ap physics: electricity & magnetism. what is potential energy? the ability to do work the energy...
TRANSCRIPT
POTENTIAL ENERGY
What is potential energy?
The ability to do work
The energy possessed by an object due to its
position in a force field.
WHAT IS WORK?
Work is an energy transfer
A constant force applied over a distance in the same
direction.
W F d
WORK AND POTENTIAL ENERGY
W Fsi
sf ds UU i U f
The work done by a conservative force equals the decrease in the potential
energy of the particle.
WORK DONE BY AN ELECTRIC FIELD
W FcA
B ds U
W q0EA
B ds U
U q0 EA
B ds
This is called a line or path integral which is independent of the path taken from point A
to B.
THE CHANGE IN POTENTIAL ENERGY PER UNIT CHARGE:
U
q0
EA
B dsV
This value is called Electric
Potential and is independent of the value of q
VU
q0
The potential at any point in the
field:
ELECTRIC POTENTIAL:
Be careful, electric potential is not potential energy.
The work per unit charge that an external agent must perform to move a test charge
from A to B without a change in kinetic energy
Units? J/C
POTENTIAL AT A POINT P
VP E
P ds
The electric field at an arbitrary point equals the work required per unit charge
to bring a positive test charge from infinity to that point
THE ELECTRON VOLT
The energy that an electron or proton gains or loses by moving through a potential difference
of 1V
1.60210 19 J 1eV
POTENTIAL DIFFERENCES IN A UNIFORM ELECTRIC FIELD:
What is the potential difference between two points if the displacement is parallel to the field
lines?
VVB VA E dsA
B
E cos0dsA
B EdsA
B E dsA
B
V Ed
EQUIPOTENTIAL SURFACE
The name given to any surface consisting of a
continuous distribution of points having the same
electric potential.
No work is done in moving a test charge between any two points on an equipotential
surface.
EQUIPOTENTIAL SURFACE Red lines are electric field lines Blue lines are equipotential surfaces Equipotential surfaces must be
perpendicular to field lines.
GAIN OR LOSS?A negative charge gains electric
potential energy when it moves in the direction of the electric field. Explain
this.A proton loses potential energy when moving in the direction of the electric field, but picks up an equal amount of
kinetic energy.Work done by a field is positive when energy is given to an object from the
field (object is lowered in a gravitational field or moved in the direction of E filed lines). This occurs when Uf <Ui so, W=-
ΔU
THE GAP BETWEEN ELECTRODES IN A SPARK PLUG IS 0.06CM, TO PRODUCE AN ELECTRIC SPARK IN A GASOLINE AIR MIXTURE, AN E FIELD OF 3X106V/M MUST BE ACHIEVED. WHEN STARTING THE CAR, WHAT MINIMUM POTENTIAL DIFFERENCE MUST BE SUPPLIED BY THE IGNITION CIRCUIT?
d0.0006m
E 3106V /m
VEd
VEd
V 3106V /m 0.0006m
V1.8kV
CALCULATE THE SPEED OF A PROTON THAT IS ACCELERATED FROM REST THROUGH A POTENTIAL DIFFERENCE OF 120V. AN ELECTRON?
V120V
v0 0
v?
mp 1.6710 27kg
me 9.1110 31kg
e1.602 19 C
VU
q
qVUKE 1
2mv2
v2qV
m
vp 1.5105m/s
ve 6.5106m/s
AN ION ACCELERATED THROUGH A POTENTIAL DIFFERENCE OF 115V EXPERIENCES AN INCREASE IN KINETIC ENERGY OF 7.37X10-17 J.CALCULATE THE CHARGE OF THE ION
V115V
KE 7.3710 17 J
q?
qVUKE
qKE
V
q6.410 19C
HOW MUCH WORK IS DONE IN MOVING AVOGADRO'S NUMBER OF ELECTRONS FROM AN INITIAL POINT WHERE THE ELECTRIC POTENTIAL IS 9V TO A POINT WHERE THE POTENTIAL IS -5V?
VA 9V
VB 5V
Na 6.021023electrons
W ?
VVA VB
V9V ( 5V)
V14V
W UqV
qNae(6.021023)( 1.60210 19C)
q 96440.4C
W qV( 96440.4C)(14V)
W 1.35MJ
POTENTIAL ENERGY DUE TO A POINT CHARGE?
What is the electric field due to a point charge (q)?
E dAQ in
0
EAQ in
0
Field is radially outward (or inward), and the Gaussian surface is a sphere chosen to match symmetry of field.
E Q in
A0
q
4r20
kq
r2
POTENTIAL ENERGY DUE TO A POINT CHARGE?
E kq
r2
VVB VA E dsrA
rB
The following simplification can be made because the potential difference is not dependant upon the path taken through the field, only the endpoints.
V EdrrA
rB
Vkq
r2 drrA
rB
V kq1
r2 drrA
rB
V kq 1
r
rA
rB
POTENTIAL ENERGY DUE TO A POINT CHARGE?
If we want to know the potential at one particular location near a point charge we chose the first point to
be infinity.
V kq 1
r
rA
rB
Vkq1
r
rB
Vkq1
r
1
Vkq
r
SUPERPOSITION PRINCIPLE
Vkq
r
Vkqi
rii
Potential at a point due to many point charges.
This is a scalar value so it is much easier to calculate than E
field.
POTENTIAL ENERGY DUE TO MANY CHARGES.
Uq2V1 kq1q2
r12
The potential energy of a pair of charged particles separated by a distance r12. this is also the work done to move q2 from
infinity to point P where q1 is located. V1 is the electric potential at
point P due to charge q1.
Notice that if the charges are like then positive work is done to bring the charges together (they
have more energy together because they will repel). Energy delivered to the system requires
positive work.
FOR MULTIPLE CHARGES:
Ukq1q2
r12
q1q3
r13
q2q3
r23
Imagine q1 is at a fixed position. We want to bring q2 and q3 from infinity to a position near q1.
Each term represents the the work required to bring each charge to a location near the other. It does not matter which
order we bring them in from infinity.
q1 q2
q3
r12
r23
r13
YOU KNOW POTENTIAL AND YOU WANT TO KNOW ELECTRIC FIELD…
V EA
B ds
How can we use this to solve for E field?
dV E ds
If the E field only has one component…
dV E xdx
E x dV
dx
YOU KNOW POTENTIAL AND YOU WANT TO KNOW ELECTRIC FIELD…
Remember that the potential is zero for displacements perpendicular to the field
If your potential is a function of all three spatial coordinates (x, y, z)
E x dV
dx
E y dV
dy
E z dV
dz
V4x2y y2 2yz
To find the electric field, we must take the partial derivative of this potential…
E x Vxx
4x2y y2 2yz
E x x
4x2y
4yx
x2
4y2x8xy
Partial derivative: Treat other variables as constant while differentiating with respect to one variable.
E x,y,z 8xy i 4x2 2y 2z j 2y ˆ k
In vector notation this is often written as…
E Vx i
y
j
z k
is called the gradient operator
If the potential is constant in some region, what is the electric field?
If the electric field is zero in some region, what is the electric potential?
QUESTIONIf the potential is constant in some region,
what is the electric field?If the electric field is zero in some region,
what is the electric potential?
vconstant, E 0
E 0, vconstant
The difference in electric potential (voltage) measured when moving from point A to point B is equal to the work which would have to be done, per unit charge, against
the electric field to move the charge from A to B.
ELECTRIC FIELD DUE TO CONTINUOUS CHARGE DISTRIBUTIONS.
The electric potential at point P due to a continuous charge distribution can be
calculated by dividing the charged body into segments of charge dq and summing the potential contributions over all segments.
Vkq
r
Vkdq
r
POTENTIAL DUE TO A UNIFORMLY CHARGED RING:
P
a
x
dq
Vkdq
r
r a2 x2
All segments of charge are equidistant from point P
Vk
a2 x2dq
VkQ
a2 x2
POTENTIAL DUE TO A UNIFORMLY CHARGED RING AT THE CENTER:
P
a
dqThe field is zero here, therefore the potential
must be constant:
VkQ
a
this potential is the work necessary to bring a test charge from infinity to the location in
the center of the ring.
POTENTIAL OF A CHARGED CONDUCTOR
A
B
This oddly shaped conductor with an
excess positive charge is in equilibrium
Along this surface path E is always
perpendicular to the displacement ds.
E ds0
VB VA E dsA
B 0
POTENTIAL OF A CHARGED CONDUCTOR
The surface of any charged conductor in equilibrium is an equipotential surface.
Furthermore, since the electric field is zero inside the conductor, we conclude that the potential is constant everywhere inside the conductor and
equal to its value at the surface.
A
B
The electric field is large near points having small convex radii of curvature and reaches
very high values at sharp points.
A CAVITY WITHIN A CONDUCTOR
AB
A cavity in a conductor with no charge in it…The electric field inside the cavity must be
zero, regardless of the charge distribution on the outside surface of the conductor.
Because the potential on the surface of the cavity
is an equipotential surface.
ELECTRIC SHIELDING Sensitive circuits and people can be
protected if placed in a cavity inside of a conductor.
CORONA DISCHARGE:
Air can become a conductor as a result of the ionization or air molecules in regions
of high electric fields. At STP this happens around 3x106
V/m
PRACTICE QUIZ 1
.A .B
.C
.D
+2Q
-Q-Q2a
Three charges are arranged in an equilateral triangle, as shown. At which of
these points (a,b,c bisect sides, d is equidistant from other points) is the
electric potential smallest?
.E
PRACTICE QUIZ 1
.A .B
.C
.D
+2Q
-Q-Q2a
C) A small positive test charge will move towards an area of low potential. Ask
yourself “Where would an small positive charge end up if released near these
charges?”
.E
PRACTICE QUIZ 2
10V
The diagram shows a set of equipotential surfaces. At point P, what is the direction
of the electric field?
40V 70V
.P
A) left B) right
C) up the slide D) down the slideE) either left or right, which one cannot be
determined
PRACTICE QUIZ 2
10V
A) A small positive charge placed at P would move to a location of low potential (left). The force that moves it is caused by an electric field which will be in the
same direction or opposite it. Equipotential surfaces are always
perpendicular to electric field lines.
40V 70V
.P
PRACTICE QUIZ 3
What is the electric potential at a point halfway between the two
charges?
a
A) kQ/a
B) 2kQ/a
C) zero
D) 4kQ/a
E) 8kQ/a
D) Electric potential is a
scalar
VkQa
2
kQa
2
2kQ
a
2
4kQ
a
PRACTICE QUIZ 4
A solid conducting sphere carries a charge +Q. Which of the following
is true of the electric field E and the electric potential V inside the
sphere?A) E=0 and V=0
B) E=0 and V≠0
C) E≠0 and V=0
D) E≠0 and V≠0
E) It cannot be determined without knowing the radius of the sphere.
B) The field in a conductor is always zero. E is the derivative of V, so to be
zero, V must have been a constant.
E dV
dr
PRACTICE QUIZ 5A negatively charged rod is brought near a metal
object on an insulating stand, as shown. When charges stop moving, the left side of the object has an excess of positive charge, and the right
side of the object, where the radius of curvature is less, has an excess of negative charge. Which
of the following best describes the electric potential on the metal object?
- - - - - - - ---
+++ +
A) It is greatest on the + sideB) It is greatest on the – sideC) It is greatest at the centerD) It is the same everywhere on the objectE) It cannot be determined from the information given
PRACTICE QUIZ 5
- - - - - - - ---
+++ +
A) It is greatest on the + sideB) It is greatest on the – sideC) It is greatest at the centerD) It is the same everywhere on the objectE) It cannot be determined from the information given
D) It is the same everywhere on the object. The surface of a conductor is an equipotential surface. The E field is 0
inside the conductor and V is constant.
PRACTICE QUIZ 6
L
Four identical charged particles, each with charge Q, are fixed in place in the shape of an equilateral pyramid with sides of length L, as shown above.
What is the potential energy of this arrangement of charges?
.Q
.Q
.Q
.Q
L
L
L
L
L
PRACTICE QUIZ 6
L
.Q
.Q
.Q
.Q
L
L
L
L
L
Each pair of charges is separated by a distance L.
Each pair contributes:
There are 6 pairings:
UQ2
40L
U6Q2
40L
Uq1q2
40rIn general:
PRACTICE QUIZ 7 What is the amount of work required to
assemble 4 identical point charges of magnitude Q at the corners of a square of side s?
s
sQ Q
Q Q
2s
U kQ2 1
s
1
s
1
s
1
s
1
2s
1
2s
U kQ2 4
s
2
2s
UkQ2
s4
2
2
U5.41kQ2
s
PRACTICE QUIZ 8
A wire that has a uniform linear charge density λ is bent into the shape shown
below. Find the electric potential at point O.
2R
2R
R.O