Download Using analogies to explain electrical relationships

Teaching Electrical Concepts via Gravitational Analogies Page 1
Using Gravitational Analogies To Introduce Elementary Electrical Field Theory Concepts
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Sue Saeli, Dept of Physics, SUNY-Buffalo State College, 1300 Elmwood Ave, Buffalo, NY 14222
Dan MacIsaac, Dept of Physics, SUNY-Buffalo State College, 1300 Elmwood Ave, Buffalo, NY 14222
<[email protected]>.
Please direct correspondence regarding this manuscript to MacIsaac.
Keywords: analogies, electric, gravity, field, potential, force
PACS codes:
01.40Gb, 01.55, 41.90
Abstract:
Familiar gravitational phenomena and conceptual analogies are useful in explaining those introductory electrical concepts
associated with field theory since the electrical ideas are unfamiliar, abstract and difficult to visualize. These analogies
emphasize the underlying continuity of fields in physics and support the spiral development of student understanding. We find
the following four tables to be particularly useful in reviewing and summarizing these comparisons after students have
conducted through appropriate touchstone activities and discourse as part of the process of making sense of electric fields.
Acknowledgements:
This manuscript addressed requirements for PHY690: Masters' Project at SUNY-Buffalo State College, and was informed by
comments of students participating in PHY622 at SUNY-BSC during summer, 2003. Portions were supported by NSF DUE
0302097. All figures by Mr. Matt Coia. Some terms were coined in discussion with Dr. David Cole of Northern Arizona
University; Mr. John Burgholzer of Amherst HS also provided significant comment. Other ideas were informed by the
comments and curricula of The ASU Modeling Physics (REF1) project and personnel, particularly Mr. Gregg Swackhamer and
Mr. Larry Dukerich, and the Workshop Physics (REF2) curriculum by Professor Priscilla Laws of Dickinson College. Errors
and omissions are the responsibility of the authors.
Introduction:
Conceptual analogies from more familiar gravitational phenomena are useful in explaining introductory electrical concepts
based upon field theory since the electrical ideas are unfamiliar, abstract and difficult to visualize. These analogies emphasize
the underlying continuity of field concepts in physics and they support the spiral development of student understanding. We
find the following four tables to be particularly useful in summarizing and reviewing these comparisons after students have
worked through appropriate activities analyzed via extended student discourse(REF3).
Teaching Electrical Concepts via Gravitational Analogies Page 2
Table 1: Introductory Analogies Between Gravitational And Electrical Forces
Gravitational
Electrical
Comments
Forces:
Newton's
Universal Law
of Gravitation
and the
Coulomb Law
for electric
forces.

Matter has a fundamental
property called mass, measured
in kg, which has just one sign:
positive.
Students may ask and need to be told that so-called
"anti-matter" has positive mass (and reversed electric
charge).
Matter has another
fundamental property called
charge, measured in
Coulombs which can have
two signs: positive or
describes the gravitational force
negative. Hence Electric
and direction, where rˆ
forces can be repulsive or
describes the direction and
attractive.
negative means attractive.
Gravitational force is therefore
qq
always attractive.
F  k 1 2 rˆ

Note we are talking about point masses and charges
or perfect spherical distributions of mass and charge.
This is a non-calculus treatment (easily extended).
mm
Fg  G 1 2 2 rˆ
r
e
The magnitude of this force is
written:
mm
Fg  G 1 2 2
r

11
G  6.67x10 Nm/kg
2

2
INSERT FIG1G.JPG HERE

r2
or in magnitude only:
Fe  k
q1q2
rˆ
r2
where in SI units:
where in SI units:

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k  9x10 9 Nm 2 /C 2
INSERT FIG1E.JPG HERE
Some have even coined the phrase "inertial charge"
for mass to exploit this analogy.
k , the gravitational

force, Fg , is usually much smaller than the

electrical force, F e (have students work both forces
Since G is much smaller than
for 2 protons and 2 electrons and compare /
discourse).
Students may not be familiar with rˆ (read as r-hat)
notation (REF3), but will need it if they go on in
physics. As well, this notation is needed to make
conceptual sense of centripetal acceleration, so if it
is new, now's the right
time to discuss it. Note the
tiny stick man in the figures defines rˆ as a unit
vector pointing to the other point mass or charge. rhat really contains direction information only. Your
students might also not recognize the "absolute
value" notation used to strip
 direction from a vector,
or the triple bar definition sign. These notations
require explicit explanation and repeated student use.
If your state exam requires a particular notation, use
that as well from the start of the year.
Teaching Electrical Concepts via Gravitational Analogies Page 3
Table 2: Introductory Analogies Between Gravitational And Electrical Fields
Gravitational
Electrical
Comments
Vector
Fields
For a small mass (compared to that of
the Earth) on or very near the surface
of the Earth we can group together
known terms and solve:
Fg  G
mearth m2
m
 mG earth2
2
rearth
rearth
further define
Fg  m g


where
g
Similarly, with the electrical
force there is a field around a
given point charge Q (or
spherically symmetrical
distribution of charge Q) and
and it is useful to talk about the
field strength around that
charge.
Gmearth
2
rearth
Fe  k
and is readily calculable, producing the
F qE
famous g = 9.8 N/kg pointing towards defining e
Q
the center of the Earth on the surface of
E k 2

the
Earth.
r , and

where
therefore, is readily calculable
 Now we can talk about the local field for uniform electric fields strength of the Earth's gravitation field say those very near a charged
at the Earth's surface, g , being the  smooth spherical shell with
ratio of the gravitational force a "test charge Q or between two
mass" (a mass much smaller than that parallel plates with opposite
charges as:
of the Earth very near the Earth's
surface). 
E
F
g
m
F
q
INSERT FIG2E.JPG HERE
INSERT FIG2G.JPG HERE

q1q2
Q
2  qo k 2
r
r
 gives
The gravitational field strength
units of force per unit mass or N/kg,
which should be shown (by students) to
be the same as the more commonly
used m/s2. Henceforth, the field units
are to be pedagogically preferred.

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Although the universal law of gravitation formula
will work with any two point or spherically
symmetric masses, we most commonly experience
the downwards force of gravity at the Earth's

surface, g . In that case one of the masses becomes
the mass of the earth and the distance is the radius of
the earth. Students should perform this calculation
of the gravitational field strength g to practice using
exponentials in their calculator under teacher

supervision. The idea is to make g slightly less
mysterious as a local value for gravitational field
strength and to permanently move to N/kg units,
which we have students show are equivalent to m/s2
and conceptually preferred.
We actually walk around the room with a plumb bob
mo ,"
and determine the strength and direction of g -- note
analogy to "a vanishingly small test charge qo ." We
– we call this "a vanishingly small test mass
stand on tables, and hold the bob in corners,
 change in
determining that g doesn't measurably
direction and size in the room. 

We start fields off by having students sketch a figure
(usually on a whiteboard) to explain the relationship
between g in the classroom, g on the surface of the
Earth at the equator and N and S poles, and g in
space near the Earth's surface. This develops a
better understanding of g and makes explicit the
The corresponding units for  E
 both a point in space and near
field analogy near
the electric field strength are,
that of a Van de
the surface of a charged shell like
therefore, force per unit
Graaf generator.
charge or N/C, again with

more common units of V/m
We want to establish and reinforce the analogies

(more on V later).

between E and g . Showing the units of g as
N/kg helps to solidify the analogy when comparing
E

An important value of
to
E (and can help clarify issues regarding
to
N/C
for
6
know is E  3x10 N/C or gravitational fields). Have students demonstrate

these
units are equivalent.
V/m – the dielectric
breakdown strength of the

Also establish the equivalence of E between
Earth's atmosphere at STP.
When this field strength is
parallel plates (REF6) and g in a room on the
exceeded, air will be torn
Earth's surface.
apart (ionized) and will
conduct; we will see static
Parallel plates can be 
attached to a Van deGraff
electricity sparks drawn
generator and ground
to demonstrate E with a

through the air. Any spark
packing peanut on a thread. This is also useful for
present means we know an
the E near a Van deGraff sphere.
instant minimum value for

E (REF5).
Note traditionally important
 numeric values of g
and E . Students should memorize these.


Teaching Electrical Concepts via Gravitational Analogies Page 4
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Tables 3 & 4: Introductory Analogies Between Gravitational And Electrical Potential Energy and Potential
Potential Gravitational
Electrical
Comments
Energy
Gravitational potential energy is the
Electric potential energy is defined analogously to
stored energy a mass can have due to
the gravitational potential energy discussed
its gravitational attraction to another Electric potential energy can previously in the course. A topographic contour
mass and depending on the
be found similarly, with more map should also be examined (REF7) and the
separation difference. At the Earth's variations possible due to
thought experiment of walking a wheelbarrow about
surface we find this by assuming a
different possible signs of
a contour lines or perpendicular to contour lines
locally uniform field strength and
charge.
should be whiteboarded & discussed. Also the path
direction:
taken by a loose ski or bowling ball free to fall down
from a mountain peak on a topo map.
PE  q E h  qV
e
PEg  m g h
when E and h are in the
same direction
when g and h are in the same
direction


INSERT FIG3G.JPG HERE

Potential
Gravitational potential (a rarely used
term) is the gravitational potential
energy per unit mass or for uniform
gravitational fields:
Vg 


PEg
m
 mgh
m  gh
where the units are J/kg and
assuming a mass displacement
parallel to g . We actually coin a
term for this we call "liftage,"

somewhat analogous to "plumbing
head" – where a scalar figure
expresses where water can flow due
to the use of a water tower in a water
distribution system:

INSERT FIG4E.JPG HERE
INSERT FIG3E.JPG HERE
Electric potential is defined as
electric potential per unit
charge, which turns out to be a
very practical measure for
electric phenomena :
Ve 
PEe
q

qEh
q
 Eh
Electric potential is more
commonly known as voltage
and for uniform fields:
Ve  Eh
the units of electrical potential
are J/C or volts.
Again, note the analogy comparing E between two
parallel plates and g within a room on the Earth's
surface. This relationship and the role of path
dependence should be explored and exploited in

activities like Arons' homework
questions or the
Modeling
Physics
work
sheets.
Students
should try

sketching lines of E and V in experimental
situations and discuss .
When we talk about potential the analogy becomes
less useful
 since we rarely discuss gravitational
potential. The idea can be explained simply in terms
of “liftage”. Use the example of a water tower that
holds the water for a municipality above the level of
all the users’ bathrooms. Therefore, the “liftage” is
dependent on the height of the water not the mass of
the water. In other words, as long as there is water
in the tower above the level of the bathroom, there
can be water flowing in the bathroom. It doesn’t
matter how much you need or how much is in the
tower.
Teaching Electrical Concepts via Gravitational Analogies Page 5
Using these tabulated analogies:
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We have used this tabulated comparative approach in several instructional contexts, and have found the least
successful way for students to learn these ideas is by presenting these complete tables in an early formal lecture, although
students are often happiest with such lectures. Rather, we suggest that the ideas be formally presented in tabular form AFTER
students have struggled with the individual ideas in discussions, after worksheets and concrete hands-on activities both
reviewing gravitational ideas and examining electrical phenomena. While the ideas in these tables could be presented as final
formal review notes for the section on introductory electric concepts, though students seem unhappy with delayed presentation
of these formalisms. Our preferred balance is to reconstruct the information one table at a time while moving through subject
as teacher-led "mini-lecture" summaries from student language and ideas. The idea is to warrant student negotiated meaning
and ideas while moving towards standardized language and formalisms when enough experience and discourse has been
completed when students are ready for the formalism and require reassurance. Hence each of these four tables usually appear
in our student notes as summary interludes amongst other activities, although we have tried constructing these kind of tables in
the back pages of student notebooks as well. Formalizing pieces only at intervals when students are ready for such (and are
requesting such) works well for us.
First, the instructor should review and enlarge notation for the universal gravitation relationship. Particularly review
calculation skills and examples using scientific notation in calculators. As Hewitt points out in Conceptual Physics, this is a
good time to have students find and compare the gravitational attraction between an electron and a proton with the gravitational
attraction. Since this would be just for review, students should be able to complete this task quickly and with very little
guidance. Next, students can calculate the electrical attraction between these same two particles. Although this calculation is
new they should have no difficulty because of the similarities to the gravitation equation. Take some time to discuss the
results. These two exercises should be done together within one presentation in order to highlight the similarities. As with any
analogies, it is important to point out any differences to help to mitigate confusion and in some cases to enhance the discussion.
For example, the comparison between the G and k allows deeper understanding of the relationship between Fg and Fe.
In the next table, is the relationship between gravitational field strength and electric field strength. Students have
difficulty seeing that “little g” is a measurement of field strength. At this point, a useful exercise would be to show how m/sec2
represents the same units as N/kg. Until students have the understanding that g , the acceleration due to gravity, is a way of
representing the field strength, the analogy is too weak. Besides, time spent with units is time well spent.
The third table deals with potential energy. This is the time to show that all energy is the same. This allows the
instructor to review the concept of energy.
Finally, the last table introduces electric potential. Without
 some development of the concept of gravitational
potential the analogy with electric potential will be useless. In order to be successful students will need to visualize how water
gets into their upstairs bathroom without a pump. A simple diagram of the local water tower next to “their” house should help
them to see that the water coming from the tower has the potential to return to the same height and that this idea does not
depend on the mass of the water. Although it is the analogies that have been the focus of this paper, this visual aid is crucial
since the idea of “liftage” is not a concept most teachers spend time teaching. As they say “a picture is worth a thousand
words.” When this idea is accepted, the concept of electric potential can be introduced.
This approach works well not only to give students a more concrete way of learning about these electrical concepts,
but also to reinforce the gravitational concepts. Our time is so limited with each individual topic there is very little opportunity
for repetition. There are numerous benefits when teachers use opportunities, like this, to overlap units or refer to previously
discussed material.
Teaching Electrical Concepts via Gravitational Analogies Page 6
References:
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REF1. D. Hestenes, G. Swackhamer & L. Dukerich, Modeling Physics Curriculum (ASU, Tempe AZ, 2004).
<http://modeling.asu.edu/Curriculum.html>. See second semester (electricity & magnetism) curricular materials (worksheets
and activities), particularly Units E1: Charge and Field and E2: Potential. A freely reproducible HS curricular resource based
on Physics Education Research (PER) touchstone curricular activities.
REF2. P. Laws, The Workshop Physics Activity Guide Module 4: Electricity & Magnetism (Wiley, Hoboken NJ, 2004). For a
comparative development see especially Unit 21: Electrical and Gravitational Energy, p. 573-593.
REF3. For an outstanding vignette demonstrating advance student discourse in the introductory physics class, see D. Desbien,
M. Joshua & K. Falconer, RTOP4 Video 4. (SUNY-BSC, Buffalo NY, 2003). Streamed QuickTime video available from
<rtsp://media1.buffalostate.edu/physics/rtop4.mov>.
REF4. R. Knight., Physics For Scientists and Engineers with Modern Physics: A Strategic Approach. (Pearson Addison
Wesley, San Francisco, 2004). P. 808 develops the r-hat notation. The workbook is an excellent source of PER-inspired
activities.
REF5. R. Chabay & B Sherwood. Matter & Interactions VolII: Electric and Magnetic Interactions. (Wiley, Hoboken NJ,
2002). Section 14.8 is a case study of sparks in air, and this is an outstanding PER-influenced curriculum for studying
electricity and magnetism.
REF6. A. Arons, Teaching Introductory Physics (Wiley, Hoboken NJ, 1996). See problem 5.17, p76 of Volume II.
REF7. B.S. Andreck, "Using Contour Maps to Teach Electric Field and Potential," Phys. Teach. 28, 499 (Oct, 1989).
Arons, A.B. (1990). Teaching Introductory Physics, Part II, problem 5.17, p 76. Reading NY: John Wiley & Sons.
Hestenes, D. Swackhamer, G, & Dukerich, L. (2004) Modeling Physics Curriculum.
http://modeling.asu.edu/Curriculum.html. See second semester (electricity & magnetism) curricular materials (worksheets and
activities), particularly Units E1: Charge and Field, and E2: Potential. Outstanding freely-reproducible HS curricular resource.
LARRY CAN THESE UNITS BE UNLOCKED ON THE MODELING SITE???
Laws, P.W. (2004). The Workshop Physics Activity Guide, Module 4. Reading NY: John Wiley & Sons. See particularly
Unit 21: Electrical and Gravitational Energy, p. 573-593.
Chabay, R. & Sherwood, B. (2002). Matter & Interactions Vol II: Electric and Magnetic Interactions. Reading NY: John
Wiley & Sons. See p484-495 for a case study of sparks in air. An outstanding resource for activities teaching introductory
electric concepts including an intelligible introduction to the use of calculus in electrostatics.
Hewitt??????
MacIsaac, Dan, Assistant Professor of Physics, SUNY- Buffalo State College.
Hewitt, Paul G., Conceptual Physics, Prentice-Hall Inc., 2002, p. 507.
Burgholzer, John Physics Teacher Amherst High School.
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