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04/21/23 Lecture V 1
Physics 122
Electric potential,Systems of charges
04/21/23 Lecture V 2
Concepts
• Primary concepts:– Electric potential– Electric energy
• Secondary concepts:– Equipotentials– Electonvolt
04/21/23 Lecture V 3
Charges in electric fields
Positive charges experience force along the direction of the field
Negative charges – against the direction of the field.
+E F
-F
EqF
04/21/23 Lecture V 4
Potential electric energy
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.
High PE High PE
Low PE
Low PE
04/21/23 Lecture V 5
Electric potential
• PE/q is a property of the field itself – called electric potential V
qqPE )(
04/21/23 Lecture V 6
Electric potential
q
PEV
• 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 V=Vb-Va is often called voltage.
04/21/23 Lecture V 7
Charges in electric fieldsE=constForce on charge q:F=qEWork done by the field to move this chargeW=Fd=dqE
W=PEa-PEb=qVa-qVb=-qV
• d E=-V
• E= -V/d, points from high potential to low• Sometimes electric field is measured in
V/m =N/C
+E F
d
a
b
04/21/23 Lecture V 8
Non-uniform electric field
b
a
baab
b
a
b
a
ldEV
WUU
EdxqFdxW
xEqxFxE
)()()(
+
04/21/23 Lecture V 9
Electric field and potential in 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.
When charges are not moving conductor is entirely at the same potential.
+
externalE
-- - - -
+ + + +
constV
EEE ernalexernal
0int
04/21/23 Lecture V 10
Electronvolt
• Energy that one electron gains when being accelerated over 1V potential difference is called electronvolt eV:
• 1eV=1.6x10-19C 1V= 1.6x10-19J• Yet another unit to measure
energy,• Commonly used in atomic
and particle physics.eVEnergy 5000
04/21/23 Lecture V 11
EquipontentialsEquipotentials • 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
04/21/23 Lecture V 12
Potential of a point chargePotential V of electric field
created by a point charge Q at a radius r is
Q>0 V>0Q<0 V<0Do not forget the signs!Potential goes to 0 at infinity.Equipotentials of a point charge
are concentric spheres.
020
0
)(r
Qk
r
drkQEdrrV
r
+
0)( V
04/21/23 Lecture V 13
Superposition of fields
Principle of superposition:
Net potential created by a system of charges is a scalar (!) sum of potentials created by individual charges:
1 2
-+
+21 VVV
....321 VVVV
Potential is a scalar no direction to worry about.
01 V
02 V
04/21/23 Lecture V 14
Electric Dipole potential
• evaluate potential at point P
•for r >> L,
2
cos
r
pkrV
p
-Q +Ql
rr
P
r
r=lcos
)( rrr
rkQ
rr
Qk
r
QkrV
04/21/23 Lecture V 15
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
04/21/23 Lecture V 16
04/21/23 Lecture V 17
The electric potential of a system of charges
evaluated is wherespacein point
to charge from distance - ,
...321
V
irr
qkV
VVVV
ii
ii
++
+ +
+
+-
-
--
-
Bunch of Charges
dVV
r
dqkdV
+ ++ ++ +++++ +
Charge Distribution
04/21/23 Lecture V 18
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
04/21/23 Lecture V 19
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.
E
a
x
xE
E
22 axr
2/322 )( ax
QxkE
04/21/23 Lecture V 20
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. a
x dV
22 axr
2/122 )(
2/
xa
QkV
r
QdkdV
04/21/23 Lecture V 21
Work to move a chargeHow much work has to be
done by an external force to move a charge
q=+1.5 C
from point a to point b?
Work-energy principle
Q1=10C-+
+
aaa VVV 21
ab PEPEPEKEW
+
bbb VVV 21
)( 21aa
aa VVqqVPE
)( 21bb
bb VVqqVPE Q2=-20C
20cm
30cm
25cm
15cm
04/21/23 Lecture V 22
Determine E from V • Think ski slopes• If V depends on one coordinate x• E is directed along x from high V to low
• If V depends on x,y,zdx
dVxE )(
z
VE
y
VE
x
VE zyx
;;
04/21/23 Lecture V 23
E near metal sphere
• Find the largest charge Q that a conductive sphere radius r=1cm can hold.
• Air breakdown E=3x106V/m
2'
1)(
r
Qk
rkQ
dr
dVrE
CErk
Q 033.01033.0)10(109
1031 7229
62
Larger spheres can hold more charge
