Transcript PHYS_2326_012909

There will be a quiz next class period, Feb 1, covering Ch 22 and
the beginning of Ch 23 (what we cover in class today)
Definitions
• Electric potential—Potential energy per unit charge at a point in an
electric field
• Path integral (line integral)—An integral performed over a path such
as the path a charge q follows as it moves from one point to another
• Volt—The unit of electric potential. 1V = 1 J/C
• Electron volt (eV)—the energy that an electron (or proton) gains or
loses by moving through a potential difference of 1 V.
• Equipotential surface—A surface consisting of a continuous
distribution of points having the same electric potential
Electric Potential
• Electric force is a conservative force, therefore there is a
potential energy associated with it.
• We can define a scalar quantity, the electric potential,
associated with it.
r
r
r r
W Efield  FE  dl  qE  dl
r r
dU  qE  dl
r
B r
U  q  E  dl
A
r
B r
U
V     A E  dl
q

Electric Potential Energy
Concepts of work, potential energy
and conservation of energy
For a conservative force, work can
always be expressed in terms of
potential energy difference
b

Wa b   F d l  U  (U b  U a )
a
Energy Theorem
For conservative forces in play,
total energy of the system is conserved
Ka  U a  Kb  U b
• The line integral used
to calculate V does not
depend on the path
taken from A to B;
therefore pick the most
convenient path to
integrate over
Electric Potential
• We can pick a 0 for the electric potential energy
U 0r 
• U is independent of any charge q that can be placed in the Electric
field
• U has a unique value at every point in the electric field
• U depends on a location in the E field only
Wa b  Fd  q0 Ed
U  q0 Ey
Wa b  U  q0 E ( ya  yb )
Potential energy U increases as the test charge q0 moves in the


direction opposite to the electric force F  q0 E: it decreases as
it moves in the same direction as the force acting on the charge
Electric Potential Energy of Two Point Charges
b
Wa b

rb
qq0
  F d l   ke 2 cos  dl
r
a
r
a
Wa b
1 1
 ke qq0   
 ra rb 
qq0
U  ke
r
Electric potential energy of two point charges
Example: Conservation of energy with electric forces
A positron moves away from an a– particle
m p  9.1  1031 kg

0
a-particle
positron
ma
7000m p
qa  2e
r0  1010 m
V0  3  106 m / s
What is the speed at the distance r  2r0  2  1010 m ?
What is the speed at infinity?
Suppose, we have an electron instead of positron.
What kind of motion we would expect?
Conservation of energy principle
K0  U 0  K1  U1
Electric Potential Energy of the System of Charges
Potential energy of a test charge q0
in the presence of other charges
U
q0
qi

4 0 i r i
Potential energy of the system of charges
(energy required to assembly them together)
U
1

qi q j
4 0 i  j r ij
Potential energy difference can be equivalently described as a work
done by external force required to move charges into the certain
geometry (closer or farther apart).


External force now is opposite to Wa b  (U b  U a )   Fext d l
the electrostatic force
Electric potential is electric potential energy per unit charge
Finding potential (a scalar) is often much easier than the field
(which is a vector). Afterwards, we can find field from a potential
U
V
q0
Units of potential are Volts [V]
1 Volt=1Joule/Coulomb
If an electric charge is moved by the electric
field, the work done by the field
Wa b
U

 (Va  Vb )
q0
q0
Potential difference if often called voltage
Two equivalent interpretations of voltage:
1.Vab is the potential of a with respect to b, equals the work done
by the electric force when a UNIT charge moves from a to b.
2. Vab is the potential of a with respect to b, equals the work that must
be done to move a UNIT charge slowly from b to a against the
electric force.
Potential due to the point charges
1
dq
V
4 0  r
Potential due to a continuous
distribution of charge
Finding Electric Potential through Electric Field
b

Wa b
 Va  Vb   E d l
q0
a

Some Useful Electric Potentials
• For a uniform electric field
r r
r
V    E  dl   E 
• For a point charge
q
V  ke
r
• For a series of point charges
qi
V  ke 
ri
r
r r
 dl  E  l
Potential of a point charge
Moving along the E-field lines means moving in the direction of
decreasing V.
As a charge is moved by the field, it loses it potential energy, whereas
if the chargeis moved by the external forces against the E-field, it
acquires potential energy
• Negative charges are a potential minimum
• Positive charges are a potential maximum
Positive Electric Charge Facts
• For a positive source charge
– Electric field points away from a positive source charge
– Electric potential is a maximum
– A positive object charge gains potential energy as it moves
toward the source
– A negative object charge loses potential energy as it moves
toward the source
Negative Electric Charge Facts
• For a negative source charge
– Electric field points toward a negative source charge
– Electric potential is a minimum
– A positive object charge loses potential energy as it moves
toward the source
– A negative object charge gains potential energy as it moves
toward the source