Transcript Electric Potential

Electric Potential
AP Physics C
Montwood High School
R. Casao
Electric Potential Energy
• When a force acts on a particle that moves from
point a to point b, the work Wab done by the force
is given by a line integral: W  b F  dx  b F  x  cos θ
a b

a

a
▫ dx is a very small distance along the particle’s path.
▫ angle  is the angle between F and dx at each point
along the path.
• If the force F is conservative, the work done by F
can be expressed in terms of potential energy U.
▫ When the particle moves from a point where the
potential energy is Ua to a point where it is Ub, the
change in potential energy is DU = Ub – Ua.
• The work done by the force is:
Wa b  Ua  Ub  (Ub  Ua )  ΔU
• When Wab is positive, Ua is greater than Ub, DU is
negative, and the potential energy decreases.
• Work is positive when the force and the direction of
motion are in the same direction (parallel;  = º);
work is negative when the force and the direction of
motion are in opposite directions (antiparallel;  =
180º).
• The work-energy theorem says that the change in
kinetic energy DK = Kb – Ka during any displacement
is equal to the total work done on the particle.
• If the only work done on the
particle is done by
conservative forces, then Kb
– Ka = -(Ub – Ua), which can
be rewritten as Ka + Ua = Kb
+ Ub.
▫ The total mechanical energy
is conserved.
Electric Potential Energy
in a Uniform Electric
Field
• A pair of charged parallel
metal plates sets up a
uniform, downward electric
field with magnitude E.
• The field exerts a downward
force F = qo·E on a positive
test charge qo.
• As the charge moves
downward a distance d from
point a to b, the force on the
test charge is constant and
independent of its location.
• Work done by the electric
field is Wab = F·d = qo·E·d.
• Work is positive because the
force is in the same direction
as the displacement.
• The electric force on the charge is constant,
therefore, the force exerted on qo is conservative.
• The work done by the electric field is independent
of the path the particle takes from point a to point
b.
• For a positive charge in a uniform electric field:
▫ If the positive charge moves in the direction of the
electric field E, the electric field does positive work
on the charge and the potential energy U decreases;
K of the charge increases.
▫ If the positive charge moves in the direction opposite
the electric field E, the electric field does negative
work on the charge and the potential energy U
increases.
▫ If a negative charge moves in the direction of the
electric field E, the electric field does negative
work on the charge and the potential energy U
increases.
▫ If a negative charge moves in the direction opposite to
the direction of the electric field E, the electric field
does positive work on the charge and the potential
energy U decreases; K of the particle increases.
• Whether the test charge is positive or negative:
▫ U increases if the test charge qo moves in the direction
opposite to the electric force F = qo·E.
▫ U decreases if the test charge qo moves in the same
direction as F = qo·E.
Electric Potential Energy of Two Point Charges
• To calculate the work done
on a test charge qo moving
in the electric field caused
by a single stationary point
charge q.
• For a displacement along
the radial line from point a
to point b, the force on qo is
given by Coulomb’s law:
k  q  qo
F
2
r
• If q and qo have the same
sign, the force F is repulsive
and F is positive; if q and qo
have opposite signs, the
force F is attractive and F is
negative.
• The force F is not constant
during the displacement
from point a to point b, so
we have to integrate to
determine Wab:
Wa b 

rb
ra
F  dr  
rb
ra
Wa b  k  q  q o  
rb
ra
rb 1
k  q  qo
 dr  k  q  q o   2  dr
2
ra r
r
rb
 r

r  dr  k  q  q o  


2

1

 ra
21
-2
rb
r
Wa b
b
r 
  1
 k  q  qo     k  q  qo   
 r  ra
  1 ra
Wa b
  1  1
1 1 
 k  q  qo  

 
  k  q  qo  
ra 
 rb
 rb ra 
Wa b
1 1
 k  q  qo    
 ra rb 
1
• The work done by the electric force for a particular
path depends only on the end points.
• The work is the same for all possible paths from a
to b.
• Consider the displacement in which a and b do not
lie on the same radial line:
• Work done on qo during the displacement given by:

rb k  q  q
o
Wa b   F  cos θ  dl  
 cos θ  dl
2
ra
ra
r
cos θ  dl  dr
rb k  q  q
o
Wa b  
 dr
2
ra
r
rb
• The work done during a small displacement dl
depends only on the change dr in the distance r
between the charges, which is the radial component
of the displacement.
• The work done on qo by the electric field E produced
by q depends only on the displacement between ra
and rb, not on the path from ra to rb.
• If qo returns to its starting point a by any path, the
total work done is zero (displacement is zero);
integral from ra to ra = .
• The potential energy Ua when qo is at any distance r
k  q  qo
from q is:
Ua 
r
▫ The potential energy is positive when q and qo have
the same sign and negative if they have opposite signs.
• Potential energy is always defined relative to some
reference point where U = .
▫ U =  when q and qo are infinitely far apart (r = ∞).
▫ U represents the work that would be done on the test
charge qo by the electric field of q if qo moved from an
initial distance r to infinity.
• Left figure: If q and qo have the same sign, the force is
repulsive and the work done by the field of q is positive and
U is positive at any finite separation.
• Right figure: If q and qo have opposite signs, the force is
attractive and the work done by the field of q is negative and
U is negative at any finite separation.
Electric Potential Energy with Several Point
Charges
• If qo moves in an electric field E produced by
point charges q1, q2, and q3 at distances r1, r2, and
r3 from qo:
the total electric field at
each point is the vector
sum of the fields due to
the individual charges, and
the total work done on qo
during any displacement is
the sum of the contributions
of the individual charges.
• Work and potential energy are scalar quantities,
not vector quantities; keep the negative signs!
• The potential energy associated with test charge qo
at point a is the algebraic sum:
 q1 q 2 q 3 

U  k  q o   

r3 
 r1 r2
• Two viewpoints on electric potential energy:
▫ In terms of the work done by the electric field on a
charged particle moving in the field, when a particle
moves from point a to point b, the work done on it by the
field is Wab = Ua – Ub.
▫ The potential energy difference Ua – Ub is equal to the
work that is done by the electric force when the particle
moves from point a to point b.
▫ When Ua is greater than Ub, the field does positive work
on the particle as it “falls” from a point of higher
potential energy a to a point of lower potential energy b.
▫ Alternative: consider how much work we would have to
do to “raise” a particle from a point b where the potential
energy is Ub to a point a where it has greater potential
energy Ua (ex. Pushing 2 protons closer together).
▫ To move the particle slowly so as not to give it any
kinetic energy, we need to exert an additional external
force that is equal to and opposite in direction to the
electric field force and does positive work.
▫ The potential energy difference Ua – Ub is then defined
as the work that must be done by an external force to
move the particle slowly from point b to point a against
the electric force.
▫ Because the external force is the negative of the electric
field force and the displacement is in the opposite
direction, the potential difference Ua – Ub is positive.
▫ This also works if Ua is less than Ub, corresponding to
“lowering” the particle (ex. moving 2 protons away
from each other); the potential difference Ua – Ub is
negative.
Electric Potential
• Electric potential is potential energy per unit charge
(J/C).
• Electric potential V at any point in an electric field is
the potential energy U per unit charge associated with
a test charge qo at that point:
U
V
qo
• Potential energy and charge are both scalar
quantities, so electric potential is a scalar too..
• SI unit of electric potential is the volt; 1 V = 1 J/C
• Dividing Wab by qo:


Wa b  Ub  Ua

 Vb  Va   Va  Vb
qo
qo
• Va is the potential at point a and Vb is the potential
at point b.
• The work done per unit charge by the electric force
when a charge moves from a to b is equal to the
potential at a minus the potential at b.
• The difference Va – Vb is called the potential of a
with respect to b; sometimes abbreviated as
Vab = Va – Vb and called the potential difference
between a and b.
• In electric circuits, the potential difference
between 2 points is called voltage.
Wa b
 Va  Vb
qo
• Vab, the potential of a wrt b, equals the work done
by the electric force when a unit charge (1 C)
moves from a to b.
• Alternative: Vab, potential of a with respect to b,
equals the work that must be done to move a unit
charge (1 C) slowly from b to a against the electric
force.
• Instrument used the measure the potential
difference (voltage) between two points is called a
voltmeter
What Good is the Electric Potential?
Two essential ideas:
• Electric potential depends only on the source charges and
their geometry. Potential is the “ability” of the source
charges to have an interaction if a charge q shows up. This
ability, or potential, is present throughout space regardless
of whether or not charge q is there to experience it.
• If we know the electric potential V throughout a region of
space, we know the potential energy U = q·V of any charge
that enters the region of space.
• Charged particles speed up or slow down as they move
through a region of changing potential.
• Conservation of energy in terms of electric potential:
Ki + q·Vi = Kf + q·Vf
Calculating the Potential from the Field
• The potential difference between any initial point i
and final point f can be determined if we know the
electric field vector E along any path connecting i
and f.
• Find the work done on a positive test charge by the
electric field as the charge moves from i to f.
• For the electric field represented by the field lines in
the figure, a positive test charge qo moves along the
path shown from point i to point f.
• At any point on the path, a force qo·E acts on the
charge as it moves through a differential
displacement ds.
• The differential work dW done on the charge by the
force during the displacement ds is:
dW = F•ds = qo·E·ds
• The total work W done on the particle by the electric
field as the particle moves from point I to point f is
the sum of the differential works done on the charge
as it moves through all the displacements ds along
the path:

W  qo  
f
i

E  ds
• From:
W
ΔV 
qo
;
ΔV  Vf  Vi


Vf  Vi    E  ds
f
i
• Electric field lines point from regions of high
potential to regions of low potential, thus the
negative sign.
• The potential difference Vf – Vi between any two
points i and f in an electric field is equal to the
negative of the line integral of E•ds from i to f.
• Because the electrostatic force is conservative, all
paths produce the same result.
f 
 Allows us to determine the
•
Vf  Vi   E  ds
i
difference in potential between any two points in
the electric field.
f 

▫ If we set Vi = , then V   E  ds gives us the
i
potential V at any point f in the electric field relative
to the zero potential at point i.
• The negative sign indicates that point b is at a
lower potential than point a; that is, Vb < Va.
• As a test charge qo moves from a to b, the
change in the potential energy is:

ΔU  q o  ΔV  q o  E  d
• If qo is positive, DU is negative.
• For a positive charge in an electric field, the
charge will lose electric potential energy when it
moves in the direction of the electric field as it
gains kinetic energy due to the electric force
qo·E it experiences in the direction of E.
• The loss of electric potential energy is equal to
the gain in kinetic energy.
• If the test charge is negative, then DU is positive. The
test charge gains electric potential energy when it
moves in the direction of the electric field.
• If a negative charge is released in the electric field, it
accelerates in the direction opposite the electric field.
The Accelerations of Positive and
Negative Charges
• A positive charge will accelerate from a region of
higher potential toward a region of lower
potential.
• A negative charge will accelerate from a region
of lower potential to a region of higher potential.
• For example: a 12 V car battery connected to
the headlights. The positive terminal has a
potential that is 12 V higher than the potential
at the negative terminal. Positive charges are
repelled from the positive terminal and travel
through the wires and headlight toward the
negative terminal.
• As the charges pass through the headlight,
almost all their potential energy is converted
into heat, which causes the filament to glow
“white hot” and emit light. When the charges
reach the negative terminal, they no longer
have any potential energy.
• The battery then provides the charges with
additional potential energy by moving them to
the higher potential terminal, and the cycle is
repeated. In raising the potential energy of
the charges, the battery does work WAB on
them, and draws from its reserve of chemical
energy to do so.
• Historically, it was believed that positive
charges flow in the wires of an electric
circuit. We now know that negative charges
flow in wires from the negative terminal
toward the positive terminal. It is very
common, particularly when referring to the
direction of current flow using Kirchhoff’s
laws, to describe the flow of negative
charges by specifying the opposite, but
equal, flow of positive charges. The
hypothetical flow of positive charges is
called the conventional electric current.
Electric Potential of a Point Charge
• The electric potential due to a point charge Q
is given by:
kQ
V
r
• If Q is positive, the potential is positive; if Q is
negative, the potential is negative.
• Voltage is a scalar quantity, therefore, only
the magnitude of the potential difference is
important, not the direction.
Equipotential Surface
• Equipotential surfaces are surfaces on
which the potential is everywhere the
same. No work is done in moving a charge
over an equipotential surface. The electric
field at an equipotential surface must be
perpendicular to the surface since
otherwise there would be a component of
the field and also therefore an electric force
parallel to the equipotential surface. Then
work would be done in moving charges over
the surface and the surface would therefore
not be an equipotential surface.
Equipotential Surface
• Equipotential Surface: A
circle of radius "r" around a
charge where the electric
field strength is the same,
therefore, the potential is
the same. An equipotential
surface consists of a
continuous distribution of
points having the same
potential. On the diagram,
the equipotential lines are
the dotted lines around the
charges.
Equipotential Surface
• Work is done by or
against the field
only when a charge
is moved from one
equipotential point
to another. This is
analogous to
moving one from
level of gravitational
potential energy to
another.
Equipotential Surface
• When moving a
charge along an
equipotential line,
DV = 0 V, therefore,
DU = qo·DV = qo·0 V.
• A unit of energy commonly used in atomic
and nuclear physics is the electron-volt. An
electron-volt (eV) is the energy that an
electron (or proton) gains when moving
through a potential difference of 1 V.
1 eV = 1.602 x 10-19 J