Basic Electrohydrodynamics (Electrostatics) of the Floating Water Bridge

Transcription

1 Basc lectohyoynamcs lectostatcs) o the loatng Wate Bge Wate has a emanent electc ole moment ue to ts molecula conguaton To scuss a ole, let us assume two ont chages, one wth chage +q an one wth chage q, seaate by a stance Ths stance s small ntamolecula) comae to the macoscoc stances n a lqu The electc ole moment o ths chage a s then ene by q eq wth the slacement vecto ontng om q to +q On the othe han, the oce on a sngle electc chage s gven by the Loentz law : q eq wth the electc el 3 In a non-unom electc el each electc ole eeences a net electc oce, whch s the sum o the oces actng on the two nvual chages n the ole: q ) q) wth the oston vecto o the negatve chage 4 eq3 g Wate ole 5 an ole n an electc el htt://enwkeaog/wk/lectc_ole_moment htt://enwkeaog/wk/loentz_law 3 htt://enwkeaog/wk/lectc_el 4 Melche JR98) Contnuum lectomechancs Cambge, MA: MIT ess, 98 Coyght Massachusetts Insttute o Technology ISBN: Also avalable onlne om MIT OenCouseWae at htt://ocwmteu 5 htt://enwkeaog/wk/le:wate-elot-tansaent-3d-ballsng Basc lectohyoynamcs lectostatcs) o the loatng Wate Bge - Wosetschläge, 44

2 Basc lectohyoynamcs lectostatcs) o the loatng Wate Bge - Wosetschläge, 44 Develong equaton 3 n the lmt o small ole slacement neglect hghe oe evatves ung sees eanson) we get o the -comonent o the oce,y,z) z y, y,z) q z y, eq4 o n geneal notaton o, eq5 an wth n the numbe o oles e unt volume we eve the Kelvn olasaton oce ensty,kelvn n n Kelvn Kelvn o, eq6 Hee, s the olazaton wthn a volume element Ths olazaton s elate to the electc slacement el 6 D by D eq7 wth = A s V - m - the vacuum emttvty an 7 the elatve emttvty o wate om equaton 7 t becomes event, that a lage ole moment wll cause a lage value o the elatve emttvty We then get om equatons 6 an 7 ) eq8 an ) ) eq9 Wth the vecto ot ouct 7 ) eq An because o the otatonal natue o the el between two lates, we en wth the ollowng equaton o the Kelvn olazaton oce ensty: ) ) Kelvn eq 6 htt://enwkeaog/wk/lectc_slacement_el 7 htt://enwkeaog/wk/vecto_calculus_enttes

3 g Numecal smulaton o the electc el wthn the wate bge usng Comsol 4 sotwae an coesonng mage o the bge Dslaye ae the absolute values o the electc slacement el Goun otental s at nnty, cathoe s at kv let), anoe at 5 kv ght) The glass beakes have a slghtly oune bm Let us now scuss the electc slacement el n the loatng wate bge n gue Ths el calculaton was one wth the AC/DC moule n Comsol 4 multhyscs sotwae, so no lu low s consee - t s the electostatc soluton o a gven geomety o the wate bge Ths geomety was taken om hotogahs o the bge ecoe at 5 kv, mm beake stance an about 3 mm bge amete om ths gue an equaton 7 we get a local electc slacement el stength aoun the beakes eges o 6 to 8-4 Cm - o a local electc el stength between 96 to 9 kvcm - We then have the elaton between oce ensty an essue 8 by oten use n lu mechancs 9, eq o the statc equlbum o the lu no lu moton) we can wte Benoull s equaton o ncomessble lus nclung the lu essue cause by the electc el as gh ) const eq 3 8 htt://enwkeaog/wk/oce 9 In tetbooks on electohyoynamcs these elatons ae geneally eve om the Mawell tenso T, wth the so-calle Mawell essue as the secon tem an k the Konecke-elta So, equaton eas: ) Kelvn Mawell ) htt://enwkeaog/wk/benoull%7s_ncle Basc lectohyoynamcs lectostatcs) o the loatng Wate Bge - Wosetschläge, 44 3

4 wth the lu ensty hee kgm -3 o wate) an h the heght o the level se cause by the hyostatc essue gh an g the acceleaton ue to gavty g =98665 ms - ) In the a above the wate suace thee s essue equlbum n ostons A an C, so that we have gue ) A C eq 4 snce a kgm -3 an,a Only ambent a essue) o ostons B an D gue ) n the wate bge we have B gh ) D eq 5 zeo level se an kv at cathoe) Wth equlbum contons we get h A C B eq 6 eq 7 D ) g eq 8 o the elevaton o the wate suace enoce by the electc el Wth an electc el stength between 96 to 9 kvcm - we calculate a ossble lu suace se o 8 to 44 cm, what s qute a lot Usually we obseve no moe than cm level se between bge an beake levels, so that we can estmate the lu velocty by ang the contbuton o the ynamc essue to equaton 8 v gh ), eq 9 htt://enwkeaog/wk/hyostatc_essue Ths equaton s oten oun n elaton wth the hstocal ellat eement, n whch two electoes ae mmese n a ola lqu wth low conuctvty eg glyceol) When a hgh voltage s ale the lqu se s gven by equaton 8 A thoough scusson o ths eement can be oun n Melche JR98) Contnuum lectomechancs Cambge, MA: MIT ess, 98 Coyght Massachusetts Insttute o Technology ISBN: Also avalable onlne om MIT OenCouseWae at htt://ocwmteu, sec 36 an eg Jones TB ) On the elatonsh o electohoess an electowettng, Langmu 8, A numbe o othe lus eements ae known n non-unom electc els, eg: ohl HA 958) Some eects o nonunom els on electcs, Jounal o Ale hyscs 9 8), 8-88 Basc lectohyoynamcs lectostatcs) o the loatng Wate Bge - Wosetschläge, 44 4

5 g 3 Bge secton enlage om g wth the lu velocty v Assumng h = mm we then get veloctes between 6 an 8 ms - o above el stengths o h = 5 mm we obtan 7 to 9 ms - esectvely Snce the electc el causes a hyostatc essue om both ses a zeo net mass low s ossble as well as a umng moton wth a eoc change tme t) between statc an ynamc essue at each se o the bge vt) ght) eq o ostons an n gue 3 the stength o the electc slacement el s between 3 an 5-4 Cm - coesonng to an electc el stength between 48 to 8 kvcm - Between ostons an we theeoe get a eence n hyostatc essue o g h h ) ), eq esultng n a hyostatc essue equvalent to a cm level se Snce the bge amete smulate n gue 3 s about 3 mm, t becomes event that o sng a small lu volume om the mle o the bge u to oston only mm elevaton) ths lu volume wll see u to a velocty o 5 ms - accong to equaton 9 When umng the lu volume om oston n gue 3 u to oston B n gue wth 3 mm se om an electc el stength o 48 kvcm - to a mamum o 9 kvcm - ),we mght easly en u wth 8 ms - lu velocty, equvalent to the velocty o the ncomng lu Ths leaves us wth the messon that n a sagge bge, ncomng an outgong wate has aomately the same ynamc essue, what makes an oscllaton n thckness lkely Ths oscllaton can be unestoo as bge umng an s oten obseve n the eement So, the lqu s ume nto the bge om both ses, whle the bge ums out the lqu back nto the beakes When the bge s not sagge, but oms a eect cylne, the el lnes wll be aallel to the suace, the solnes eencula to the as, the Kelvn olasaton oce ensty wll be hozontal, all enegy wll be ully knetc, only ynamc essue wll bul u The bge wll um wate though om both ses Basc lectohyoynamcs lectostatcs) o the loatng Wate Bge - Wosetschläge, 44 5