Electricity & Magnetism
Lecture 5: Electric Potential Energy
Today...
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Recap of Unit 19 grading
This unit’s (21) wri=en Homework
Electric PotenDal Energy
Unit 21 session GravitaDonal and Electrical PE
Electricity & MagneDsm Lecture 5, Slide 1
Stuff you asked about:
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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
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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
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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 locaDon 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
B. kQqR/r12
C. kQqR/r22
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 locaDon 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
B. kQqR/r12
C. kQqR/r22
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:
W=
R~r2
~r1
F~ · d~r
F
dr
W>0
Object speeds up ( ΔK > 0 )
W<0
Object slows down ( ΔK < 0 )
F
dr
or
F
dr
F
dr
W=0
Constant speed ( ΔK = 0 )
Electricity & MagneDsm Lecture 5, Slide 3
Dot Product (review)
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If you know A and B in rectangular coordinates, then you can find the angle between them.
Potential Energy
If gravity does negaDve work, potenDal energy increases!
Same idea for Coulomb force… if Coulomb force does negaDve work, potenDal energy increases.
+
+
+
+
Δx
F
Coulomb force does nega2ve 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
Δx
Work done by force is posi:ve
ΔU = −Work is nega:ve
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
East
FH
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 conservaDve force.
Does not have any ΔU.
ConservaDve force: ΔU = − WE
FE
FH
E
dr
B) WH > 0 and WE < 0
Is ΔU posiDve or negaDve?
A) PosiDve
B) NegaDve
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 separaDon of “d”
d
q1
q2
𝛜 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
Case B
d
2d
In case A two negaDve 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 configuraDon 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)
r
Electricity & MagneDsm Lecture 5, Slide 10
0
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
(superposiDon)
d
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 negaDve. Now what is the total energy required to bring the three charges in infinitely far away?
Q
A) 0
B) d
C) 2
d
D) E) Q
1
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 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 negaDve 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 negaDve
C. YES, no ma=er what the sign of the third charge
D. NO
LATION!
U
LC
A
C
E
H
T
O
D
’S
T
E
L
HOW?
““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 negaDve 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=0
X=d
x
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
A
X=0
A)
B)
C)
D)
E)
A & B
A & C
B & C
B & D
A & D
B
C
−2Q
X=d
D
x
Let’s calculate the posiDons of A and B
Electricity & MagneDsm Lecture 5, Slide 20
Lets work out where A is
d
r
A
Q
−2Q
X=0
X=d
x
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
d−r
r
Q
X=0
−2Q
B
X=d
x
Serng 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
Q
X=0
1
r+d
=
2
r
r
−2Q
X=d
(r must be posiDve)
D
x
Summary
For a pair of charges:
r
Q1
Just evaluate
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