straight pipe model. Orifice model. * Please refer to Page 3~7 Reference Material for the theoretical formulas used here. Qa/Qw=(ηw/ηa) (P1+P2)/(2 P2) Air lek test equivlent to IX7nd IX8 In order to perform quntittive tests Fig. shows the reltionship between ir lek mount nd wter lek mount. Wter lek mount cn be converted into ir lek mount. By performing

More information

Solenoid Operated Proportional Directional Control Valve (with Pressure Compensation, Multiple Valve Series)

Solenoid Operated Proportional Directional Control Valve (with Pressure Compensation, Multiple Valve Series) Solenoid Operted Proportionl Directionl Control Vlve (with Pressure Compenstion, Multiple Vlve Series) Hydrulic circuit (Exmple) v Fetures hese stcking type control vlves show pressure compensted type

More information

Chapter Outline How do atoms arrange themselves to form solids? Types of Solids

Chapter Outline How do atoms arrange themselves to form solids? Types of Solids Chpter Outline How do toms rrnge themselves to form solids? Fundmentl concepts nd lnguge Unit cells Crystl structures Fce-centered cubic Body-centered cubic Hexgonl close-pcked Close pcked crystl structures

More information