THE PHYSICS TEACHER ◆ Vol. 45, January 2007 579THE PHYSICS TEACHER ◆ Vol. 45, January 2007 579Using Gravitational Analogies to Introduce Elementary Elec- trical Field Theory ConceptsSusan Saeli and Dan MacIsaac, SUNY-Buffalo State College, Buffalo, NYSince electrical field concepts are usually unfa-miliar, abstract, and difficult to visualize, con-ceptual analogies from familiar gravitational phenomena are valuable for teaching. Such analogies emphasize the underlying continuity of field con-cepts 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.1Gravitational Electrical CommentsForces: Newton’s Universal law of gravi-tation and the Coulomb law for elec-tric forces.Matter has a fundamen-tal property called mass, measured in kg, which has just one sign: posi-tive.12g2ˆmmGrr=−Fdescribes the gravitational force and direction, where ˆris a unit vector describing the direction and negative means attractive. Gravitational force is there-fore always attractive. The magnitude of this force is written: Fg= Gmmr122where in SI units: G = 6.67 x 10-11 N.m2/kg2Matter has another fundamental property called charge, measured in coulombs, which can have two signs: positive or negative. Hence elec-tric forces can be repul-sive or attractive.kor in magnitude only:where in SI units:k = 9 x 109 N.m2/C2Students may not know that so-called “anti-matter” has positive mass (but reversed electric charges).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.Some use the phrase “gravitational charge” for mass to exploit this analogy.Since G is much smaller than k, the gravita-tional force Fg is usually much weaker than the electrical force Fe (have students work both forces for 2 protons and 2 electrons and com-pare).Students may not yet be familiar with ˆr(read aloud as r-hat) notation2 but will need it in later physics. This notation is also used in dis-cussing centripetal acceleration so review or introduce it. Note the tiny stick man in the fig-ures defines ˆr as a unit vector pointing to the other point mass or charge. ˆr really contains direction information only. Notation requires lots of student practice and explicit explana-tion; use your state physics exam notation from the start of the course.rm2m1rrq2q1rTable I. Introductory analogies between gravitational and electrical forces.580 THE PHYSICS TEACHER ◆ Vol. 45, January 2007Gravitational Electrical CommentsVectorFieldsFor a small mass (com-pared to that of the Earth) on or very near the surface of the Earth, we can group together known terms and solve:Fgearthearthearthearth==GmmrmGmr222then further group asFgg= mdefining gearthearth≡Gmr2,which is readily calcula-ble, 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 sur-face, |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.gF=mThe 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:Fg = mg =m [-9.8 N/kg]ˆy , where ˆy is a unit vector pointing upward.Now g should hold less conceptual mystery.Similarly, with the electrical force there is a field around a given point charge Q (or spherically symmetrical dis-tribution of charge Q), and it is useful to talk about the field strength around that charge.Fe==kqqrqkQr12202can be rewritten asFEe= qdefining E ≡ kQr2.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: .EF=qThe 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 con-duct; we see sparks drawn through the air. Presence of electric sparks means we know an instant minimum value for |E|. 3We explicitly state the use of particular sub-scripts 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 for-mula will work with any two point or spheri-cally 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 com-pare 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 sur-face of a charged shell like the dome of a Van de Graaff generator.We want to establish and reinforce the analo-gies 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
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