Electrostatics: Is this a fair question?

Question

This problem from a recent edition of Halliday & Resnick (& whoever)...

In the x-y plane there are equal charges +q placed on the y axis at (0,D) and (0,-D).  Let E(x) be the field at a point (x,0) on the x axis.  Let a = x/D be the x position scaled to the y distance D.

(a) Find the value of a where E(x) = E(aD) is a maximum.

Note: This is eminently fair, and the answer is a = 1/√2.  It's a pretty good elementary problem.  The questionable part comes next:

Find the values of a where E is one half the maximum value.  (b) and (c) ask for these two values.  The Wiley website asks for something like 2% accuracy, and accepted a numerical solution obtained with the use of technology.  

Is it fair to have a problem that can't be solved analytically using methods that the student has studied?  Unless I'm missing something, this involves solving an unfactorable cubic, and nobody tortures lower-division math students with the cubic formula.  They get an introduction to numerical integration in Calc I, typically, but most won't have taken a numerical analysis class.

So, is there a way to get a 2% answer to this without technology using methods that a student would be expected to have covered?  

If not, is it sensible to even ask such a question?  By today's standards, of course.  When I took the course, it would have been unthinkable.

nealjking · Accepted Answer

OK, along the x-axis, the E-field will be:
Ex = 2Qx/(sqrt(D^2 + x^2))^3
Ey = 0

and:
(d/dx)Ex = (D^2 - 2x^2)/(sqrt(D^2 + x^2))^5
so indeed the maxima are at x = ±D/sqrt(2), where
Ex = ± 2Q/sqrt(3)

So the question is: When is
2Qx/(sqrt(D^2 + x^2))^3 = Q/sqrt(3) , which implies:
12x^2 = (D^2 + x^2)^3 or using y = x^2:
12y = (D^2 + y)^3 : a cubic, as you stated.

Now to the issues:
- Is it fair to pose a problem that cannot be solved analytically? 
Yes, only a tiny fraction of problems in the real world can be solved analytically, so people should learn to use whatever tools are available. I think every physics or engineering student should have a laptop computer, so if s/he doesn't want to use Newton's method....

- Is there a way to get a 2% answer?
Use a spreadsheet! Plot Ex vs. x and compare with 1/sqrt(3). I can slap this up on Excel in 5 minutes, and immediately see that the answers are near 0.34 to 0.35, and 1.32 to 1.33. By refining the increment, I can get as much accuracy as I need. Since 2% of .35 is .007, I'll make the increment .005, and I find that the answer is between 0.34 and 0.345.

Science is not about "following the rules": It is about finding things out, using whatever tools you have. A spreadsheet is a useful tool!

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Electrostatics: Is this a fair question?

2 Answers