Teaching Physics with the Physics Suite

Edward F. Redish

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Functional dependence and the electric field*

1. (a) Suppose you want to purchase a sweater, in Maryland, which has a list price of $40 for which you pay $2 in sales tax. Suppose your friend bought the same sweater, in Maryland, but it had a list price of $80 for which she paid $4 in sales tax. How does the ratio of sales tax to price of the sweater compare for you and your friend (i.e. compare the ratios (sales tax)/(sweater price)? What does that ratio tell us? What is that ratio defined as?

(b) Suppose a charge exerts a repulsive force of 4 newtons on a test charge of 0.2 microcoulombs that is brought within 2 cm of it. However, the charge exerts a repulsive force of 8 newtons on a test charge of 0.4 microcoulombs that is brought within 2 cm of it. How does the ratio of the force on the test charge to the test charge, itself, compare in each case (i.e. compare (force felt by test charge)/(test charge) in each case)? What does that ratio tell us? What is that ratio defined as?

(c) Suppose a charge Q exerts a force F on a test charge q that is brought near to it. By how much would the force exerted by Q increase if the test charge increased by a factor of a, where a can be any constant (i.e. a = -17 or 5 or 7.812, etc.)? By how much would the ratio of the force on the test charge to the test charge, itself, increase if the test charge increased by a factor of a? Explain.

(d) When the value of one quantity depends on the value a second quantity (and perhaps on others), we say that the first quantity is a function of the second. How the first quantity changes when the second changes is called the functional dependence. For example, if t = As, we say that t has a linear functional dependence on s. When s doubles, so does t. If s is divided by 10, so is t. As a second example, if we had y = Bx2, we would say that y depends quadratically on x. If x doubles, y quadruples. If x is divided by 10, then y is divided by 100. (Try this with some numbers, picking whatever values of the constants A and B you would like.)

(i) What is the functional dependence of the sales tax paid on the price of the sweater? Explain. Write an equation that relates the tax paid (t) to the cost of the sweater (s).

(ii) What is the functional dependence of the sales tax percentage rate on the price of the sweater? Explain.

(iii) In part (c), what is the functional dependence of the force, F on the magnitude of the test charge, q? Explain.

(iv) What is the functional dependence of the electric field established by Q, EQ, on the test charge, q? Explain.

* Problem written by Jonathan Tuminaro

Note to the instructor: Students sometimes have a great deal of difficulty with the idea that the electric field is independent of the test charge used to measure it. Writing E = F/q hides the dependence of the force on the charge. This problem sets up an everyday analogy and explicitly discusses the concept of functional dependence .

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Page last modified October 15, 2002: E01