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Electric Potential Energy

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Electric Potential Energy Electricity & Magnetism Lecture 5: Electric Potential Energy Today... Ø Recap of Unit 19 grading Ø This unit’s (21) wri=en Homework Ø Electric PotenDal Energy Ø Unit 21 session Gravitaonal and Electrical PE Electricity & MagneDsm Lecture 5, Slide 1 Stuff you asked about: Ø recap all the formulas so far and how they can be used in difficult quesDons please. And do so perhaps every 5 lectures (if we have Dme) because a lot are popping up. I'll buy a dozen donuts for the class if need be. There’s now a directory of Main Points Ø In the first prelecture quesDon: where did gravity come from? Isn't gravity perpendicular to the moDon, thus has no effect. Ø And don't we have a 10minute break at 1.20? My schedule says so • ;( You can take a break, if you want. :) Electricity & MagneDsm Lecture 5, Slide 2 CheckPoint Result: EPE of Point Charge A charge of +Q is fixed in space. A second charge of +q was first placed at a distance r1 away from +Q. Then it was moved along a straight line to a new posiDon at a distance R away from its starDng posiDon. The final locaon of +q is at a distance r2 from +Q. What is the change in the potenDal energy of charge +q during this process? A. kQq/R 2 B. kQqR/r1 2 C. kQqR/r2 D. kQq((1/r2)-(1/r1)) E. kQq((1/r1)-(1/r2)) “kQq/r is the poten2al at a point so the difference in these two poten2als will be found by doing kQq((1/r2)-(1/r1))” “Since the par2cle is moved away from the fixed charge, the poten2al energy must increase. The part 1/r1-1/r2 would yield a posi2ve answer because 1/r1>1/r2” Electricity & MagneDsm Lecture 5, Slide 12 CheckPoint Result: EPE of Point Charge A charge of +Q is fixed in space. A second charge of +q was first placed at a distance r1 away from +Q. Then it was moved along a straight line to a new posiDon at a distance R away from its starDng posiDon. The final locaon of +q is at a distance r2 from +Q. What is the change in the potenDal energy of charge +q during this process? A. kQq/R 2 B. kQqR/r1 2 C. kQqR/r2 D. kQq((1/r2)-(1/r1)) E. kQq((1/r1)-(1/r2)) Note: +q moves AWAY from +Q. Its Poten2al energy MUST DECREASE ΔU < 0 Electricity & MagneDsm Lecture 5, Slide 13 Recall from Mechanics: ~r2 W = F~ d~r · ~r1 R F dr W > 0 Object speeds up ( ΔK > 0 ) F dr or W 0 F < Object slows down ( ΔK < 0 ) dr F dr W = 0 Constant speed ( ΔK = 0 ) Electricity & MagneDsm Lecture 5, Slide 3 Dot Product (review) If you know A and B in rectangular coordinates, then you can find the angle between them. Potential Energy If gravity does negave work, potenDal energy increases! Same idea for Coulomb force… if Coulomb force does negave work, potenDal energy increases. + + F Δx Coulomb force does negave work + + Poten2al energy increases Electricity & MagneDsm Lecture 5, Slide 4 CheckPoint: Motion of Point Charge Electric Field A charge is released from rest in a region of electric field. The charge will start to move A) In a direc:on that makes its poten:al energy increase. B) In a direc:on that makes its poten:al energy decrease. C) Along a path of constant poten:al energy. It will move in the same direc:on as F F Work done by force is posi:ve Δx ΔU = −Work is negave Nature wants things to move in such a way that PE decreases Electricity & MagneDsm Lecture 5, Slide 5 Clicker Question You hold a posiDvely charged ball and walk due west in a region that contains an electric field directed due east. FE FH East dr West WH is the work done by the hand on the ball WE is the work done by the electric field on the ball Which of the following statements is true: A) WH > 0 and WE > 0 B) WH > 0 and WE < 0 C) WH < 0 and WE < 0 D) WH < 0 and WE > 0 Electricity & MagneDsm Lecture 5, Slide 6 Clicker Question Not a conservave force. Conservave force: ΔU = − WE Does not have any ΔU. FE FH E dr B) WH > 0 and WE < 0 Is ΔU posiDve or negave? A) PosiDve B) Negave Electricity & MagneDsm Lecture 5, Slide 7 Example: Two Point Charges Calculate the change in potenDal energy for two point charges originally very far apart moved to a separaon of “d” d q1 q2 � is another epsilon Charged parDcles with the same sign have an increase in potenDal energy when brought closer together. For point charges open choose r = infinity as “zero” potenDal energy. Electricity & MagneDsm Lecture 5, Slide 8 Clicker Question Case A d Case B 2d In case A two negave charges which are equal in magnitude are separated by a distance d. In case B the same charges are separated by a distance 2d. Which configuraon has the highest potenDal energy? A) Case A B) Case B Electricity & MagneDsm Lecture 5, Slide 9 Clicker Question Discussion As usual, choose U = 0 to be at infinity: Case A d Case B 2d U(r) UA > UB U(d) U(2d) 0 r Electricity & MagneDsm Lecture 5, Slide 10 Potential Energy of Many Charges Two charges are separated by a distance d. What is the change in potenDal energy when a third charge q is brought from far away to a distance d from the original two charges? Q2 d d (superposion) Q1 q d Electricity & MagneDsm Lecture 5, Slide 14 Potential Energy of Many Charges What is the total energy required to bring in three idenDcal charges, from infinitely far away to the points on an equilateral triangle shown. A) 0 Q B) C) d d D) E) Q d Q Work to bring in first charge: W1 = 0 Work to bring in second charge : Work to bring in third charge : Electricity & MagneDsm Lecture 5, Slide 15 Potential Energy of Many Charges Suppose one of the charges is negave. Now what is the total energy required to bring the three charges in infinitely far away? Q 2 A) 0 B) d d C) D) Q d Q E) 1 Work to bring in first charge: W1 = 0 Work to bring in second charge : Work to bring in third charge : Electricity & MagneDsm Lecture 5, Slide 16 CheckPoint: EPE of a System of Point Charges 1 Two charges which are equal in magnitude, but opposite in sign, are placed at equal distances from point A as shown. If a third charge is added to the system and placed at point A, how does the electric potenDal energy of the charge collecDon change? A. PotenDal energy increases B. PotenDal energy decreases C. PotenDal energy does not change D. The answer depends on the sign of the third charge If the new charge has a poten2al energy caused by charge 1 equal to charge 2, when we sum them up, it will add up to 0, resul2ng in no change in total poten2al energy. third charge will add kq1q3/d and kq2q3/d to the total U, unless q3 is zero, it will definitely change the total poten2al energy of the system. Electricity & MagneDsm Lecture 5, Slide 17 CheckPoint: EPE of a System of Point Charges 2 Two point charges are separated by some distance as shown. The charge of the first is posiDve. The charge of the second is negave and its magnitude is twice as large as the first. Is it possible find a place to bring a third charge in from infinity without changing the total potenDal energy of the system? A. YES, as long as the third charge is posiDve B. YES, as long as the third charge is negave C. YES, no maer what the sign of the third charge D. NO HOW? LET’S DO THE CALCULATION! ““As long as the third charge is twice as far from the larger negative charge as it is the smaller positive charge, the total potential energy of the system will be unaffected. “ “The potential energies the third charge will contribute to the system have opposite signs but NOT equal magnitudes. So the net potential energy will not equal 0.” Electricity & MagneDsm Lecture 5, Slide 18 Example A posiDve charge q is placed at x = 0 and a negave charge −2q is placed at x = d. At how many different places along the x axis could another posiDve charge be placed without changing the total potenDal energy of the system? Q −2Q x X = 0 X = d A) 0 B) 1 C) 2 D) 3 Electricity & MagneDsm Lecture 5, Slide 19 Example At which two places can a posiDve charge be placed without changing the total potenDal energy of the system? Q −2Q A x X = 0 B C X = d D A) A & B B) A & C C) B & C Let’s calculate the posiDons of A and B D) B & D E) A & D Electricity & MagneDsm Lecture 5, Slide 20 Lets work out where A is r d Q −2Q A x X = 0 X = d Set ΔU = 0 Makes Sense! Q is twice as far from −2q as it is from +q Electricity & MagneDsm Lecture 5, Slide 21 Lets work out where B is r d − r Q −2Q x X = 0 B X = d Seng ΔU = 0 Makes Sense! Q is twice as far from −2q as it is from +q Electricity & MagneDsm Lecture 5, Slide 22 What about D? Can you prove that is not possible to put another charge at posiDon D without changing U? d r Q −2Q x X = 0 X = d D 1 2 (r must be posiDve) r+d = r Summary For a pair of charges: r Q Just evaluate 1 Q2 (We usually choose U = 0 to be where the charges are far apart) For a collecDon of charges: Sum up for all pairs Electricity & MagneDsm Lecture 5, Slide 23.
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