**Unformatted text preview: **Physics 212
Lecture 5
Electric Potential Energy Physics 212 Lecture 5, Slide 1 Recall from physics 211: W = F dr r1
W>0 r2 WTOT = K F dr F dr F dr F dr
9 Object speeds up ( K > 0 ) or W<0 Object slows down ( K < 0 ) W=0 Constant speed ( K = 0 )
Physics 212 Lecture 5, Slide 2 Potential Energy U -W conservative If gravity does negative work, potential energy increases! Same idea for Coulomb force... if Coulomb force does negative work, potential energy increases.
+ + F x
+ + Coulomb force does negative work Potential energy increases
Physics 212 Lecture 5, Slide 3 Analogy to gravity
m Exert force Fext = - Fg over distance h. Fext
m h Fg Work by me = Fext h = - Fgh = mgh = U h F d F d ext r = - g r = U g ( h ) -U g ( 0 ) = U g
h 0 0 Work by me - (Work by gravity) Change in Potential energy A charge is released from rest in a region of electric field. The charge will start to move A) in a direction that makes its potential energy increase B) in a direction that makes its potential energy decrease C) along a path of constant potential energy Checkpoint 4 F
x It will move in the same direction as F Work done by force is positive U = -Work is negative Nature wants things to move in such a way that PE decreases
34 Physics 212 Lecture 5, Slide 5 Example: Charge in External Field
You hold a positively charged ball and walk due west in a region that contains an electric field directed due east. FE FH E dr 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) W < 0 and W > 0 14 Physics 212 Lecture 5, Slide 6 Conservative force: U = - WE Not a conservative force. Does not have any U. FE FH E
B) WH > 0 and WE < 0 Is U positive or negative? A) Positive B) Negative dr 16 Physics 212 Lecture 5, Slide 7 Move charge q2 from r1 r2. Find the change in potential energy. Fel
q2 q1 ^ dr q1 q2 ^ Fel = r12 4 0
q1
r2 Fel
q2 ^ dr r 12 r2 ^ q1 q2 r12 U = U ( r2 ) - U ( r1 ) = - r = - Fel d d 2 r 4 0 r1 r12 r1 From the diagram,
r2 ^ d r12 r = d r so, r2 q1 q2 d r q1 q2 1 q1 q2 1 1 U = - =- - = 2 r = 4 - r k q1 q2 4 0 r1 r 4 0 r2 1 r1 0 1 1 - r 2 r1 Calculate the change in potential energy for two point charges originally very far apart, moved to a separation of d.
d
q1 q2 q1q2 U = - k 2 dr = kq1q2 r12 d 1 1 q1q2 1 q1q2 = - k d = 4 d d 0 Charged particles w/ same sign have an increase in potential energy when brought closer together. For point charges we often choose r = as "zero" potential energy. So the potential energy of this configuration is, U ( d ) = U (
19 d) +U ( ) = U ( d) = k q1 q 2 d
Physics 212 Lecture 5, Slide 9 Example: Getting the signs right
Case A Case B d 2d In case A two negative 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 configuration has the highest potential energy? A) Case A B) Case B 22 Physics 212 Lecture 5, Slide 10 Example: Getting the signs right As usual, choose U = 0 to be at infinity: Case A Case B U(r) q1q2 1 U (r ) = 4 0 r d 2d q2 1 UA = 4 0 d
q2 1 UB = 4 0 2d UA > UB
r r
Physics 212 Lecture 5, Slide 11 U(d) U(2d) 0 0 23 Checkpoint 1 A B C D E U initial = 1 Qq 4 0 r1 U final = 1 Qq 4 0 r2 U U f - U i = Qq 1 1 - 4 0 r2 r1 Note: +q moves AWAY from +Q. Its potential energy MUST DECREASE U < 0
34 Physics 212 Lecture 5, Slide 12 Potential Energy of Many Charges
Two charges are separated by a distance d. What is the change in potential energy when a third charge q is brought from far away to a distance d from the original two charges? Q2 qQ1 1 qQ2 1 U = + 4 0 d 4 0 d
(superposition)
Q1 d d q d 25 Physics 212 Lecture 5, Slide 13 Potential Energy of Many Charges
What is the total energy required to bring in three identical charges, from infinitely far away to the points on an equilateral triangle shown ? Q 1 4 0 d Q2 1 U = 2 4 0 d Q2 1 U = 3 4 0 d Q2 1 U = 6 4 0 d U = 2 Q d d W (by me) = 3 Q2 Wi = 4 0 d Q d
1 Q2 W2 = 4 0 d Q 3 Q2 U = + 40 d Work (by me) to bring in first charge: W1 = 0 Work (by me) to bring in second charge : Work (by me) to bring in third charge :
27 1 Q2 1 Q2 2 Q2 W3 = + = 4 0 d 4 0 d 4 0 d
Physics 212 Lecture 5, Slide 14 Potential Energy of Many Charges
Suppose one of the charges is negative. Now what is the total energy required to bring the three charges in infinitely far away? A) 0
Q2 1 B) U = +1 4 0 d Q2 1 C) U = -1 4 d 0 2 Q 1 D) U = +2 4 d 0 2 Q 1 E) U = -2 4 d 0 3 -Q d d W ( by me ) = 1 Q2 Wi = - 4 0 d 1 Q d Q 2 1 Q2 U = - 40 d Work to bring in first charge: W1 = 0 Work to bring in second charge : 1 Q2 W2 = + 40 d 1 Q2 1 Q2 2 Q2 - =- Work to bring in third charge : W3 = - 4 0 d 4 0 d 4 0 d
29 Physics 212 Lecture 5, Slide 15 Checkpoint 2 A B C D 31 Physics 212 Lecture 5, Slide 16 Checkpoint 3 LET'S DO THE CALCULATION !!
34 Physics 212 Lecture 5, Slide 17 Example: Potential Energy Changes A positive charge q is placed at x=0 and a negative charge 2q is placed at x=d. At how many different places along the x axis could another positive charge be placed without changing the total potential energy of the system? q X=0 -2q X=d x A) B) C) D) 0 1 2 3 37 Physics 212 Lecture 5, Slide 18 Example: Potential Energy Changes At which two places can a positive charge be placed without changing the total potential energy of the system? A A) B) C) D) E) A&B A&C B&C B&D A&D q X=0 B C -2q X=d D x Let's calculate the positions of A and B 40 Physics 212 Lecture 5, Slide 19 Lets work out where A is
r d q X=0 -2q X=d x A 1 Qq 1 2Qq U = + - 40 r 40 r + d Set U = 0
1 2 = r r+d
Makes Sense! Q is twice as far from -2q as it is from +q
Physics 212 Lecture 5, Slide 20 r=d
43 Lets work out where B is
r q X=0 d-r B -2q X=d x Setting U = 0 1 2 = r d -r 2r = d - r r= d 3
Physics 212 Lecture 5, Slide 21 46 Makes Sense! Q is twice as far from -2q as it is from +q Summary
For a pair of charges:
Just evaluate q1q2 U =k r r
q1 q2 (We usually choose U = 0 to be where the charges are far apart) For a collection of charges:
Sum up q1q2 U =k r for all pairs Physics 212 Lecture 5, Slide 22 ...

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