Physics 2102
Jonathan Dowling
Physics 2102
Lecture: 07 TUE 09 FEB
Electric Potential II
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PHYS2102 FIRST MIDTERM
EXAM!
6–7PM THU
11 FEB 2009
Dowling’s Sec. 4 in ???
YOU MUST BRING YOUR STUDENT ID!
YOU MUST WRITE UNITS TO GET FULL CREDIT!
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/spring2010/phys2102/formulasheet.pdf
KNOW SI PREFIXES n, cm, k, etc.
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

v
kq2
E12  2
r
q1
P1
P2


U12    F12dr
Electric
Potential
[J/C]=[V]
=Volt
kq2
V12 
r
dV12
E12  
dr
dU12
F12  
dr
q2
Electric
Field
[N/C]=[V/m]
q2
P2
P1
 V  
12
E
12
dr
Continuous Charge Distributions
• Divide the charge distribution into
differential elements
• Write down an expression for
potential from a typical element —
treat as point charge
• Integrate!
• Simple example: circular rod of
radius r, total charge Q; find V at
center.

r
dq
V
 dV  
k

r
k
 dq  r Q
kdq
r
Nm2 C  Nm J 
Units :  2
     V
 C m  C  C
Potential of Continuous Charge
Distribution: Line of Charge
• Uniformly charged
rod
• Total charge Q
• Length L
• What is V at position
P?
x
P
dx
L
a
 Q/L
dq   dx
kdq
kdx
V 

r
(
L

a

x
)
0
L
 k  ln(L  a  x)
L
0
L  a
V  k ln 

a


Units: [Nm2/C2][C/m]=[Nm/C]=[J/C]=[V]
Electric Field & Potential:
A Simple Relationship!
Focus only on a simple
case: electric field that
points along +x axis but
whose magnitude varies
with x.
Notice the following:
• Point charge:
E = kQ/r2
V = kQ/r
• Dipole (far away):
E = kp/r3
V = kp/r2
• E is given by a DERIVATIVE of V!
• Of course!
f
V    E  ds
i
dV
Ex  
dx
Note:
• MINUS sign!
• Units for E:
VOLTS/METER (V/m)
E from V: Example
• Uniformly charged
rod
• Total charge Q
• Length L
• We Found V at P!
• Find E from V?
x
dx
L
Units:
a
 Q/L
L  a 
V  k ln 
 a 

 kln L  a  ln a
dV
d
E 
 k ln L  a  ln a
da
da

 1
1 
 
P  k
L  a a 
a  L  a
 L 
 k
 k

 L  aa 
L  aa 
Nm2 C m  N  V 


 2
2     


 C m m  C m
√ Electric Field!
Electric Field & Potential: Question
•
•
Hollow metal sphere of radius R
has a charge +q
Which of the following is the
electric potential V as a function
of distance r from center of
sphere?
V
1

r
(a)
V
r
r=R
V
+q

(c)
r=R
(b)

r=R
1
r
r
1
r
r
Electric Field & Potential: Example
+q
E
V
Outside the sphere:
• Replace by point charge!
Inside the sphere:
• E = 0 (Gauss’ Law)
• E = –dV/dr = 0 IFF V=constant
1
 2
r

1
r
dV
E
dr
d  Q
  k 
dr  r 
Q
k 2
r
Equipotentials and
Conductors
• Conducting surfaces are
EQUIPOTENTIALs
• At surface of conductor, E is normal
(perpendicular) to surface
• Hence, no work needed to move a
charge from one point on a
conductor surface to another
• Equipotentials are normal to E, so
they follow the shape of the
conductor near the surface.
V
E
Conductors Change the
Field Around Them!
An Uncharged Conductor:
A Uniform Electric Field:
An Uncharged Conductor in the
Initially Uniform Electric Field:
Sharp Conductors
• Charge density is higher at
conductor surfaces that have small
radius of curvature
• E = s/e0 for a conductor, hence
STRONGER electric fields at
sharply curved surfaces!
• Used for attracting or getting rid of
charge:
– lightning rods
– Van de Graaf -- metal brush
transfers charge from rubber belt
– Mars pathfinder mission -tungsten points used to get rid of
accumulated charge on rover
(electric breakdown on Mars
occurs at ~100 V/m)
(NASA)
Ben Franklin Invents the Lightning Rod!
LIGHTNING SAFE CROUCH
If caught out of doors during an approaching storm and
your skin tingles or hair tries to stand on end, immediately
do the "LIGHTNING SAFE CROUCH ”. Squat low to the
ground on the balls of your feet, with your feet close
together. Place your hands on your knees, with your head
between them. Be the smallest target possible, and
minimize your contact with the ground.
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 field and electric potential: E= -dV/dx
• Electric potential energy: work used to build the system, charge by charge.
Use W = U = qV for each charge.
• Conductors: the charges move to make their surface equipotentials.
• Charge density and electric field are higher on sharp points of conductors.