Download Study Guide 2

PH 1120
Electricity and Magnetism
P.K.Aravind/T.H. Keil
Term B04
STUDY GUIDE #2
In this part of the course we will study the following topics:
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Electric potential difference and electric potential for a uniform field
Electric potential difference and electric potential for a point charge
Calculation of the potential of a system of point charges by the principle of superposition
Motion of a charge through a region of varying potential
Equipotential surfaces and their relationship to electric field lines
Capacitors as circuit elements and as devices for storing electrical energy
The electric potential provides an alternative way of thinking about electric fields that is appealing
because the potential is a scalar and not a vector. Nevertheless the potential encodes complete
information about the electric field, including its vector character. Understanding how it does this is
one of the major goals of this part of the course. The potential is also important because of its
connection with the all-important concept of energy.
Definition 1. The potential difference, VB – VA, between two points B and A is the negative of the
work done by the electric field on a unit positive charge that is taken from A to B.
Remark: The potential difference between two points is independent of the path along which the
charge is moved from one point to the other. The truth of this statement is not at all obvious. A
proof of it will be given in lecture for the case of a uniform electric field. Later in your careers you
will learn that it is true of all (conservative) electric fields. The notion of potential difference is so
useful precisely because it is independent of the path between the points in question.
Definition 2. The absolute potential at any point is the potential difference between that point and a
suitable reference point. The absolute potential at the reference point is taken to be zero.
Objective 6: Electric Potential Difference and Electric Potential for a Uniform Field
(a) Understand how Definition 1 implies the following result for the magnitude and sign of the
potential difference between two points in a uniform electric field: the magnitude is the product of
the electric field and the “stream distance” between the points, and the sign is positive or negative
according as the final point is “upstream” or “downstream” of the initial point. The figure below
shows two points A and B in a uniform electric field of magnitude E directed horizontally to the
right, with B being downstream of A. The stream distance between the points (i.e. the distance
measured along the field) is d. The potential difference VB – VA is equal to – Ed, with the negative
sign occurring because B is downstream of A.
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A
E
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B
(b) To be able to use Definition 2 to assign absolute potentials to arbitrary points in a uniform field,
given the absolute potential at any one point.
Objective 7: Equipotentials of a Uniform Electric Field
To know the equipotential surfaces for a uniform electric field and their relationship to the field
lines, and to be able to pass back and forth between the field lines and the equipotential surfaces.
Suggested Study Procedure for Objectives 6 and 7:
A discussion of the properties of uniform fields can be found in Ch.23-1, pp.870-1 and in Example
23.9 on p.887. This somewhat fragmented description will be brought together and amplified in
class. Your best approach is to make sure you understand all the ideas laid out in the Objectives and
test your understanding by working your way through all the assigned problems. The first half of
Lab 1, in which you will determine the equipotentials of a uniform field, will also make you very
familiar with this material.
Suggested Problems for Objectives 6 and 7:
Problems 23.15,17,31,35,36(a),(b),37(a)
Objective 8: Electric Potential Difference and Potential of a point charge
Know how to calculate the absolute potential and the potential difference between two points in
space for a point charge. Also know how to calculate these quantities for a set of point charges
fixed at specified locations in space.
Objective 9: Motion of a charge in a potential
Know how to solve for the motion of a charge in a region of varying potential using the principle of
conservation of energy.
Suggested Study Procedure for Objectives 8 and 9:
Read Ch.23-2,3 but you can ignore all the worked examples on pp.888-9 that involve integration
(however this can be optional reading for those of you who are comfortable with integral calculus).
Note: much of Ch.23-1 deals with the concept of the potential energy of a set of point charges.
While this is an interesting concept with many applications, we will not deal with it in this course.
Suggested Problems for Objectives 8 and 9:
Problems 23.13,20,21,22,80,33*(study Example 23.11 first).
SP#1. Two positive charges of +q are at (-a,0) and (a,0). A negative charge of –q is released from
rest at the point (0,d). Find the speed of the negative charge when it gets to the origin.
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Objective 10: Equipotential surfaces and their relationship to electric field lines
Know the equipotential surfaces for a single point charge and for a pair of point charges. Also have
an appreciation for more complex equipotential surfaces, including a knowledge of how to pass
from such surfaces to electric field lines and vice-versa. In particular, given a set of labeled
equipotentials, you should be able to determine roughly the magnitude and direction of the electric
field in the region between the equipotentials.
Suggested Study Procedure for Objective 10:
Read Ch.23-4, looking carefully at the pictures.
Suggested Problems for Objective 10:
SP#2. An excess charge placed on an isolated metal sphere always distributes itself uniformly on
the surface of the sphere, producing zero electric field inside and the same field and potential
outside as if the entire charge were concentrated at the center. Suppose an isolated metal sphere of
radius 10cm is charged to a potential of -100V (this is the absolute potential at the surface of the
sphere, with the potential taken to be zero very far away from the sphere). (a) Calculate the charge
on the sphere, (b) what is the potential at the center of the sphere? (takes no calculation, only a little
thought), (c) describe the shape and position of the -50V and -10V equipotentials.
SP#3. A charge of 5nC is at (0,5cm) and one of -5nC at (0,-5cm). Sketch roughly the positions of
the 0V, +10V and -5V equipotentials. Find where each equipotential cuts the y-axis.
Optional Reading Ch.23-5 is recommended for those of you who are comfortable with the
calculus. You can test your understanding by attempting Problem 23.41. In this problem, take A =
4, B = -5, C = 3, x and y in meters and the potential in volts, and calculate the magnitude and
direction of the electric field at the origin and at the point ( 2,-4,0).
Objective 11: Capacitance
(a)Know the definition of capacitance in terms of charge and potential difference.
(b)Know how to calculate the capacitance of a parallel plate capacitor.
(c)Know about capacitors in parallel and series and how to calculate the equivalent capacitance of
an arbitrary parallel-series combination.
(d)For an arbitrary parallel-series combination, know how to calculate the charge and potential drop
across each of the capacitors.
Suggested Study Procedures for Objective 11:
Read Ch.24-1, but ignore the worked examples involving the spherical and cylindrical capacitors
on pp.912-3. Read Ch.24-2.
Suggested problems for Objective 11:
Problems 24-3,4,8,18,19,20.
Objective 12: Capacitors and Energy
(a)Know the energy stored in a capacitor with a given charge or potential difference.
(b)Know how energy is distributed among an arbitrary series-parallel combination of capacitors
connected to voltage sources.
(c)Know the energy density stored in an electric field.
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Suggested Study Procedures for Objective 12:
Read Ch.24-3. Study all the worked examples carefully.
Suggested problems for Objective 12:
Problems 24-25,27,31,61.
SP#4. Two charges of 2nC and -4nC are 10cm apart. Calculate the energy density stored in the
electric field at the point midway between them.
SP#5. An air capacitor consisting of two closely spaced parallel plates has a capacitance of 1000pF.
The charge on each plate is 5µC. (a) What is the potential difference between the plates? (b) If the
charge is kept constant, what will the potential difference between the plates be if their separation is
doubled? (c) How much work is required to double the plate separation? (hint: think in terms of the
change in the stored energy).
Optional Reading. The notion of dielectrics will be touched on only lightly in the lectures.
However those of you wanting to learn more about this important topic should read Ch.24-4.
Homework Assignment #5 (due by 5pm on Mon 11/8)
Go to www.masteringphysics.com and log in. Click on Assignment list and select
Assignment #5. You will get 4 chances to submit the correct answer. If your first answer is
incorrect, you should consider making use of the hints.
Homework Assignment #6 (due by 5pm on Wed 11/10)
Go to www.masteringphysics.com and log in. Click on Assignment list and select
Assignment #6. You will get 4 chances to submit the correct answer. If your first answer is
incorrect, you should consider making use of the hints.
Homework Assignment #7 (due by 5pm on Fri 11/12)
Go to www.masteringphysics.com and log in. Click on Assignment list and select
Assignment #7. You will get 4 chances to submit the correct answer. If your first answer is
incorrect, you should consider making use of the hints.
Homework Assignment #8 (due Mon 11/15)
Problem 24-15. Which of the capacitors stores the maximum energy, and what is it?
Problem 24-26. Also calculate the energy density of the electric field between the plates.
Problem 24-30.
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