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Electricity & Magnetism Lecture 7: Conductors and Capacitance

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Electricity & Magnetism Lecture 7: Conductors and Capacitance Electricity & Magnetism Lecture 7: Conductors and Capacitance Today’s Concept: A) Conductors B) Capacitance Today’s Experiment: Baery and Bulb Circuits (we return to capacitance soon) Electricity & MagneAsm Lecture 7, Slide 1 Stuff you asked about: “THey never said wHat capacitance is or wHat its used for” “How do I "cHarge" a capacitance? if I don't put a wire in between, the circuit is not complete. If I put a wire in between, I won't be able to build up a cHarge difference. Also, if I cHarge them as is (with no wire in between), won't they discHarge immediately? Don't understand How this works.” “WHat is the difference between capacitors and baeries, althougH they both stores energy, can we say that capacitors are a kind a baeries or vice versa?” “I'm not looking forward to the midterm.” Below is a list of topics I would like to discuss in explicit detail during the lecture: 1) Capacitors 2) Capacitors 3) Capacitors ... Electricity & MagneAsm Lecture 7, Slide 2 A battery will create a potential difference across a capacitor which is equal to the potential difference in the battery. Capacitance andCapacitors Capacitors vs. Batteries (30.5) ‣ A baery uses an electrocHemical reacAon to separate WecHarges. Δ have beenV between terminals is called “EMF”, using capacitors a lot without defining capacitance or describing how to charge-up these devices. symbol: E or E. ‣ Capacitors store cHarge on its plates. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 4 / 14 Some capacitors 0 Our Comments WE BELIEVE THERE ARE ONLY THREE THINGS YOU NEED TO KNOW TO DO ALL OF HOMEWORK! 1. E = 0 within the material of a conductor: (at stac equilibrium) CHarges move inside a conductor in order to cancel out the fields that would be there in the absence of the conductor. THis principle determines the induced cHarge densiAes on the surfaces of conductors. 2. Gauss’ Law: Q ~ ~ = enclosed E dA · ✏0 If cHarge distribuAons Have sufficient symmetry (spHerical, surfaceI cylindrical, planar), then Gauss’ law can be used to determine the electric field everywHere. b ∆Ua b ~ ∆Va b ⇤ = E~ dl ⇤ ⇥ q − · Za CONCEPTS DETERMINE THE CALCULATION ! Electricity & MagneAsm Lecture 7, Slide 3 Conductors THe Main Points CHarges free to move E = 0 in a conductor Surface = EquipotenAal E at surface perpendicular to surface Electricity & MagneAsm Lecture 7, Slide 4 CheckPoint: Two Spherical Conductors 1 Two spHerical conductors are separated by a large distance. THey eacH carry the same posiAve cHarge Q. Conductor A Has a larger radius than conductor B. Compare the potenAal at the surface of conductor A with the potenAal at the surface of conductor B. “THe potenAal is the same as from a point A. VA > VB cHarge at the center of A or B. So, following B. VA = VB V=kQ/r, A would Have a smaller potenAal “ C. VA < VB Electricity & MagneAsm Lecture 7, Slide 5 CheckPoint: Two Spherical Conductors 2 Two spHerical conductors are separated by a large distance. THey eacH carry the same posiAve cHarge Q. Conductor A Has a larger radius than conductor B. THe two conductors are now connected by a wire. How do the potenAals at the conductor surfaces compare now? “THe potenAals will become equal since the A. VA > VB cHarges will want to go to places of lower B. VA = VB potenAal, unAl it balances out. “ C. VA < VB Electricity & MagneAsm Lecture 7, Slide 6 CheckPoint: Two Spherical Conductors 3 Two spHerical conductors are separated by a large distance. THey eacH carry the same posiAve cHarge Q. Conductor A Has a larger radius than conductor B. WHat Happens to the cHarge on conductor A aer it is connected to conductor B by the wire? “Since B initially has a higher potential, A. QA increases charges move from B to A. “ B. QA decreases C. QA does not cHange Electricity & MagneAsm Lecture 7, Slide 7 CheckPoint: Charged Parallel Plates 1 Two parallel plates of equal area carry equal and opposite cHarge Q0. THe potenAal difference between the two plates is measured to be V0. An uncHarged conducAng plate (the green thing in the picture below) is slipped into the space between the plates without toucHing either one. THe cHarge on the plates is adjusted to a new value Q1 sucH that the potenAal difference between the plates remains the same. Compare Q1 and Q0. o Q1 < Q0 o Q1 = Q0 o Q1 > Q0 THE CAPACITOR QUESTIONS WERE TOUGH!THE PLAN: We’ll work througH the example in the PreLecture and then do the prefligHt quesAons. Electricity & MagneAsm Lecture 7, Slide 8 Capacitance Capacitance is defined for any pair of spaally separated conductors. How do we understand this definiAon ? Consider two conductors, one with excess cHarge = +Q and the other with excess cHarge = −Q +Q d E V −Q THese cHarges create an electric field in the space between them We can integrate the electric field between them to find the potenAal difference between the conductor THis potenAal difference sHould be proporAonal to Q ! • THe rao of Q to the potenAal difference is the capacitance and only depends on the geometry of the conductors Electricity & MagneAsm Lecture 7, Slide 9 Example (done in Prelecture 7) First determine E field produced by charged conductors: +Q y WHat is σ ? d E x −Q A = area of plate Second, integrate E to find the potenAal difference V d V = E~ d~y − · Z0 As promised, V is proporAonal to Q ! C determined by geometry ! Electricity & MagneAsm Lecture 7, Slide 10 Clicker Question Related to CheckPoint +Q0 d IniAal cHarge on capacitor = Q0 −Q0 +Q1 Insert uncHarged conductor d t CHarge on capacitor now = Q1 −Q1 How is Q1 related to Q0 ? Plates not connected to anything A) Q1 < Q0 B) Q1 = Q0 CHARGE CANNOT CHANGE ! C) Q1 > Q0 Electricity & MagneAsm Lecture 7, Slide 11 Where to Start ? +Q1 d t −Q1 WHat is the total cHarge induced on the bo9om surface of the conductor? A) +Q0 B) −Q0 C) 0 D) PosiAve but the magnitude unknown E) Negave but the magnitude unknown Electricity & MagneAsm Lecture 7, Slide 12 Why ? +Q0 E −Q0 E = 0 +Q 0 E −Q0 WHAT DO WE KNOW ? E must be = 0 in conductor ! CHarges inside conductor move to cancel E field from top & boom plates. Electricity & MagneAsm Lecture 7, Slide 13 Calculate V y Now calculate V as a funcAon of distance from V( y) = E~ d~y − · the bo9om conductor. Z0 d +Q0 E t y y -E d E = 0 t 0 V −Q0 WHat is ΔV = V(d)? y A) ΔV = E0d B) ΔV = E0(d − t) THe integral = area under the curve C) ΔV = E0(d + t) Electricity & MagneAsm Lecture 7, Slide 14 CheckPoint Results: Charged Parallel Plates 1 Two parallel plates of equal area carry equal and opposite cHarge Q0. THe potenAal difference between the two plates is measured to be V0. An uncHarged conducAng plate (the green thing in the picture below) is slipped into the space between the plates without toucHing either one. THe cHarge on the plates is adjusted to a new value Q1 sucH that the potenAal difference between the plates remains the same. Compare Q1 and Q0. A. Q1 < Q0 B. Q1 = Q0 C. Q1 > Q0 “THe distance for Q1 is smaller therefore the cHarge must decrease to compensate for the cHange in distance “ “Since the potenAal remains the same, there is no cHange in the cHarge of Q. “ “the field througH the conductor is zero, so it Has constant potenAal. Because of this it must Have greater cHarge so the total V is that same. “ Electricity & MagneAsm Lecture 7, Slide 15 CheckPoint Results: Charged Parallel Plates 2 Two parallel plates of equal area carry equal and opposite cHarge Q0. THe potenAal difference between the two plates is measured to be V0. An uncHarged conducAng plate (the green thing in the picture below) is slipped into the space between the plates without toucHing either one. THe cHarge on the plates is adjusted to a new value Q1 sucH that the potenAal difference between the plates remains the same. Compare the capacitance of the two configuraons in the above problem. A. C1 > C0 B. C1 = C0 C. C1 < C0 “THe distance for Q1 is smaller therefore the cHarge must decrease to compensate for the cHange in distance “ “Since the potenAal remains the same, there is no cHange in the cHarge of Q. “ “the field througH the conductor is zero, so it Has constant potenAal. Because of this it must Have greater cHarge so the total V is that same. “ Electricity & MagneAsm Lecture 7, Slide 16 CheckPoint Results: Charged Parallel Plates 2 Two parallel plates of equal area carry equal and opposite cHarge Q0. THe potenAal difference between the two plates is measured to be V0. An uncHarged conducAng plate (the green thing in the picture below) is slipped into the space between the plates without toucHing either one. THe cHarge on the plates is adjusted to a new value Q1 sucH that the potenAal difference between the plates remains the same. Compare the capacitance of the two configuraons in the above problem. A. C1 > C0 B. C1 = C0 C. C1 < C0 We can determine C from either case same V (prefligHt) same Q (lecture) C depends only on geometry ! E0 = Q0 /ε0A C0 = Q0 /E0d C0 = ε0A /d C1 = Q0 /[E0(d − t)] C1 = ε0A /(d − t) Electricity & MagneAsm Lecture 7, Slide 17 Energy in Capacitors BANG Electricity & MagneAsm Lecture 7, Slide 18 Calculation cross-secon a4 A capacitor is constructed from two conducAng a3 cylindrical sHells of radii a1, a2, a3, and a4 and length a 2 L (L >> ai).
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