Using Gravitational Analogies to Introduce Elementary Electrical Field Theory Concepts Susan Saeli and Dan MacIsaac SUNY Buffalo State College Buffalo NY S ince electrical field concepts are usually unfamiliar abstract and difficult to visualize conceptual analogies from familiar gravitational phenomena are valuable for teaching Such analogies emphasize the underlying continuity of field concepts in physics and support the spiral development of student understanding We find the following four tables to be helpful in reviewing gravitational and electrical comparisons after students have worked through hands on activities analyzed via extended student discourse 1 Table I Introductory analogies between gravitational and electrical forces Forces Newton s Universal law of gravitation and the Coulomb law for elec Gravitational Electrical Comments Matter has a fundamental property called mass measured in kg which has just one sign positive Matter has another fundamental property called charge measured in coulombs which can have two signs positive or negative Hence electric forces can be repulsive or attractive Students may not know that so called antimatter has positive mass but reversed electric charges Fg G m1m2 r2 r Some use the phrase gravitational charge for mass to exploit this analogy k describes the gravitational force and direction where r is a unit vector describing or in magnitude only the direction and negative means attractive Gravitational force is there where in SI units fore always attractive tric forces The magnitude of this force is written Fg G m1m2 r2 r in SI units G 6 67 x 10 11 N m2 kg2 r m1 k 9 x 109 N m2 C2 where These are point masses and charges or perfect spherical distributions of mass and charge Tinker toy arrangements are later extended to real objects via calculus or symmetry q1 q2 r Since G is much smaller than k the gravitational force Fg is usually much weaker than the electrical force Fe have students work both forces for 2 protons and 2 electrons and compare Students may not yet be familiar with r read aloud as r hat notation2 but will need it in later physics This notation is also used in discussing centripetal acceleration so review or introduce it Note the tiny stick man in the figures defines r as a unit vector pointing to the other point mass or charge r really contains direction information only Notation requires lots of student practice and explicit explanation use your state physics exam notation from the start of the course m2 r 104 The Physics Teacher Vol 45 February 2007 Table II Introductory analogies between gravitational and electrical fields Gravitational Vector Fields Electrical 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 Similarly with the electrical force there is a field around a given point charge Q or spherically symmetrical distribution of charge Q and it is useful to talk about the field strength around that charge mearthm2 mG r 2 earth mearth Fe k r 2 earth defining g Gmearth r 2 q0k Q r2 defining E k earth F g m EARTH The gravitational field strength has units of force per unit mass or N kg which is the same as the more commonly used m s2 Field units are preferred and we wind up explicitly re stating y r2 Fe q E which is readily calculable producing the famous g 9 8 N kg pointing down toward the center of the Earth on the surface of the Earth Now we can talk about the local field strength of the Earth s gravitation field at the Earth s surface g being the ratio of the gravitational force on a test mass a mass much smaller than that of the Earth very near the Earth s surface to its mass Fg mg m 9 8 N kg q1q2 can be rewritten as then further group as Fg m g Comments where y is a unit vector pointing upward Now g should hold less conceptual mystery The Physics Teacher Vol 45 February 2007 Q r 2 This is readily calculable for uniform electric fields say those very near a charged smooth spherical shell with charge Q or between two parallel plates with opposite charges as F E q q q0 The corresponding units for the electric field strength are therefore force per unit charge or N C again with alternatively more common units of V m Table IV An important value of E to know is E 3 x 106 N C or V m the dielectric breakdown strength of the Earth s atmosphere at STP When this field strength is exceeded air will be torn apart ionized and will conduct we see sparks drawn through the air Presence of electric sparks means we know an instant minimum value for E 3 We explicitly state the use of particular subscripts and capitalization for letters m and q what is inferred in the use of each and when and why we change subscripts Although the universal law of gravitation formula will work with any two point or spherically symmetric masses we most commonly experience the downward force of gravity at the Earth s surface In that case one of the masses becomes the mass of the Earth and the distance is the radius of the Earth Students perform this calculation of the gravitational field strength g We walk around the class with a plumb bob a vanishingly small test mass m0 and compare the strength and direction of g Note analogy to a vanishingly small test charge q0 First we stand on tables and then we hold the bob in different corners of the room rudely determining by touch and vision that g doesn t measurably change in direction and size regardless of location We start fields off with students sketching 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 around the Earth This develops a better understanding of g and makes explicit the E field analogy near both a point in space and near the surface of a charged shell like the dome of a Van de Graaff generator We want to establish and reinforce the analogies between E and g Stressing the units of g as N kg helps to solidify the analogy when comparing to N C for E and can help clarify issues regarding gravitational fields Students should show N kg is equivalent to m s2 and later do the same for N C and V m Also establish the similarity of E between two charged parallel plates4 and g in a room on the Earth s surface Parallel charged plates e g aluminum pie pans can be attached to a Van de Graaff generator to explore E with a packing peanut on a stick and thread or Christmas foil streamers Also compare to
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