Electricity & Magnetism Lecture 5: Electric Potential Energy

Transcription

1 Electricity & Magnetism Lecture 5: Electric Potential Energy Toay... Ø Ø Electric Poten1al Energy Unit 21 session Gravita1onal an Electrical PE Electricity & Magne/sm Lecture 5, Slie 1

2 Stuff you aske about: Ø Ø recap all the formulas so far an how they can be use in ifficult ques/ons please. An o so perhaps every 5 lectures (if we have /me) because a lot are popping up. I'll buy a ozen onuts for the class if nee be. In the first prelecture ques/on: where i gravity come from? Isn't gravity perpenicular to the mo/on, thus has no effect. Ø An on't we have a 10minute break at 1.20? My scheule says so ;( There s now a irectory of Main Points You can take a break, if you want. :) Electricity & Magne/sm Lecture 5, Slie 2

3 CheckPoint Result: EPE of Point Charge A charge of +Q is fixe in space. A secon charge of +q was first place at a istance r 1 away from +Q. Then it was move along a straight line to a new posi/on at a istance R away from its star/ng posi/on. The final loca/on of +q is at a istance r 2 from +Q. What is the change in the poten/al energy of charge +q uring this process? A. kqq/r B. kqqr/r 1 2 C. kqqr/r 2 2 D. kqq((1/r 2 )-(1/r 1 )) E. kqq((1/r 1 )-(1/r 2 )) kqq/r is the poten1al at a point so the ifference in these two poten1als will be foun by oing kqq((1/r2)-(1/r1)) Since the par1cle is move away from the fixe charge, the poten1al energy must increase. The part 1/r1-1/r2 woul yiel a posi1ve answer because 1/r1>1/r2 Electricity & Magne/sm Lecture 5, Slie 12

4 CheckPoint Result: EPE of Point Charge A charge of +Q is fixe in space. A secon charge of +q was first place at a istance r 1 away from +Q. Then it was move along a straight line to a new posi/on at a istance R away from its star/ng posi/on. The final loca/on of +q is at a istance r 2 from +Q. What is the change in the poten/al energy of charge +q uring this process? A. kqq/r B. kqqr/r 1 2 C. kqqr/r 2 2 D. kqq((1/r 2 )-(1/r 1 )) E. kqq((1/r 1 )-(1/r 2 )) Note: +q moves AWAY from +Q. Its Poten1al energy MUST DECREASE ΔU < 0 Electricity & Magne/sm Lecture 5, Slie 13

5 W = R~r 2 ~r 1 ~F ~r Recall from Mechanics: F r W > 0 Object spees up ( ΔK > 0 ) or F r F r W < 0 Object slows own ( ΔK < 0 ) F r W = 0 Constant spee ( ΔK = 0 ) Electricity & Magne/sm Lecture 5, Slie 3

6 Dot Prouct (review) If you know A an B in rectangular coorinates, then you can fin the angle between them.

7 Potential Energy If gravity oes nega1ve work, poten1al energy increases! Same iea for Coulomb force if Coulomb force oes nega1ve work, poten1al energy increases. + + F + + Δx Coulomb force oes nega1ve work Poten1al energy increases Electricity & Magne/sm Lecture 5, Slie 4

8 CheckPoint: Motion of Point Charge Electric Fiel A charge is release from rest in a region of electric fiel. The charge will start to move A) In a irec1on that makes its poten1al energy increase. B) In a irec1on that makes its poten1al energy ecrease. C) Along a path of constant poten1al energy. It will move in the same irec1on as F F Work one by force is posi1ve Δx ΔU = Work is nega1ve Nature wants things to move in such a way that PE ecreases Electricity & Magne/sm Lecture 5, Slie 5

9 Clicker Question You hol a posi1vely charge ball an walk ue west in a region that contains an electric fiel irecte ue east. East F E F H r West W H is the work one by the han on the ball W E is the work one by the electric fiel on the ball Which of the following statements is true: A) W H > 0 an W E > 0 B) W H > 0 an W E < 0 C) W H < 0 an W E < 0 D) W H < 0 an W E > 0 Electricity & Magne/sm Lecture 5, Slie 6

10 Clicker Question Conserva/ve force: ΔU = W E Not a conserva/ve force. Does not have any ΔU. E F E F H r B) W H > 0 an W E < 0 Is ΔU posi1ve or nega1ve? A) Posi1ve B) Nega1ve Electricity & Magne/sm Lecture 5, Slie 7

11 Checkpoint Masses M 1 an M 2 are ini1ally separate by a istance R a. Mass M 2 is now move further away from mass M 1 such that their final separa1on is R b. Which of the following statements best escribes the work W ab one by the force of gravity on M 2 as it moves from R a to R b? A) W ab > 0 B) W ab = 0 C) W ab < 0

12 Checkpoint ~r ~F g ~F g ~r > 0 ~F g ~r = 0 ~F g ~r < 0 A) W ab > 0 B) W ab = 0 C) W ab < 0

13 Example: Two Point Charges Calculate the change in poten/al energy for two point charges originally very far apart move to a separa/on of q 1 q 2 Charge par/cles with the same sign have an increase in poten/al energy when brought closer together. ε is another epsilon For point charges oeen choose r = infinity as zero poten/al energy. Electricity & Magne/sm Lecture 5, Slie 8

14 Clicker Question Case A Case B 2 In case A two nega/ve charges which are equal in magnitue are separate by a istance. In case B the same charges are separate by a istance 2. Which configura/on has the highest poten/al energy? A) Case A B) Case B Electricity & Magne/sm Lecture 5, Slie 9

15 Clicker Question Discussion As usual, choose U = 0 to be at infinity: Case A Case B U(r) 2 U A > U B U() U(2) 0 r Electricity & Magne/sm Lecture 5, Slie 10

16 Potential Energy of Many Charges Two charges are separate by a istance. What is the change in poten/al energy when a thir charge q is brought from far away to a istance from the original two charges? Q 2 (superposi/on) Q 1 q Electricity & Magne/sm Lecture 5, Slie 14

17 Potential Energy of Many Charges What is the total energy require to bring in three ien/cal charges, from infinitely far away to the points on an equilateral triangle shown. A) 0 B) Q C) D) E) Q Work to bring in first charge: W 1 = 0 Work to bring in secon charge : Q Work to bring in thir charge : Electricity & Magne/sm Lecture 5, Slie 15

18 Potential Energy of Many Charges Suppose one of the charges is nega/ve. Now what is the total energy require to bring the three charges in infinitely far away? Q 2 A) 0 B) C) D) E) 1 Q Q Work to bring in first charge: W 1 = 0 Work to bring in secon charge : Work to bring in thir charge : Electricity & Magne/sm Lecture 5, Slie 16

19 CheckPoint: EPE of a System of Point Charges 1 Two charges which are equal in magnitue, but opposite in sign, are place at equal istances from point A as shown. If a thir charge is ae to the system an place at point A, how oes the electric poten/al energy of the charge collec/on change? A. Poten/al energy increases B. Poten/al energy ecreases C. Poten/al energy oes not change D. The answer epens on the sign of the thir charge If the new charge has a poten1al energy cause by charge 1 equal to charge 2, when we sum them up, it will a up to 0, resul1ng in no change in total poten1al energy. thir charge will a kq 1 q 3 / an kq 2 q 3 / to the total U, unless q 3 is zero, it will efinitely change the total poten1al energy of the system. Electricity & Magne/sm Lecture 5, Slie 17

20 CheckPoint: EPE of a System of Point Charges 2 Two point charges are separate by some istance as shown. The charge of the first is posi/ve. The charge of the secon is nega/ve an its magnitue is twice as large as the first. Is it possible fin a place to bring a thir charge in from infinity without changing the total poten/al energy of the system? A. YES, as long as the thir charge is posi/ve B. YES, as long as the thir charge is nega/ve C. YES, no maker what the sign of the thir charge D. NO HOW? LET S DO THE CALCULATION! It is possible to bring a thir char in from infinity without changing the total poten1al energy of the system if this thir char is place a istance twice as far way from the nega1ve charge than from the posi1ve charge (an q1 an q2 must be equal in magnitue?). The potential energies the thir charge will contribute to the system have opposite signs but NOT equal magnitues. So the net potential energy will not equal 0. Electricity & Magne/sm Lecture 5, Slie 18

21 Example A posi/ve charge q is place at x = 0 an a nega/ve charge 2q is place at x =. At how many ifferent places along the x axis coul another posi/ve charge be place without changing the total poten/al energy of the system? A) 0 B) 1 C) 2 D) 3 Q X = 0 2Q X = x Electricity & Magne/sm Lecture 5, Slie 19

22 Example At which two places can a posi/ve charge be place without changing the total poten/al energy of the system? Q 2Q A X = 0 B C X = D x A) A & B B) A & C C) B & C D) B & D E) A & D Let s calculate the posi1ons of A an B Electricity & Magne/sm Lecture 5, Slie 20

23 Lets work out where A is r A Q X = 0 2Q X = x Set ΔU = 0 Makes Sense! Q is twice as far from 2q as it is from +q Electricity & Magne/sm Lecture 5, Slie 21

24 Lets work out where B is r r Q X = 0 B 2Q X = x Seeng ΔU = 0 Makes Sense! Q is twice as far from 2q as it is from +q Electricity & Magne/sm Lecture 5, Slie 22

25 What about D? Can you prove that is not possible to put another charge at posi/on D without changing U? r Q X = 0 2Q X = D x 1 r+ = 2 r (r must be posi/ve)

26 Summary For a pair of charges: r Just evaluate Q 1 Q 2 (We usually choose U = 0 to be where the charges are far apart) For a collec1on of charges: Sum up for all pairs Electricity & Magne/sm Lecture 5, Slie 23