A Solution to RWP1 - Gel Electrophoresis

Gather information:  We know from the statement of the problem that the DNA fragments are negatively charged, and that the amount of charge is proportional to the length of the fragment, but we do not know for certain the amount of charge per base pair.  The smallest possible charge per base pair is: q = -2e = -2(1.6e-19 C).
The molecular "weight" of an average base pair = 635 daltons = 635(1.67e-27 kg) = 1.06e-24 kg
Fragments that can be analyzed range from 200 bp to 50 kb, corresponding to:  m = 2.1e-22 kg to 5.3e-20 kg
Note:  The largest DNA fragment that can be analyzed with this technique is a small fraction of the DNA in a haploid nucleus:  (5e4)(635)/1.9e12 = 1/60,000
Based on the time required for the DNA fragments to travel, they must not be accelerating, but are migrating at a terminal speed of about 1 ft/hr. 
Altough not explicitly stated, the applied electric field must be uniform, created by two parallel linear electrodes at a difference of 50 to 150 V.
Assuming the gels are run the length of the box, then the strength of the electric field must be approximately E = (100 V)/(20 cm) = 5 V/cm = 500 V/m.

Organize:  This problem is primarily an application of a uniform electric field.  The important aspect that is different from previous textbook problems is the fact that the gel provides a resistive force which opposes the electrostatic force on the negatively-charged DNA fragments.

Analyze:  The NET force acting on a DNA fragment must be zero, otherwise it would accelerate and take much less than 30 minutes to travel ~20 cm.
Verification: Assuming only the minimum electric charge of -2e per base pair, the electric force on a typical DNA fragment of length 1000 bp would be:
    F = qE = -2(1.6e-19 C)(500 N/C) = -1.6e-16 N
    With no opposing force, this would result in an acceleration: a = F/m = (1.6e-16 N)/(1000*1.06e-24 kg) = 1.5e5 m/s^2 = 15,000 g !!!
    If this were true, then the DNA fragments would travel the length of the gel box in a fraction of a second!
It is clear from the the Physlet simulation that the smaller DNA fragments do travel faster and migrate farther than the larger molecules, despite the fact that the larger molecules should have a greater electric force since the charge is proportional to the number of base pairs.  Evidently, the resistive force on these molecules depends greatly on their size, much like objects falling through air where density is more important than size in determining terminal speed. 
The Physlet simulation does not show that the larger DNA fragments migrate proportionally faster in higher fields, but instead there is simply a constant shift to the right in the location of the bands as the applied voltage is increased.  If this non-linear effect does occur in real life (as indicated in the problem statement), then it could be due to the non-linear drag forces acting on the larger molecules.
From this simulation, it appears that the rate of migration increases with the applied field strength, but not linearly.  Instead, the rate of migration increases quickly for lower fields and more slowly for higher fields, yielding a curve that looks similar to the I-V plot for a light bulb (rising up and to the right as it begins to level off).

Learn:  The movement of the DNA fragments in a gel electrophoresis box is very similar to the movement of electrons in a wire.  In both cases, the negatively-charged particles move in response to an external electric field, but these mobile particles are slowed to a "snail's pace" by obstacles that impede their motion.  If the electrodes were points instead of lines, the resulting electric field would not be uniform, and the rate of migration would be greater in the center (where the field is strongest) than near the edges of the gel box.  Even with the parallel-line geometry, the electric field is somewhat weaker near the sides of the box, so hopefully the analysis is confined to the rectangular region between the linear electrodes.

Grading rubric: (10 pts. total)
2 - Gather information:  charge, mass, speed, electric field
1 - Organize: approach using constant E with resistive force
2 - Analyze:  Net force = 0, electric force calculated for typical DNA fragment
1 - rate of migration is non-linear, graph of migration versus E increases but more slowly (concave down)
1 - recognition that smaller fragments travel faster despite less electric force
1 - effect of non-uniform electric field from pointed electrodes
2 - Learn:  relevant physical insights (similarity to electric current, field fringing effects, etc.)
+1 - Overall effort, organization, neatness, and citation of references

Key to comments marked on papers:

1. Failed to gather sufficient information about the experiment. Needed to say something about charge/mass/speed/Electric field. See the "G' section of the solution posted online for an example of what was expected.

2. Did not give a satisfactory explanation of why the rate of motion is non-linear.

3. Did not sufficiently explain the effect of non-uniform electric field from point electrodes.

4. Did not adequately discuss how DNA fragments of different sizes experience different drag forces.

5. Did not conclude that net force on DNA is 0 N

6. Relevant insights insufficient. See solution for an example of what was expected.