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Transcript
```Physics C
Electric Potential
Is the electrical force conservative or nonconservative?
Name:__________________
Electric Potential
Definition:
What characterizes a conservative force?
Mathematical relationship to potential
energy:
How does the change in a electrical
potential energy relate to work?
Write an equation that expresses the
relationship between electrical potential
energy and work done by an electrostatic
force.
High
Potential
(voltage)
+
+
+
+
+
A
+
+
+
+
+
+
+
+
+
+
+
Field Refresher
Equi-potential Surfaces
B
+
C
+
Sample Problem: A proton is accelerated
through a potential difference of -20,000 V.
What is the potential and kinetic energy
change of the proton? How much work did
the electric field do on the proton?
Low
Potential
(voltage)
-
Sample Problem: How fast is a 2 MeV alpha
particle moving? What potential difference is
needed to stop this alpha particle?
Electric Field
The work done by an electrical force in
moving a charge q from point A to point B is
given by:
Uniform field:
Sample Problem: An electric field is given by
E = 250 i V/m. A 12.0 C charge moves
from the origin to (0.20 m, 0.50 m). What is
the change in potential energy and electric
potential for the charge?
Non-uniform field:
This results in a potential energy change of
Uniform field:
Non-uniform field:
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Bertrand
Sample Problem: An electron is released
from rest in an electric field of magnitude by
5900 V/m. Through what potential difference
will it have passed after moving 1.00 cm?
How fast will it be moving at this time?
Sample Problem: Two point charges (5.0 nC
and -3.0 nC) are separated by 35 cm. What
is the potential energy of the pair? What is
the electric potential at a point midway
between the charges?
Equi-potential or Iso-potential surfaces
Definition:
Mathematically, how do you determine the
electric field from the electric potential if you
know how the potential changes with
position?
Characteristics:
Draw the field lines and equi-potential
surfaces around a positive point charge
What can you say about how field strength
relates to the rate at which the electric
potential is changing?
Sample Problem: Find the potential at a
distance of 1.00 cm from a proton. Repeat
for an electron.
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Sample Problem: The electric potential in a
region is given by the function
V = -9.0 x V/m – 3.0 x2 V/m2.
What is the magnitude and direction of the
electric field at x = 2.0 m?
2
Bertrand
Sample Problem: Over a region of space,
the electric potential is given by
V = -3.0 x + 6.0 x y – 2.0 x2y.
Derive the vector representing the electric
field at (1, 2) m.
Sample Problem: Find the potential at a
point inside a charged non-conducting solid
sphere of radius R as a function of its
distance from the center of the sphere.
Assume charge Q is distributed uniformly.
Equation for potential due to multiple point
charges:
Where do conductors bear excess charge?
Equation for potential due to multiple point
charges:
Does it matter if the conductor is solid or
hollow?
Sample Problem: Determine the electric
potential at a point P located on the
perpendicular axis of a uniformly charged
ring of radius R and total charge Q.
Sample Problem
Two charged conductors are connected by a
long conducting wire, and a charge of 10 nC
is placed on the combination. Sphere A has
a diameter of 10 mm, and sphere B has a
diameter of 5 mm. How much charge is on
each sphere? What is the electric potential
of each sphere?
Sample Problem: Determine the electric
potential along the perpendicular central
axis of a uniformly charged disk of radius R
and surface charge density .
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Bertrand
What is a Capacitor?
Parallel plate capacitance equation:
Deriving Capacitance -- Steps
What does it mean when we say a capacitor
is “charged”?
1.
2.
3.
4.
5.
6.
Draw a parallel plate capacitor
Draw the capacitor; identify symmetry
Draw Gaussian surface
Write Gauss’ Law
Solve Gauss’ Law for E
Develop function for V from E
Develop function for C from V
Derive C for Parallel Plate Capacitor
Draw a cylindrical capacitor
Derive C for Charged Sphere
Capacitance equation:
Upon what does capacitance depend?
Sample Problem: How much charge is on
each plate of a 4.00
connected to a 12-V battery?
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Bertrand
Equivalent capacitance – series capacitors
Derive C for a Spherical Capacitor
What is the same for all capacitors in
series?
Why is the above statement true?
Equivalent capacitance – parallel capacitors
What is the same for all capacitors in
parallel?
Why is the above statement true?
Derive C for a Cylindrical Capacitor
Sample problem: Determine equivalent capacitance of the
configuration shown
C
C
C
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C
C
C
Bertrand
What is a dielectric?
Sample problem: Determine equivalent capacitance between a
and b. If the potential difference between a and b is 10V, what
charge is stored on C3? (C1=5F, C2=10F, C3=2F)
C1
a
C1
C3
C2
C2
C2
How is capacitance changed when a
dielectric is inserted between the capacitor
plates? Write the relationship here.
C2
b
How does a dielectric work?
How do capacitors store energy?
Sample Problem
Find the capacitance of a parallel plate
capacitor that uses Bakelite as a dielectric, if
each of the plates has an area of 5.0 cm 2
and the plate separation is 2.00 mm.
Equation for energy stored in capacitor:
Sample Problem: A 3.00 mF capacitor is
connected to a 12.0 V battery. How much
energy is stored in the capacitor?
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Bertrand
Physics C
Electric Potential
Name:__________________
FR Problem 1
A006 E1. The square of side a at right contains a positive point charge +Q fixed at
the lower left corner and negative point charges -Q fixed at the other three corners of
the square. Point P is located at the center of the square.
a. On the diagram, indicate with an arrow the direction of the net electric field at
point P.
b.
Derive expressions for each of the following in terms of the given quantities and fundamental
constants.
i. The magnitude of the electric field at point P
ii. The electric potential at point P
c.
A positive charge is placed at point P. It is then moved from point P to point R, which is at the
midpoint of the bottom side of the square. As the charge is moved, is the work done on it by the
electric field positive, negative, or zero?
Positive
Negative
Zero
Explain your reasoning.
d.
i: Describe one way to replace a single charge in this configuration that would make the electric
field at the center of the square equal to zero. Justify your answer.
ii.
Describe one way to replace a single charge in this configuration such that the electric potential
at the center of the square is zero but the electric field is not zero. Justify your answer.
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Bertrand
FR Problem 2
S116 E1. Consider the electric field diagram above.
a. Points A, B, and C are all located at y = 0.06 m .
i. At which of these three points is the magnitude of the electric field the greatest? Justify your answer.
ii. At which of these three points is the electric potential the greatest? Justify your answer.
b.
An electron is released from rest at point B.
i. Qualitatively describe the electron's motion in terms of direction, speed, and acceleration.
ii. Calculate the electron's speed after it has moved through a potential difference of 10 V.
c.
Points B and C are separated by a potential difference of 20 V. Estimate the magnitude of the electric
field midway between them and state any assumptions that you make.
d. On the diagram, draw an equipotential line that passes through point D and intersects at least three
electric field lines.
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Bertrand
FR Problem 3
S336E1. A spherical cloud of charge of radius R contains a total charge +Q with a non-uniform volume
charge density that varies according to the equation
(r) = o(1 – r/R) for r < R and
 = 0 for r > R,
where r is the distance from the center of the cloud. Express all algebraic answers in terms of Q, R,
and fundamental constants.
a. Determine the following as a function of r for r > R .
i. The magnitude E of the electric field
ii.
The electric potential V
b.
A proton is placed at point P shown above and released. Describe its motion for a long time after its
release.
c.
An electron of charge magnitude e is now placed at point P, which is a distance r from the center of
the sphere, and released. Determine the kinetic energy of the electron as a function of r as it strikes the
cloud.
d.
Derive an expression for o .
e.
Determine the magnitude E of the electric field as a function of r for r < R .
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Bertrand
FR Problem 4
S118E2.
In the figure at right, a non-conducting solid sphere of radius a with charge +Q
uniformly distributed throughout its volume is concentric with a non-conducting
spherical shell of inner radius 2a and outer radius 3a that has a charge –Q uniformly
distributed throughout its volume. Express all answers in terms of the given
quantities and fundamental constants.
(a) Using Gauss's law, derive expressions for the magnitude of the electric field as a
function of radius r in the following regions.
i. Within the solid sphere (r < a )
ii. Between the solid sphere and the spherical shell (a < r < 2a )
iii. Within the spherical shell (2a < r < 3a )
iv. Outside the spherical shell (r > 3a )
(b) What is the electric potential at the outer surface of the spherical shell (r = 3a )? Explain your
reasoning.
(c) Derive an expression for the electric potential difference Vx – Vy between points X and Y shown in the
figure.
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Bertrand
FR Problem 5
A277E1. An isolated conducting sphere of radius a = 0.20 m is at a
potential of -2,000 V.
a. Determine the charge Q0 on the sphere.
The charged sphere is then concentrically surrounded by two uncharged conducting hemispheres of inner
radius
b = 0.40 m and outer radius c = 0.50 m, which are joined together as shown above, forming a
spherical capacitor. A wire is connected from the outer sphere to ground, and then removed.
b. Determine the magnitude of the electric field in the following regions as a function of the distance r
from the center of the inner sphere.
i. r <a
ii. a < r < b
iii. b < r < c
iv. r > c
c.
Determine the magnitude of the potential difference between the sphere and the conducting shell.
d.
Determine the capacitance of the spherical capacitor.
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Bertrand
FR Problem 6
A277E3. The nonconducting ring of radius R shown at right lies in the yz-plane and
carries a uniformly distributed positive charge Q.
a. Determine the electric potential at points along the x-axis as a function of x.
b.
i. Show that the x-component of the electric field along the x-axis is given by
Ex 
Qx
3
4 0  R 2  x 2  2
ii. What are the y- and z- components of the electric field along the x-axis?
c.
Determine the following.
i. The value of x for which Ex is a maximum
ii. The maximum electric field Ex max
d.
On the axes below, sketch Ex versus x for points on the x-axis from x = -2R to x = +2R.
e.
An electron is placed at x = R/2 and released from rest. Qualitatively describe its subsequent motion.
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Bertrand
Use this page for
Extra Notes
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Bertrand
Use this page for
Extra Notes
5/7/2017
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Bertrand
```
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