A Trick to Generate Exam Problems

Intermediate Physics for Medicine and Biology.

When teaching a class based on Intermediate Physics for Medicine and Biology, instructors need to write problems for their exams. My goal in this post is to explain a trick for creating good exam problems. 

One of my favorite homework problems in IPMB is from Chapter 4.

Problem 37. The goal of this problem is to estimate how large a cell living in an oxygenated medium can be before it is limited by oxygen transport. Assume the extracellular space is well stirred with uniform oxygen concentration C₀. The cell is a sphere of radius R. Inside the cell oxygen is consumed at a rate Q molecule m⁻³ s⁻¹. The diffusion constant for oxygen in the cell is D.

(a) Calculate the concentration of oxygen in the cell in the steady state.

(b) Assume that if the cell is to survive the oxygen concentration at the center of the cell cannot become negative. Use this constraint to estimate the maximum size of the cell.

(c) Calculate the maximum size of a cell for C₀ = 8 mol m⁻³, D = 2 x 10⁻⁹ m² s⁻¹, Q = 0.1 mol m⁻³ s⁻¹. (This value of Q is typical of protozoa; the value of C₀ is for air and is roughly the same as the oxygen concentration in blood.)

Homework problems for Chapter 4 in
Intermediate Physics for Medicine and Biology.

I usually work this problem in class. Not only do students practice solving the steady-state diffusion equation, but also they estimate the maximum size of a cell from some basic properties of oxygen. In the Solution Manual—available to instructors only (email us)—we explain the purpose of each problem in a preamble. Here is what the solution manual says about Problem 37:

This important “toy model” considers the maximum size of a spherical cell before its core dies from lack of oxygen. One goal of biological physics is to show how physics constrains evolution. In this case, the physics of diffusion limits how large an animal can be before needing a circulatory system to move oxygen around.

How do you create an exam problem on this subject? Here's the trick: Do Problem 37 in class and then put a question on the exam identical to Problem 37 except “sphere” is replaced by “cylinder”. The problem is only slightly changed; just enough to determine if the student is solving the problem from first principles or merely memorizing. In addition, nerve and muscle fibers are cylindrical, so the revised problem may provide an even better model for those cells. Depending on the mathematical abilities of your students, you may need to provide students with the Laplacian in cylindrical coordinates. (If the exam is open book then they can find the Laplacian in Appendix L).

Here’s a second example: Chapter 1 considers viscous flow in a tube; Poiseuille flow. On the exam, ask the student to analyze viscous flow between two stationary plates.

Section 1.17

Problem 36 ½. Consider fluid flow between two stationary plates driven by a pressure gradient. The pressure varies in the x direction with constant gradient dp/dx, the plates are located at y = +L and y = -L, and the system is uniform in the z direction with width H. The fluid has viscosity η.

(a) Draw a picture the geometry.

(b) Consider a rectangular box of fluid centered at the origin and derive a differential equation like Eq. 1.35 governing the velocity v_x(y).

(c) Solve this differential equation to determine v_x(y), analogous to Eq. 1.37. Assume a no-slip boundary condition at the surface of each plate. Plot v_x(y) versus y.

(d) Integrate the volume fluence and find the total flow i. How does i depend on the plate separation, 2L? How does this compare with the case of flow in a tube?

An interesting feature of this example is that i depends on the third power of L, whereas for a tube it depends on the fourth power of the radius. Encourage the student to wonder why.

Third example: A problem in Chapter 7 compares three different functions describing the strength-duration curve for electrical stimulation. On your exam, have the students analyze a fourth case.

Section 7.10

Problem 46 ½. Problem 46 analyzes three possible functions that could describe the strength-duration curve, relating the threshold current strength required for neural excitation, i, to the stimulus pulse duration, t. Consider the function i = A/tan^-1(t/B). Derive expressions for the rheobase i_R and chronaxie t_C in terms of A and B. Write the function in the form used in Problem 46. Plot i versus t.

And still more: Problem 32 in Chapter 8 examines magnetic stimulation of a nerve axon using an applied electric field E_i(x) = E₀ a²/(x² + a²). Give a similar problem on your exam but use a different electric field, such as E_i(x) = E₀ exp(-x²/a²).

And yet another: Chapter 10 examines the onset of cardiac fibrillation and chaos. The action potential duration APD is related to the diastolic interval (time from the end of the previous action potential to the start of the following one) DI by the restitution curve. Have the student repeat Problem 41 but using a different restitution curve: APD_i+1 = 300 DI_i/(DI_i + 100).

Final example: Problem 36 in Chapter 9 asks the student to calculate the electrical potential inside and outside a spherical cell in the presence of a uniform electric field (Figure 9.19). On your exam, make the sphere into a cylinder.

I think you get the point. On an exam, repeat one the of homework problems in IPMB, but with a twist. Change the problem slightly, using a new function or a modified geometry. You will be able to test the knowledge and understanding of the student without springing any big surprises on the exam. Many problems in IPMB that could be modified in this way.

Warning: This trick doesn’t always work. For instance, in Chapter 1 if you try to analyze fluid flow perpendicular to a stationary object, you run into difficulties when you change the sphere of Problem 46 into a cylinder. The cylindrical version of this problem has no solution! The lack of a solution for low Reynolds number flow around a cylinder is known as Stokes’ Paradox. In that case, you’re just going to have to think up your own exam question.