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Electric potential,
Systems of charges
Physics 122
Lecture V 1

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Concepts
•
Primary concepts:
–
Electric potential
–
Electric energy
•
Secondary concepts:
– Equipotentials
–
Electonvolt
Lecture V 2


E
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+

F

F
Charges in electric fields
Positive charges experience force along the direction of the field
Negative charges – against the direction of the field.

F

 q E
Lecture V 3

Potential electric energy
High PE
High PE Low PE
Low PE
Just like gravity electric force can do work work does not depend on the path it depends only on the initial and final position
 there is a potential energy associated with electric field.
4/15/2020 Lecture V 4

Electric potential
PE ( q )
 q
•
PE/q is a property of the field itself – called electric potential V
5 4/15/2020 Lecture V

Electric potential
V

PE q
• V – electric potential is the potential energy of a positive test charge in electric field, divided by the magnitude of this charge q.
•
Electric potential is a scalar (so much nicer!).
•
Electric potential is measured in Volts (V=J/C).
•
Potential difference between two points
D
V=V b
-V a is often called voltage .
4/15/2020 Lecture V 6


E
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Charges in electric fields b
+

F a d
E=const
Force on charge q:
F=qE
Work done by the field to move this charge
W=Fd=dqE
W=PE a
-PE b
=qV a
-qV b
=-q
D
V
• d E=-
D
V
• E= -
D
V/d, points from high potential to low
• Sometimes electric field is measured in
V/m =N/C
Lecture V 7

Non-uniform electric field

E ( x )


F ( x )

 q E ( x )
W
U b
 b
 a

U a
Fdx

 q b
 a

W ba
Edx
V
  b
 a

E d l

4/15/2020 Lecture V
+
8

Electric field and potential in

E external conductors
E=0 in good conductors in the static situation.
+ + +
+
+
E is perpendicular to the surface of conductor.
-
Metal hollow boxes are used to shield electric fields.

E
V


E exernal
 const
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

E int ernal

0
When charges are not moving conductor is entirely at the same potential .
Lecture V 9

Electronvolt
Energy

5000 eV
• Energy that one electron gains when being accelerated over 1V potential difference is called electronvolt eV:
•
1eV=1.6
x 10 -19 C 1V= 1.6
x 10 -19 J
•
Yet another unit to measure energy,
• Commonly used in atomic and particle physics.
4/15/2020 Lecture V 10

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Equipontentials
Equipotentials
• are surfaces at the same potential;
• are always perpendicular to field lines;
•
Never cross;
•
Their density represents the strength of the electric field
•
Potential is higher at points closer to positive charge
Lecture V 11

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Potential of a point charge
V (

)

0 Potential V of electric field
+ created by a point charge Q at a radius r is
V ( r
0
)
  
Q>0

V>0
Edr
  kQ

 r
0 dr r
2
 k
Q<0

V<0
Q r
0
Do not forget the signs!
Potential goes to 0 at infinity.
Equipotentials of a point charge are concentric spheres.
Lecture V 12

Superposition of fields
V
1

0
+
V

V
1

V
2
V
2

0
+
1
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2
-
Principle of superposition:
Net potential created by a system of charges is a scalar (!) sum of potentials created by individual charges:
V

V
1

V
2

V
3

....
Potential is a scalar
 no direction to worry about.
Lecture V 13

r
D r
D r=lcos


 p
-Q l
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Electric Dipole potential
+Q r
P
V
• evaluate potential at point P
 k
Q
 k r r
Q
 D r

D r kQ r ( r
 D r )
Lecture V
• for r >> L ,
V
 k p cos
 r
2
14

Test problem
• What is wrong with this picture?
–
A Equipotentials must be parallel to field lines
– B Field lines cannot go to infinity
– C Some field lines point away from the negative charge
– D Equipotentials cannot be closed
4/15/2020 Lecture V 15

4/15/2020 Lecture V 16

The electric potential of a system of charges
Bunch of Charges
Charge Distribution
V

V
1

V
2

V
3

...
V i
 k q i r i
, r i
distance from charge i to point in space where V is evaluated
V dV
 dq
 k
 dV r
+
-
+
-
-
+
+
-
-
+
+
+
+
+ + +
4/15/2020 Lecture V 17

Symmetry and coordinate systems
•
Coordinate systems are there to help you
•
You have a choice of
–
System type
• Cartesian
• Cylindrical
• Spherical
–
Origin (0,0), Direction of axis
•
A good choice (respecting the symmetry of the system) can help to simplify the calculations
4/15/2020 Lecture V 18

Ring of charge
•
A thing ring of radius a holds a total charge Q .
Determine the electric field on its axis, a distance x from its center.
a r
 x x
2  a
2

E
E x

E

E
 k
Qx
( x
2  a
2
)
3 / 2
4/15/2020 Lecture V 19

Ring of charge
•
A thing ring of radius a holds a total charge Q .
Determine the electric potential on its axis, a distance x from its center.
dV
 k
Qd

/ 2
 r
V
 k
( a
2 
Q x
2
)
1 / 2
4/15/2020 Lecture V a r
 x
2  a
2 x dV
20

Work to move a charge
V a 
V
1 a 
V
2 a
20cm
+
+
Q
1
=10 m
C
V
+ b 
V
1 b 
V
2 b
-
15cm
Q
2
=-20 m
C
How much work has to be done by an external force to move a charge q=+1.5 m
C from point a to point b?
W
Work-energy principle
 D
KE
 D
PE

PE b

PE a
PE
PE a b
 qV b
 qV a
 q ( V
1 b
 q ( V
1 a

V
2 b
)

V
2 a
)
4/15/2020 Lecture V 21

Determine E from V
• Think ski slopes
•
If V depends on one coordinate x
• E is directed along x from high V to low
E ( x )

•
If V depends on x,y,z
E x
 

V
 x
; E y

 dV dx


V
 y
; E z
 

V
 z
4/15/2020 Lecture V 22

E near metal sphere
• Find the largest charge Q that a conductive sphere radius r=1cm can hold.
•
Air breakdown E=3x10 6 V/m
E ( r )
  dV dr
 
 kQ

1 r


'
 k
Q r 2
Q

1
Er
2 k

3

10
6
9

10
9
( 10

2
Larger spheres can hold more charge
)
2 
0 .
33

10

7 
0 .
033 m
C
4/15/2020 Lecture V 23