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Electric Fields
We have studied many forces up until this point which are contact forces. Gravitation and
the electric force are not contact forces. Objects do not have to touch for these forces to
act. They can act over a distance.
The idea of a field was devised to help explain how these forces can act over a distance.
The field refers to the area surrounding an object. In the case of an electric field, the field
extends outward from every charge and permeates the all of space. When a second charge
is placed in this field, it feels a force because of the electric field at that point.
Electric field
The area around a charged object. This field exerts a force on any charged object
in its vicinity. The closer the charged object is brought to the charged object
creating the field, the greater the force exerted on it.
An Animation Explaining Electric Fields
http://www.sciencejoywagon.com/physicszone/lesson/07elecst/fieldint/efield.htm
Test charge
a positive charge of very small magnitude. The test charge is used to determine
the direction of the electric field. The electric field is deined as the force on a test
charge with the test charge being so small that it approaches zero. Defining the
electric field in this manner means that the electric field only describes the effect
of the charges creating the electric field at that point.
Electric field lines
The electric field can be represented with electric field lines. Their density is a
measure of the strength of the electric field at that point. Their direction is one
that a positive test charge would take in the field. Field lines are always directed
from the positive charge and toward a negative charge.
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The fild lines indicate the direction of the electric field; the field points in
the direction tangent to the field line at any point.
Electric field lines start on a positive charge and end on a negative charge.
The number of lines starting on a positive charge, or ending on a negative
charge, is proportional to the magnitude of the charge.
The closer the lines are drawn together, the stronger the electric field is in
that region.
The field lines between two parallel plates are parallel and equally spaced,
except near the edges. Thus the electric field is uniform between two
parallel plates (except at the edges).
The field lines indicate the direction of the electric field.
Electric field lines never cross. To do so would imply that a positive test
charge would have two directions simultaneously at that point.
Electric field lines are drawn perpendicular to a surface outside of a
conductor.
Electric Field mapping
http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html
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Locate one or more charge by clicking in the screen.
The applet will draw the equipotential lines and the electric field lines
(which are white).
Locate charges and draw field lines http://www.gel.ulaval.ca/~mbusque/elec/
Electric field strength (or intensity)
symbol is E and SI unit is N/C; the force on a test charge.
Electric field strength of a point charge:
where E is the electric field, k is Coulomb’s constant, and d is the distance
between the charge and the test charge. Note: the force depends only upon the
magnitude of the point charge producing the field, not on the value of the test
charge.
Electric field is a vector quantity; it has both magnitude and direction. The
resultant electric field due to several point charges can be determined using the
same method as was used in Coulomb's Law problems. Calculate the strength of
the electric field due to each point charge at point P. Determine the direction of
the electric field by determing the direction that a test charge placed at point P
would take. Use vectors to determine the magnitude and direction of the resultant
electric field at point P.
Superposition principle for electric fields If the field is due to more than one
charge, the resultant field at at point is found by adding the individual fields due
to each charge at that point vectorially.
Electric Fields and Conductors
 The electric field inside a good conductor is zero when the charges are at rest.
 Any net charge on a good conductor is distributed equally on the surface.
 The electric field is always perpendicular to the surface outside of a conductor.
 Inside a nonconductor (which does not have electrons free to move), an electric
field can exist. And, the electric field outside a nonconductor does not necessarily
make a perpendicular angle to the surface.
AP Multiple Choice questions:
1. Electric field questions are very common on the AP multiple choice test.
2. You may be asked to describe how the charge is distributed inside, on, or
around a charged sphere.
3. You may be asked the magnitude of the electric field inside, on, or around
a charged sphere. Or, you may be given several points around, on, or
inside a charged sphere and asked where the electric field is the greatest or
the least.
4. You may be asked to perform a simple calculation predicting the
magnitude of the electric field at some distance from a point charge.
5. You may be given a drawing indicating several distances between or
outside of two charged spheres (usually along the x-axis) and asked where
the field is zero, the least, the greatest, etc. Remember, E=kq/d2 where q is
the charge of the point charge. Remember, mentally put a positive test
charge at that point to determine the direction of the electric field at that
point.
6. You may be asked to perform simple calculations involving the electric
field. Remember, F=Eq.
7. You may be asked to predict the magnitude and direction of the electric
field between two charged plates. Remember, this field is uniform. At all
points between the plates, if edge effects are ignored, the magnitude and
the direction are the same.
8. You may be asked to compare the magnitude of the electric field at one
location to another location. The charge and/or the distance may be
changed.
9. You may be given a diagram with multiple charges in a geometric
configuration (triangle or square) and asked to give the direction of the net
electric field at point P.
10. They may ask simple questions involving the characteristics of electric
field lines.
AP Free Response questions:
1. A very common problem is a charge hanging by a thread in an electric
field. You may be asked to calculate the electric field intensity or the
charge (depends on what is given). Remember, draw a free-body diagram
which only includes forces! The force exerted by the electric field on the
charge is given by F=Eq. If hanging from a thread, this acts horizontally.
It is offset by the horizontal component of the tension. The vertical
component of the tension is offset by the weight of the charge.
2. You may be given a configuration of charges in a triangle or a square. You
may be asked where to place an additional charge so that the net electric
field on it is zero. Or, you may be asked to calculate the net electric field
at point P (and its direction).
3. You may be given two charges along the x-axis and asked to calculate
where the electric field is zero due to the two charges.
4. You may be given a charge moving between two charged plates. The
charge enters the area between the two plates and initially is moving
horizontally. It is very common to be given the voltage. Remember, in a
uniform electric field, V=Ed. You can use this to calculate E. Also,
remember qV=W. The product of charge and voltage is thus energy. They
use this a lot to have you calculate a speed using 1/2mv2.
Electric Field Lines
http://www.sciencejoywagon.com/physicszone/lesson/07elecst/static/fieldmap.ht
m
Electric Potential
Two points are said to differ in electric potential if work is done to move a charge
from one point to another point in an electric field.
Potential (symbol is V; SI unit is volt)
work done on a charge; or the electric potential is the potential energy per unit
charge. Only differences in potential can be measured.
V=W/q
In a uniform electric field (a parallel plate capacitor):
V=Ed
where E is the electric field strength and d is the separation between the plates in
meters
Electric Potential Energy When a charge q moves from point B to point A in an
electric field, the change in electric potential energy is simply the negative of the
work done to move the same charge from point A to point B. Just as we defined
the electric field as the force per unit charge, we will define the electric potential
(or potential) as the potential energy per unit charge. If a point charge q has
electric potential energy of PEA at some point A, the electric potential at point A is
given by
VA = PEA / q
Since only differences in potential are measurable, the potential at point A would
simple be the difference in potential energy, or the work done, to move the charge
from some point B to point A.
VBA = VB - VA = WBA/q
Electric potential is a scalar term. When finding the electric potential due to a
collection of point charges, you need only add the potentials together with no
concern for direction. Include a sign for the potential corresponding to the sign of
the charge.
Clarifying the Difference Between Electric Energy and Potential
http://www.sciencejoywagon.com/physicszone/lesson/07elecst/potentil/epotenti.h
tm
Absolute Potential The electric potential at a distance r from a single point
charge can be derived from the expression for electric field due to a point charge.
Also called electric potential of a point charge. The expression for absolute
potential:
V=kq/r
Notes about Absolute Potential:
1. The potential of infinitiy is defined to be zero.
2. If a point charge is positive, the absolute potential of the charge is
positive. When moving a charge from infinity to this point, the potential
energy increases above a zero level.
3. If a point charge is negative, the absolute potential of the charge is
negative. When moving a charge from infinity to this point, the potential
energy decreased below a zero level.
4. To find the absolute potential of a configuration of multiple charges when
working problems, calculate the separate absolute potentials of each
charge. The absolute potential is thus negative if the charge is negative
and positive if the charge is positive. Add the absolute potentials with
these signs corresponding to the sign of the charge.
Electrostatic energy (U) for point charges can be found. It is simply the same
thing as "work" in the definition of voltage. Since the electric potential is defined
as the potential energy per unit charge, then the change in potential energy of
charge q moved between points a and b is simply equal to qVab. In other words,
U=qV
If dealing with point charges, U = qV becomes U = k (q1q2)/d. If one is trying to
find the total electrostatic energy due to a system of charges, one finds the sum of
the electrostatice energies between each charge. As in abolute potential, one
includes the sign of the charge.
Relationship Between Electric Potential and Electric Field One can describe
the effects of charge distribution using either electric field or electtric potential.
Electric potential can be easier to use than electric fields because it is a scalar
quantity rather than a vector quantity.
1. In a uniform field (such as between two parallel plates), the units for
electric field (N/C) can be written as V/m. We find that E = V/d in a
uniform field.
2. In a nonuniform field (such as produced by a point charge), the electric
field ina given direction at any point in space is equal to the rate at which
the electric potential changes over distance in that direction.
Equipotential Lines Just as electric field lines represented the electric field
around a charge, equipotential lines represent the electric potential about a charge.
In three dimensions, they become equipotential surfaces.
Equipotential Surface An equipotential surface is one on which all points are at
the same potential. The potential difference between any two points on an
equipotential surface is zero; there is no work done to move a charge between
these two points.
An Animation Showing Equipotential Lines
http://www.sciencejoywagon.com/physicszone/lesson/07elecst/potentil/potlines.ht
m
Characteristics of Equipotential Surfaces
1. No work is done to move a charge between two points on the same
equipotential surface.
2. Electric field lines are perpendicular to equipotential surfaces.
3. The surface of a conductor is an equipotential surface. (A conductor must
be entirely at the same potential in statics. There is no electric field within
a conductor in statics because otherwise an electron would experience a
force and would move.)
Electron Volt (eV)A unit used to deal with the energy of electrons. One electron
volt is defined as the energy acquired by an particle carrying a charge equal to
that of the electron as a result of moving through a potential difference of 1 V. It
is not an SI unit, just an easier unit to use than Joules sometimes.
1 eV = 1.6 x 10-19 J
Millikan’s oil drop experiment
accurately measured the charge of an electron
Eq=mg
where E is the electric field, q is the charge in Coulombs, and mg is the weight in
Newtons
AP Multiple Choice questions:
1. You may be asked to perform a simple calculation determining the electric
potential at some point P a distance d from a point charge (remember to
use V=kq/d).
2. Remember the definition of potential difference - it is the work done on a
charge. If they tell you how much work is done on a certain charge, you
can apply this definition to determine the potential difference.
3. They can give you two charges on the x-axis, noting several points either
outside or between them, and ask you at which point the electric potential
is the greatest, the least, or zero.
4. They may give you charges in a square or triangle and ask you what is the
potential of a charge held at point P. (Read notes above on electrostatic
energy, U).
AP Free Response questions:
1. They could give you an array of charges (in a square or a triangle) and ask
you to determine the electrostatic potential of the array at some point P.
They may also ask you to compare the work done to move a charge to this
point P compared to another array of charges.
2. A very common type of problem is to have a charge entering the region
between two parallel and charged plates of known voltage between them.
Again, remember the two important formulas, V=Ed and qV=W. The first
is used to calculate E when V and d are known. The second is used to
calculate speed when V is known.
3. In this same type of problem, they may ask you to draw the forces on the
charge using a free-body diagram. They may ask you to draw the path of
the charge between the two plates. They may apply mechanics concepts,
asking you to calculate speed, time, or vertical displacement of the charge
as it travels between the two plates. Remember, the vertical acceleration is
due to the electric field (you can ignor gravity because the charge isn't
between the plates long enough for it to have an effect). The horizontal
acceleration is zero so the time required to move through the region is
found using d=vt.
Sharing of Charge
In a conductor, charges move until all parts of a conductor are at the same
potential. If a large and a small sphere have the same total charge, the large sphere
will have a lower potential. If a large and a small sphere have the same potential,
the large sphere will have the greater charge.
Grounding
the potential of the earth is zero. Any object connected to the earth will have its
excess charge flow into the earth. It is considered to be grounded.
Electrostatic charges are only found on the outside of conductors.
Capacitors
Capacitor
a device (sometimes called a condenser) that stores charge in the electric field
between its plates. Each plate carries the same amount of charge, one plate being
negative and the other being positive. A potential difference exists between the
two plates.
Capacitance
symbol is C and SI unit is the Farad, F
q=CV
where q is the charge in Coulombs, C is the capacitance, and V is the potential
difference.
Capacitance for a parallel-plate capacitor Capacitance is a proportionality
constant. It is a constant for a given capacitor. It does not depend upon charge or
voltage. Its value only depends upond the structure and dimensions of the
capacitor itself. For a parallel-plate capacitor with plates of area A separated by a
distance d of air, the capacitance is given by: This relationship makes sense.
Plates with a larger area will have less repulsion between charges (they're further
apart) for a given amount of charge q. Thus, more charge can be held. A greater
separation means that the charge on each plate exerts less attractive force on the
other plate. Less charge is drawn from the battery, and the capacitance is less.
Notice the use of the permitivity of free space constant (We learned in our
previous unit how Coulomb's constant was related to the permitivity of free
space.)
Dielectric An insulating sheet found in most capacitors between the plates. A
dielectric allows higher voltages to be applied without charge crossing the gap. A
dielectric allows the plates to be placed closer together without touching, allowing
an increased capacitance. A dielectric increases the capacitance by a factor K,
which is known as the dielectric constant. For a parallel-plate capacitor, C = K Co,
where Co is the capacitance without the capacitor.
It requires energy to place charges on the plates of a capacitor. When the
capacitor is discharged, this electrical energy is released. The energy stored in a
capacitor is equal to the work done to charge it.
Energy = ½ C V2 = 1/2 q V
where C is the capacitance, q is the charge, and V is the voltage
When a DC voltage source is connected across an uncharged capacitor, the rate at
which the capacitor charges up decreases as time passes. At first, the capacitor is
easy to charge because there is little charge on the plates. But, as the charge
accumulates, more and more work is needed to move additional charges on the
plates because the plates already have charge of the same sign on them. As a
result, the capacitor charges exponentially, quickly at the beginning and more
slowly as the capacitor becomes fully charged. At any time, the charge on the
plates is given by:
Half-life
The time it takes the capacitor to reach half full is called the half-life and is
related to the time capacitive time constant in the following way:
half-life = RC ln 2
where R is resistance in ohms and C is capacitance in Farads
Important things to remember about capacitors on the AP test:
1. The electric field between the two charged plates is uniform, the same
magnitude and direction at all points, neglecting edge effects.
2. You can use the direction of the electric field to predict the path a charged
particle will take between the two plates. For example, if it is an electron
and the top plate is positive, it will be deflected upward.
3. Important formulas to remember: V=Ed and qV=W. The first can be used
to calculate the electric field intensity between the two plates. The second
can be used to determine the increase in kinetic energy of the particle as it
passes between the two plates. This allows you to calculate speed.
Capacitors in Series and Parallel Combinations in Circuits
Equivalent CapacitanceThe capacitance of a single capacitor that can be
substituted for a combination of capacitors
1. Parallel combination of capacitors:
o To find the equivalent capacitance, add the individual
capacitances.
2. Series combination of capacitors:
o To find the reciprocal of the equivalent capacitance, add the
reciprocals of the individual capacitances.
AP Multiple Choice questions:
1. There are a surprising number of capacitor questions on the AP test.
2. They ask questions in which the separtion between the plates and/or the
area of the plates is changed and how that affects charge and/or voltage.
3. They give you two capacitors in parallel (almost always, but sometimes in
series) and ask you what is the equivalent capacitance or how much charge
is stored in one of the capacitors. Remember, capacitors add in parallel
and q=CV.
4. You might be asked to calculate the energy stored in a capacitor.
AP Free Response questions:
1. Not very common except as two charged plates.
2. You could be given a capacitor of known capacitance and voltage and
asked to calculate the charge stored on it.
3. As part of the same problem, a dielectric may be inserted between the
plates of the capacitor. They will ask you what the potential difference is
(same) and what the electric field is (can be calculated using V=Ed).
4. They can ask you for calculations for a capacitor requiring that you know
C=kA/d.
AP Physics B - Electric Field Objectives
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Students should understand the the concept of electric field so they can:
1. define it in terms of force on a test charge.
2. calculate the magnitude and direction of the force on a positive or negative
charge placed in a specified field.
3. given a diagram on which an electric field is represented by flux lines,
determine the direction of the field at a given point. Identify locations
where the field is strong and where it is weak, and identify where positive
or negative charges must be present.
4. analyze the motion of a particle of specified charge and mass in a uniform
electric field.
Students should understand Coulomb's Law and the principle of superposition so
they can:
1. determine the force that acts between specified point charges, adn describe
the electric field of a single point charge.
2. use vector addition to determine the electric field produced by two or more
point charges.
Students should know the fields of highly symmetric charge distributions so they
can describe the electric field of parallel charged plates.
Students should understand the nature of electric fields in and around conductors
so they can:
explain the mechanics responsible for the absence of electric field inside a
conductor, and why all excess charge must reside on the surface of the conductor.
explain why a conductor must be an equipotential, and apply this principle in
analyzing what happens when conductors are joined by wires.
determine the direction of the forces on a charge particle brought near an
uncharged or grounded conductor.
Student should be able to describe and sketch a graph of the electric field and
potential inside and outside a charged conducting sphere.
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Students should understand induced charge and electrostatic shielding so they
can"
1. describe qualitatively the process of charging by induction.
2. determine the direction of the force on a charged particle brought near an
uncharged or grounded
AP Physics B - Electric Potential Objectives
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Students should understand the concept of electric potential so they can;
1. calculate the electical work done on a positive or negative charge that
moves through a specified potential difference.
2. given a sketch of equipotentials for a charge configuration, determine the
direction and approximate magnitude of the elctric field at various
positions.
3. apply conservation of energy to determine the speed of a charged particle
that has been accelerated through a specified potential difference.
4. calculate the potential difference between two points in a uniform electric
field, and state which is at the higher potential.
Students should know the potential function for a point charge so they can
determine the eelctric potential in the vicinity of one or more point charges.
Student should be able to describe and sketch a graph of the electric field and
potential inside and outside a charged conducting sphere.
AP Physics B - Capacitors & Dielectrics Objectives
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Students should know the definition of capacitance so they can relate stored
charge and voltage for a capacitor.
Students should understand energy storage in capacitors so they can:
1. relate voltage, charge, and stored energy for a capacitor.
2. recognize situations in which energy stored in a capacitor is converted to
other forms.
Students should understand the physics of the parallel plate capacitor so the can:
1. describe the electric field inside the capacitor, and relate the strength of
this field to the potential difference between the plates and the plate
separation.
determine how changes in dimension will affect the value of the
capacitance. Electric Fields, Potential, and Capacitors Sample Problems
1. A +40 C test charge is placed in an electric field produced by a +5 C charge. It
experiences a 6 N force. What is the strength of the electric field at this point?
2. A +3 C test charge is placed in a 20,000 N/C field. What force does it
experience?
3. A test charge experiences an electric field of 400,000 N/C at a point 1 cm away.
What is the magnitude of the charge producing the field?
4. 30 J of work is done to move a +5 C charge from one point to another in an
electric field. What is the difference in potential between the two points?
5. How much work is done to move an electron across a potential of 3 V?
6. Two parallel plates are separated by 0.5 m. An electric field of 6000 N/C exists
between the plates. What is the potential difference between the plates?
7. A potential difference of 60 V exists between two parallel plates separated by 3
cm. What electric field exists between the plates?
8. An oil drop carrying a charge of -8 x 10-19 C is suspended between two plates
separated by 8 mm. A 1200 V potential difference exists between the plates. What
is the weight of the suspended drop?
9. An oil drop weighs 5.8 x 10-14 N. It is suspended in a 3000 N/C field. What is the
charge of the drop? How many electrons does it carry?
10. What is the capacitance of a capacitor carrying a charge of 0.00012 C and having
a 6 V potential difference?
11. What charge exists on a 30 F capacitor with a 120 V potential difference
between its plates?
12. What energy is stored in a 330 F capacitor connected across 9 V?
13. A capacitor stores 0.005 J of energy. It is connected across a 12 V potential
difference. What is its capacitance?
14. Lab-type problem: A 330 F capacitor and a 100  resistor are used. The signal
generator is set for 4 V.
beginning time: 1.2471 sec
time to reach 2 V: 1.2671 sec
maximum voltage: 3.95 V
area under curve: 0.0015303 A sec
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Calculate the theoretical value for the charging half-life.
Calculate the experimental value for the half-life.
Calculate an experimental value for the capacitance using your charging
half-life.
Use the area under the curve and the maximum voltage to calculate the
energy stored.
Use the maximum voltage and your experimental value for the capacitance
to calculate the energy stored.
Electric Fields and Potential Homework
1. A test charge of -2 x 10-8 C experiences a force of 0.06 N when placed in an
electric field. What is the electric field intensity? Ans: 3 x 106N/C
2. A test charge of +5 x 10-4 C is in an electric field which exerts a force of 2.5 x 104
N upon it. What is the strength of the electric field? Ans: 0.5 N/C
3. A test charge of +80 C is placed in a 50 N/C field. What force does it
experience? Ans: 0.004 N
4. A +4.9 C charge produces an electric field of 3.6 x 104 N/C upon a positive test
charge. How far away is the charge? Ans: 1.11 m
5. A 19 C charged sphere produces a 1.7 x 10 3 N/C upon a 0.5 C test charge.
What force is the charge subjected to? How far away is the test charge from the
sphere? Ans: 0.00085 N; 10.03 m
6. The electric field intensity between two parallel, charge plates is 8000 N/C. The
plates are 0.05 m apart. What potential difference exists between the plates? Ans:
400 V
7. A spark will jump across dry air when the electric field is larger than 1 x 106 N/C.
If two parallel plates have a potential difference of 5000 V, how far apart must
they be to prevent a spark from jumping across them? Ans: 0.005 m
8. What work is done on a 5 C charge when its electric potential is increased by 1.5
V? Ans: 7.5 J
9. A charge of 50 C is raised in potential by 110 V. What work is done in raising the
potential of the charge? Ans: 5500 J
Capacitors and Oil Drop Homework
1. An oil drop carries a charge of 5 electrons and is balanced in a field of 4.7 x 104
N/C. What is the drop's weight? Ans: 3.76 x 10-14 N
2. A negatively charged oil drop weighs 8.5 x 10-15 N. The drop is suspended in an
electric field of 5300 N/C. What is the drop's charge? How many excess electrons
does it carry? Ans: 1.6 x 10 -18 C; 10 electrons
3. What is the charge on a 6F capacitor with a potential difference of 0.60 V? Ans:
3.6 C
4. A capacitor has a charge of 3 C and a potential difference of 45 V. What is its
capacitance? Ans: 0.067 F
5. A 0.002 F capacitor has a 6 V potential difference across it. How much energy
does it store? Ans: 0.036 J
6. A +26 C charged sphere is touched to a -19 C charged sphere and then removed.
What charge remains on each? Ans: +3.5 C
Data Analysis -- Charging and Discharging a Capacitor
1. Calculate a theoretical value for the charging half-life. Show all work.
2. Calculate a theoretical value for the discharging half-life. Show all work.
3. Using the experimental value for the charging half-life, calculate an experimental
value for the capacitance.
4. Using the experimental value for the discharging half-life, calculate an
experimental value for the capacitance.
Questions -- Charging and Discharging a Capacitor
1. Calculate a percentage difference between the theoretical and the experimental
value for the charging half-life. Repeat for the discharging half-life. Show all
work.
2. Calculate a percentage difference between the theoretical and the experimental
value for the capacitance using charging data. Repeat for discharging data. Show
all work. Remember to use the value for maximum voltage that you found for
each in your calculations (procedure 1 data).
3. The industry standard for capacitance allows a 20% difference between the listed
capacitance value and the actual value. Does your percentage difference fall
within this range? List two factors that could account for a high percentage
difference. Answer for both the charging and discharging procedures.
4. Why does the area under the curve for the current vs time graph represent the
charge stored in the capacitor? Use dimensional analysis to support your answer.
5. Knowing the charge stored on the capacitor and the maximum voltage (procedure
one), calculate the energy stored in the capacitor. Do this for charging and for
discharging. Show all work. (This will be a theoretical value for the energy.).
6. Calculate a value for the energy stored in the capacitor knowing your
experimental value for the capacitance and the maximum voltage (procedure one).
Do this for charging and for discharging. Show all work. (This will be an
experimental value for the energy.).
7. Compare your answers to 5 and 6 using a percentage difference. Do this for
charging and for discharging. Show your setups.
8. Why did the voltage across the capacitor start at a minimum and reach a
maximum as the capacitor charged? Why did the voltage across the resistor start
at a maximum and reach a minimum as the capacitor charged? In your answer,
define voltage. Relate the definition of voltage to the process of charging a
capacitor.
9. Why did the current start at a maximum and reach a minimum as the capacitor
charged? In your answer, define current. Relate the definition of current to the
process of charging a capacitor.
10. Compare the appearance of the two voltage graphs. Explain the difference in the
appearance of the two voltage graphs.