PhyzExamples: Advanced Electrostatics
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1 PyzExaples: Avance Electrostatics Pysical Quantities Sybols Units Brief Definitions Carge or Q coulob [KOO lo]: C A caracteristic of certain funaental particles. Eleentary Carge e C Te uantity of carge carrie by protons an electrons. Electric iel E newton per coulob: N/C or volt per eter: V/ Te electric force experience by eac unit of carge in a particular location. Typically, te unit of carge is te coulob. Electric Potential Energy PE joule: J Energy of position arising fro electric forces acting on electric carge. Electric Potential V joule per coulob: J/C or volt: V Te electric potential energy el by eac unit of carge in a particular location. Typically, te unit of carge is te coulob. Capacitance C coulob per volt: C/V or fara: Te uantity of carge el by eiter of two parallel plates for eac unit of electric potential ifference between te two plates. Coulob Constant k N 2 /C 2. Perittivity of ree Space ε C 2 /N 2 Masses Electron: kg Proton: kg Neutron: kg Euations E / electric fiel electric force / carge E kq/r 2 electric fiel near a point or sperical carge coulob constant carge on point or spere / suare of istance fro point or center of spere E 4πkQ/A unifor electric fiel between plates 4 π coulob constant plate carge / plate area PE E electric potential energy carge of boy in an electric fiel electric fiel istance tat carge is ove troug electric fiel V PE/ electric potential electric potential energy / carge V kq/r electric potential near a sperical carge coulob constant carge / istance fro sperical carge V E electric potential electric fiel istance between enpoints of te fiel C Q/V capacitance te carge on eiter of two plates / electric potential ifference between te plates C ε 0 A/ capacitance te perittivity of free space area of one plate / istance between te plates PE 1 /2QV energy store in a capacitor alf te carge separate potential across te plates PE 1 /2CV 2 energy store in a capacitor alf te capacitance suare of te potential across te plates PE Q 2 /2C energy store in a capacitor suare of te carge separate / two ties te capacitance Soot Operations Exaples 1. At a istance of 7.4 fro te center of a sperical carge tere is an electric fiel of 25kN/C. Wat is te carge on te spere? 1. R7.4 E25,000N/C Q? EkQ/R 2 QER 2 /k Q(25,000N/C)(7.4) 2 / 9x10 9 N 2 /C 2 Q C 152µC 2. Wat is te raius of eac of two circular plates wit opposite carges of 200nC wit a 100,000V/ electric fiel between te? 2. Q200x10 9 C E100,000V/ E4πkQ/A Aπr 2 E4πkQ/πr 2 r (4kQ/E) r [(4)(9x10 9 N 2 /C 2 )(200x10 9 C) / 100,000V/] r /3/02 3:01 PM
2 3. Wat is te strengt of a unifor electric fiel in wic 1.3J of energy are reuire to ove an object wit 8.6C of carge a istance of 34? 3. PEW1.3J 8.6x10 3 C PEE EPE/ E 1.3J / 8.6x10 3 C E 4400J/C 4400N/C 5. How far fro te center of a sperical carge of 1.0C woul te electric potential be 9.0V? 5. Q1.0C V9.0V R? VkQ/R RkQ/V R 9x10 9 N 2 /C 2 1.0C / 9.0V R 1.0x10 9 (>620,000i) 7. Wat is te capacitance of two parallel plates if eac ols a carge of 12C wen tere is an electric potential ifference of 3.0V between te? 7. Q12x10 3 C V3.0V C? C Q/V C 12x10 3 C/3.0V C 4.0x10 3 C/V Wat is te separation between two parallel plates if tey eac ave an area of an ave a capacitance of 7300p? 9. A0.472 C7300x10 12? C ε 0 A/ ε 0 A/C 8.85x10 12 C 2 /N / 7300x Wat is te capacitance of a capacitor tat stores 70J of energy wen carge wit a 6.0V battery? 11. PE70x103 V6.0V C? PE CV 2 /2 C 2PE/V 2 C 2(70x10 3 J)/(6.0V) 2 C µ 4. Wat is te electric potential at a point in space in wic an object wit a 2.5µC carge as 4.7J of potential energy? x10 6 C PE4.7J V? VPE/ V 4.7J / 2.5x10 6 C V 1,900,00V 1.9MV 6. Two parallel carge plates ave a potential of 7500V an a unifor electric fiel of 125,000V/ between te. How far are te plates fro eac oter? 6. V7500V E125,000V/ VE V/E 7500V/125,000V/ c 8. How uc carge can be el on eac plate of a 25µ capacitor carge to 120V? 8. C25x10 6 C V120V Q? C Q/V Q CV Q 25x V Q 0.003V 3.0C 10. How uc energy is store in a capacitor if it ols 3.6C of carge separate at a potential of 1.4V? 10. Q3.6C V1.4V PE? PE QV/2 PE 3.6C 1.4V / 2 PE 2.5J 12. How uc carge is separate in a 1200µ capacitor if it stores 6.4J of energy? 12. C1200x J PE Q 2 /2C Q (2C PE) Q (2 1200x J) Q 0.12C
3 PyzGuie: IELDS a sie-by-sie coparison of gravitational an electric fiels A fiel is a tree-iensional escription of a certain region of space. A particular type of fiel provies a escription of ow tat particular uantity varies trougout space. iels o not consist of fiel substance, tey are not anyting aterial, rater tey are eans for escribing a istortion in space: a fiel is a tool uc like a vector is a tool. orce fiels inicate ow uc force acts on any particle susceptible to tat fiel. Since force is a vector uantity, a force fiel ust represent te irection of force as well as te uantity of force. GRAVITY gravitational fiel gravitational force ass electric fiel electric force carge I. THE IELD AROUND A SPHERICAL MASS Gravitational force exists between any two objects wit ass. If a sall (test) ass is place in te vicinity of te large (fiel-creating) ass M, a gravitational force will act on te test ass. If a test ass of 3 is place in te sae place, te gravitational force will be tree ties as great. Te fiel concept allows a escription of a point in space tat specifies te uantity an irection of gravitational force per unit of ass. I. THE IELD AROUND A SPHERICAL CHARGE Electric force exists between any two objects wit carge. If a sall (test) carge is place in te vicinity of te large (fiel-creating) carge Q, an electric force will act on te test carge. If a test carge of 3 is place in te sae place, te electric force will be tree ties as great. Te fiel concept allows a escription of a point in space tat specifies te uantity an irection of electric force per unit of carge. M 3 Gravity is always an attractive force, so te irection of te fiel is always towar te ass M. 3 Wee! Q 3 - Positive carges experience a force in te irection of te fiel; negative carges (like electrons) experience a force in te opposite irection. -3 Te irection of an electric fiel is efine as te irection a positive carge woul ove in tat fiel Q Quantitatively, to calculate te fiel strengt g (te aount of force per ass: /), we use our unerstaning of universal gravitation. Quantitatively, to calculate te fiel strengt E (te aount of force per carge: /), we use our unerstaning of Coulob s Law. g G M R 2 G M R 2 g G M R 2 k Q E R 2 k Q R 2 E k Q R 2 Te fiel aroun a ass is proportional to te uantity of ass M an inversely proportional to te suare of te istance R between te center of ass of M an te point in space were te fiel is being easure. Te fiel aroun a carge is proportional to te uantity of carge Q an inversely proportional to te suare of te istance R between te center of carge of Q an te point in space were te fiel is being easure. 3/3/02 3:03 PM
4 GRAVITY II. A UNIORM GRAVITATIONAL IELD suc as one near te surface of te eart If, instea of looking at a fiel-generating ass fro a istance, we exaine a sall region of space in te vicinity of te ass, te fiel as a constant value (instea of aving an inversesuare epenence). Again, a larger ass in tis fiel experiences a greater force, but te ratio of force per ass (i.e. te fiel) is constant. II. A UNIORM ELECTRIC IELD suc as one between parallel plates If, instea of looking at a fiel-generating carge fro a istance, we exaine a sall region of space in te vicinity of te carge, te fiel as a constant value (instea of aving an inversesuare epenence). Again, a larger carge in tis fiel experiences a greater force, but te ratio of force per carge (i.e. te fiel) is constant. Q g g - E --- E Q To ouble te force acting on a given particle, one woul ave to ouble te ass of te eart witout increasing te volue of te eart. In oter wors, one woul ave to ouble te ensity of te eart. Doubling te ensity of te eart woul tus ouble te strengt of te gravitational fiel. g g IELD UNITS: Since te gravitational fiel is a escription of gravitational force per unit of ass, te units are units of force ivie by units of ass. In te SI syste, te unit of gravitational fiel strengt is N/kg (newton per kilogra, wic can be siplifie to /s 2 ). Since te electric fiel is a escription of electric force per unit of carge, te units are units of force ivie by units of carge. In te SI syste, te unit of electric fiel strengt is N/C (newton per coulob). To ouble te force acting on a given particle, one woul ave to ouble te carge on te plates witout increasing te area of te plates. In oter wors, one woul ave to ouble te carge ensity of te plates. Doubling te carge ensity of te plates woul tus ouble te strengt of te electric fiel. - E --- E "COOKIE SHEET" CALCULATIONS... Te electric fiel between two carge plates is E 4πkQ/A were k is te electrostatic force constant N 2 /C 2, Q is te carge on te surface on one plate (te positive one), an A is te surface area of one of te plates. NOTICE tat te fiel strengt as no epenence on te istance fro eiter plate: it s unifor between te plates! Q Q
5 PyzGuie: POTENTIAL ENERGY sie-by-sie coparison of gravitational an electrical potential energy Wen a force is exerte on an object over a certain istance, we say tat work as been one. Wen an object is ierse in a force fiel, we can o work on tat object by oving it against te force exerte troug te fiel. Te work we o in suc a case becoes te potential energy of te object. If te object is release, te fiel will o work on it exerting a force on te object in accorance wit te strengt of te fiel an te susceptibility of te object to tat fiel. or instance, a aer roppe fro a eigt of one eter is acte on by gravity. Te force acting on te aer woul be stronger on te eart tan on te oon because te eart s gravitational fiel is stronger tan te oon s. If a penny were also roppe fro 1, it woul not experience as uc force because it as less ass. Mass is wat akes an object susceptible to a gravitational fiel. orce exerte on a ass by te gravitational fiel. GRAVITY Wen lifting, work ust be one to overcoe te gravitational force. Wen roppe, te eart oes work on te ass troug tat sae force. g Wen lifting, work ust be one to overcoe te electric force. Wen roppe, te plates o work on te carge troug tat sae force. orce exerte on a carge by te electric fiel. E - - To lift te ass to a eigt, a force eual to te gravitational force ust be exerte. Te force g is exerte troug te istance. Work one against te fiel becoes te potential energy of te ass. g W g PE g To lift te carge by a istance, a force eual to te electric force ust be exerte. Te force E is exerte troug te istance. Work one against te fiel becoes te potential energy of te carge. E W E PE E - - Wen te ass is release, te gravitational fiel oes work on it, accelerating it towar te groun. Potential energy is transfore into kinetic energy. PE g KE 0 PE 0 KE g Wen te carge is release, te electric fiel oes work on it, accelerating it towar te negative plate. Potential energy is transfore into kinetic energy. PE E KE 0 PE 0 KE E - - 3/3/02 3:02 PM
6 PyzGuie: POTENTIAL a sie-by-sie coparison of gravitational an electrical potential An object wit ass elevate above a reference point on Eart as gravitational potential energy. An object wit carge elevate above an euilibriu point in an electric fiel as electrical potential energy. (Electric euilibriu is attaine, for exaple, wen a positive carge reaces a negative plate, or vice versa.) In te case of electricity, it is useful to know ow uc potential energy per unit of carge is associate wit a point in space. Potential energy per unit carge is calle electrical potential (also calle voltage an electrootive force). Gravitational potential is a less useful concept tat escribes ow uc potential energy per unit of ass is associate wit a point in space. We stuy gravitional potential at tis point only so tat we can copare it to electrical potential. GRAVITY grav. potential gravitational potential energy ass electrical potential electrical potential energy carge Potential in a Unifor Gravitational iel A ass elevate above euilibriu in a gravitational fiel as gravitational potential energy. Te gravitational potential can be calculate fro te PE an te ass of te object. PE GP GP g g < EQUILIBRIUM: PE 0 All asses elevate above euilibriu by a istance ave eual gravitational POTENTIALS, altoug tey ay ave ifferent gravitational potential ENERGIES. PE g GP g/ g 4 PE 4g GP 4g/4 g 10 GP g PE 10g GP 10g/10 g Potential in a Unifor Electric iel A carge elevate above euilibriu in an electric fiel as electric potential energy. Te electric potential can be calculate fro te PE an te carge on te object. PE E V V E < EQUILIBRIUM: PE 0 All carges elevate above euilibriu by a istance ave eual electric POTENTIALS, altoug tey ay ave ifferent electric potential ENERGIES PE E V E/ E PE 4E V 4E/4 E V E PE 10E V 10E/10 E Regarless of ass, all objects ave te sae gravitational potential at a given eigt: GP g. Regarless of carge, all objects ave te sae electrical potential for a given plate separation istance: V E. Since a tiny carge coul traverse te entire istance between two plates, is consiere te entire istance between te plates. 3/3/02 3:02 PM
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