Transcript Lecture 4

Lecture 4: May 27th 2009
Physics for Scientists and Engineers II
Physics for Scientists and Engineers II , Summer Semester 2009
Work Done on Charge by Electric Force Along a Closed Path
•
Imagine a charge in an electric field is moved by external force as shown in a
closed path.
E
 x1
 x2
 x3
+
Fe
 x5
 x4
a
WFe 
F
e
 d s  F e   x1  F e   x 2  F e   x 3  F e   x 4  F e   x 5   Fe a  Fe a  0
path
(Work done by Fe  0 actually for any closed path)
•
In other words: Electrostatic force is a conservative force.
 An electric potential (a scalar quantity) can be defined.
(just like we did with the gravitational force / potential)
Physics for Scientists and Engineers II , Summer Semester 2009
Remember from Physics 2215
Fnet
v 2  v02
For constant net force : a 
and a 
m
2 x
Fnet v 2  v02


m
2 x

1 2 1 2
mv  mv0  Fnet x
2
2
KE  WFnet
KE  WFnonconservative  WFconservative
KE  WFconservative  WFnonconservative
KE  U
 WFnonconservative
,
where : U  WFconservative
U can be a change in gravitatio nal potential energy or
it can be a change in electrical potential energy or
a change in both of these.
Let' s consider t he case of change in electric potential energy now.
Physics for Scientists and Engineers II , Summer Semester 2009
Change in Electric Potential Energy
•
Imagine a charge q0 is moved in an electric field along some path
ds
E
Fe
+ q0
Work done by electric force : WFe 
F
e
 d s  q0
path
 E ds
path
B
Change in potential energy :
U   q0
 E  d s  q  E  d s
0
path
A
Because F e is a conservati ve force , the exact path is not relevant.
Only the starting and end points are relevant.
Physics for Scientists and Engineers II , Summer Semester 2009
Electric Potential Energy
•
An absolute potential energy can be defined relative to some position, where
arbitrarily the potential energy is set to be U = 0 (like in gravitational PE)
Electric Potential
•
•
The electric potential energy depends not only on the location of the charge
and the location of U = 0, but also on the amount of charge itself.
A quantity independent of charge can be defined: The electric potential V.
Electric potential energy differnce of charge q 0 between places A and B :
U  U B  U A
U
V 
  E d s
q0
A
B
Electric potential difference between A and B :
Never confuse potential and potential energy !!!
Physics for Scientists and Engineers II , Summer Semester 2009
Work – Energy Relationship
Work done BY external agent ON a charge q when moving the charge from A to B :
W  U  K
When moving it with constant v elocity (KE  0) :
W  U  q V
Units of Electric Potential
The unit of electric potential is " Volt" (V) :
J
1V 1
C
( to move a charge of 1 Coulomb through a potential difference of 1 Volt,
1 Joule of work must be done)
Physics for Scientists and Engineers II , Summer Semester 2009
Other Useful Units
N
V
1
(Unit of electric field)
C
m
(Electric field strength describes how much the electric potential changes with position)
1
1eV  1.60 10 19 C V  1.60 10 19 J
(1eV is the amount of energy the charge - field system gains or looses when a charge
of amount " e" (electron or proton) is moved through 1 Volt of potential difference )
Physics for Scientists and Engineers II , Summer Semester 2009
Potential Difference in a Uniform Field
B
A
E
B
B
B
A
A
A
VB  VA  V    E  d s    E cos 0ds    Eds
E  constant
B
 V   E  ds   Ed
A
Negative sign means :
* A displaceme nt in the same direction as the electric field lines
results in a loss of electric potential.
* In other word s : The electric potential at point B is less than that at point A.
Physics for Scientists and Engineers II , Summer Semester 2009
Potential Energy Difference in a Uniform Field
B
A
E
B
V   E  ds   Ed
A
 U  q E d
Negative sign means :
* A displaceme nt of a positive charge in the same direction as the electric field lines
results in a loss of electric potential energy.
* A displaceme nt of a negative charge in the same direction as the electric field lines
results in a gain of electric potential energy.
Physics for Scientists and Engineers II , Summer Semester 2009
Non-parallel displacement
s
B

A
E
B
V    E  d s   E  s   E s cos 
A
For   90 : V  0
 All points in a plane perpendicu lar to a uniform electric field are at
the same electric potential.
Equipotent ial surface  any surface consisting of a continuous distributi on
of points havning the same electric potential.
Physics for Scientists and Engineers II , Summer Semester 2009
Electron “Gun”
E
- A
E
+
+
++
Fe
d
v
B
B
- +
V   E  ds   E  s   Ed cos 180  Ed
A
U  q E  s  qEd cos 180  qEd , where q  -e
The electron starts at rest just outside of the negative plate.
What is v final of the electron at the positive plate?
KE  U  Wnonconservative forces  0
1 2
2eEd
mv final  (e) Ed  0  v final 
2
m
Physics for Scientists and Engineers II , Summer Semester 2009
Electron “Gun”
E  5.0 103
V
m
- -
E
+
+
++
Fe
v
5.0cm
V  Ed  5.0 103
V
 0.050m  2.5 10 2 V
m
2eEd
2eV
2 1.6 10 19 C  2.5 10 2 V
v final 


m
m
9.110 31 kg
 9.4 10
6
CV
Nm
 9.4 106
 9.4 106
kg
kg
Physics for Scientists and Engineers II , Summer Semester 2009
kg m
2
s 2  9.4 106 m
kg
s
Electric Potential and Potential Energy due to Point Charges

E
B
ds
rB
A
rA
B
VB  V A    E  d s
,
where E  ke
A
q
q
rˆ
r2
B
q
rˆ  d s
2
r
A
VB  V A   k e 
,
where rˆ  d s  ds cos   dr
1 1
 ke q  
A
 rB rA 
1 1
U B  U A  q0 VB  VA   ke q0 q  
 rB rA 
B
dr
q
VB  V A   k e q  2  k e
r
r
A
Again, U is independent of the path
chosen, confirming that the electric
force is conservative.
B
Physics for Scientists and Engineers II , Summer Semester 2009
Potential energy change when moving
a charge q0 in the electric field of charge
q from A to B.
Electric Potential due to Point Charges; Choosing Zero Point

E
B
ds
rB
A
rA
V    Ed s  ke 
q
q
dr
r2
q
 const .
r
Convention :
V(r  )  0 for the electric potential of a point charge (const.  0)
q
 V(r)  ke
r
 V(r)  ke
Physics for Scientists and Engineers II , Summer Semester 2009
Electric Potential due to Multiple Point Charges
V   Vi  ke 
i
i
qi
ri
Potential Energy of a System of Two Point Charges
Assume charge q2 is already in place. Then, a charge q1 is brought from a very large distance
(“infinity”) near charge q2 using an external force.
 U  q1 V  q1ke
q2
qq
 ke 1 2
r12
r12
Electric potential due to the
electric field of charge q2.
Since U  U final  U   U  Wexternal force (assuming KE  0) we conclude :
1)
2)
Bringing two like charges together requires positive work by an external force.
When two unlike charges are brought together, negative work is done by an external force.
Physics for Scientists and Engineers II , Summer Semester 2009
F external
+
+
Fe
-
Fe
ds
F external
-
ds
Potential energy increases (work done by external force is positive)
F external
Fe
-
+
ds
Potential energy decreases (work done by external force is negative)
Physics for Scientists and Engineers II , Summer Semester 2009
Electric Potential Energy of Collection of Point Charges
1) Initially, we have only one charge (q1):
U 0
q1
2) Next, a second charge (q2) is brought near the charge already in place (q1) :
q1
r12
q2
q2
q1
q1q2
U  q2V1  q2 ke
 ke
r12
r12
Physics for Scientists and Engineers II , Summer Semester 2009
Adding the Third Charge ….
3) Finally, a third charge (q3) is brought near the charges already in place (q1and q2) :
q1
q2
r12
r23
q1q2
U  ke
 q3V1 2
r12
r13
q q 
qq
 ke 1 2  q3 ke  1  2 
r12
 r13 r23 
 q1q2 q1q3 q2 q3 

 ke 


r13
r23 
 r12
Physics for Scientists and Engineers II , Summer Semester 2009
q3
q3
Electric Field Calculation from Electric Potential
dV   E  d s  E x dx  E y dy  E z dz 
If E y  E z  0 :
dV   E x dx  E x  
dV
dx
For fields with spherical symmetry (e.g., field of a point charge) :
dV   E  d s   Er dr
 Er 
dV
dr
In general : E x  
V  x, y, z 
x
Ey  
V  x, y, z 
y
Ez  
 


or, using vector notation : E  V   iˆ  ˆj  kˆ  V
y
z 
 x
 is called the gradient operator.
Physics for Scientists and Engineers II , Summer Semester 2009
V x, y, z 
z