Transcript Physics 212 - Louisiana State University

Physics 2102
Jonathan Dowling
Physics 2102
Lecture: 09 MON 02 FEB
Electric Potential II
Ch24.6-10
PHYS2102 FIRST MIDTERM EXAM!
6–7PM THU 05 FEB 2009
Dowling’s Sec. 2 in Lockett Hall, Room 6
YOU MUST BRING YOUR STUDENT ID!
The exam will cover chapters 21 through 24, as
covered in homework sets 1, 2, and 3. The formula
sheet for the exam can be found here:
http://www.phys.lsu.edu/classes/spring2009/phys2102/formulasheet.pdf
THERE WILL BE A REVIEW SESSION 6–7PM
WED 04 FEB 2009 in Williams 103
Conservative Forces, Work, and Potential Energy
W
 Frdr
Work Done (W) is
Integral of Force (F)
U  W
Potential Energy (U)
is Negative of Work
Done
dU
F r  
dr
Hence Force is
Negative Derivative
of Potential Energy
Coulomb’s Law for Point Charge
kq1q2
F12  2
r
Force [N]
= Newton
kq1q2
U12 
r
Potential
Energy
[J]=Joule

dU12
F12  
dr
q2

q1
P2
P1

 U   F dr
12
12
v
kq2
E12  2
r
Electric
Field
[N/C]=[V/m]
Electric
Potential
[J/C]=[V]
=Volt
kq2
V12 
r
dV12
E12  
dr
q2
P2
P1
 V  
12
E
12
dr
Electric Potential of a Point
Charge
f
P
i

V    E  ds    E ds 
R
R
kQ
kQ
kQ
   2 dr  

r
r 
R

Note: if Q were a
negative charge,
V would be negative
Electric Potential of Many Point Charges
• Electric potential is a
SCALAR not a vector.
q4
• Just calculate the potential
due to each individual point
charge, and add together!
(Make sure you get the
SIGNS correct!)
qi
V  k
ri
i
r3
r4
q5
r5
Pr2
q2
r1
q1
q3
Electric Potential and Electric
Potential Energy
U  Wapp  qV
+Q
a
What is the potential energy of a dipole?
–Q
+Q
a
–Q
• First: Bring charge +Q: no work involved, no potential energy.
• The charge +Q has created an electric potential everywhere, V(r) = kQ/r
• Second: The work needed to bring the charge –Q to a distance a from the
charge +Q is Wapp = U = (-Q)V = (–Q)(+kQ/a) = -kQ2/a
• The dipole has a negative potential energy equal to -kQ2/a: we had to do
negative work to build the dipole (electric field did positive work).
Positive Work
+Q
a
+Q
Negative Work
+Q
a
–Q
Potential Energy of A System of Charges
• 4 point charges (each +Q and
equal mass) are connected by
strings, forming a square of side
L
+Q
+Q
• If all four strings suddenly snap,
what is the kinetic energy of
each charge when they are very
far apart?
+Q
+Q
• Use conservation of energy:
– Final kinetic energy of all four
charges = initial potential energy
stored = energy required to
assemble the system of charges
Do this from scratch!
Potential Energy of A System of
Charges: Solution
• No energy needed to bring in
first charge: U1=0
+Q
• Energy needed to bring in
3rd charge =
kQ2 kQ2
U3  QV  Q(V1  V2 ) 

L
2L
• Energy needed to bring in
4th charge =
2kQ2 kQ2
U 4  QV  Q(V1  V2  V3 ) 

L
2L
+Q
2L
• Energy needed to bring in
2nd charge:
kQ 2
U 2  QV1 
L
L
+Q
+Q
Total
potential energy is sum of all
the individual terms shown on left
hand side =
2
kQ
L
4  2 
So, final kinetic energy of each charge
=
2


kQ
4 2
4L

Example
qi
V  k
ri
i
Positive and negative charges of equal magnitude Q
–Q
+Q
are held in a circle of radius R.
1. What is the electric potential at the center of each
circle?
• VA =
• VB =
• VC =
k3Q  2Q/r  kQ/r
k2Q  4Q/r  2kQ/r
k 2Q  2Q / r  0

2. Draw an arrow representing the approximate

direction of the electric field at the center of each
A
B
circle.
3. Which system has the highest electric potential
energy?
UB =Q•Vb has largest un-canceled charge
C
Electric Potential of a Dipole (on axis)
What is V at a point at an axial distance r away from the
midpoint of a dipole (on side of positive charge)?
V k
Q
k
Q
a
a
(r  )
(r  )
2
2
a
a 

 (r  2 )  (r  2 ) 
 kQ 

a
a
 (r  )(r  ) 
2
2 


Qa
2
a
2
4 0 (r  )
4
p
a
–Q +Q
r
Far away, when r >> a:
V 
p
4 0 r 2
Electric Potential on Perpendicular
Bisector of Dipole
You bring a charge of Qo = –3C
from infinity to a point P on the
perpendicular bisector of a
dipole as shown. Is the work
that you do:
a) Positive?
b) Negative?
c) Zero?
U = QoV = Qo(–Q/d+Q/d) = 0
a
-Q +Q d
P
–3C
Summary:
• Electric potential: work needed to bring +1C from infinity; units
[V] = Volt
• Electric potential uniquely defined for every point in space -independent of path!
• Electric potential is a scalar — add contributions from individual
point charges
• We calculated the electric potential produced by a single
charge: V = kq/r, and by continuous charge distributions :
V =  kdq/r
• Electric potential energy: work used to build the system,
charge by charge. Use W = qV for each charge.