Physics 106b-2014 Assignment 4
Due: Fri., Feb.28, 4:00 pm (turn in to envelope on my sill)
Reading: Ch. 25
A) Mastering Physics Problem Set 4
B) Handwritten problems:
4A. Electrostatics & biochemistry grab bag:
The relevant values of dielectric constant,  are 1 for vacuum, ~3 for oils, ~8 for a cell membrane, and about 80
for water. (Here ~ or tilde means “approximately equal to”.)
a) Calculate the magnitude of force between two elementary electronic charges, e, separated by 1.00 nm
(molecular dimensions) in vacuum, oil and water. How much greater or smaller are these forces than the ~ 6
pN forces we see are often provided by molecular motor proteins (the little chemical machines that move things
around in cells.) (Comment: Molecules have dimensions on the nanometer scale, and often have patches of
charge about this size on their surface, so this comparison is a very important one for biochemistry.)
b) The energy available in the environment of our proteins due to just thermal energy ½ kB T (where kB is
Boltzmann’s constant and T is in units of Kelvin, those of absolute temperature.). Compute the “Bjerrum
length”, lB, defined as the distance between two elementary charges e where their electrical potential energy
equals this thermal energy, assuming T = 300 K. Assume a dielectric constant  = 76.5 for the cell’s interior.
(This length tells you over what distances thermal energy can overcome electrostatic interactions.)
4B. As we discussed in lecture, in my research group
we measure the electronic properties of self-assembled
nanowires. We use substrates made from silicon that
has been “doped” (altered by adding impurities) so that
it behaves as a metal. The silicon is coated with a thin
layer of SiO2, which is an insulator. We use electron
beam lithography to create a set of metal electrodes on
top of the SiO2; these electrodes are made from a goldpalladium alloy, AuPd. For this problem, assume that
each electrode is a square that is 1 mm on a side; the
thickness of the AuPd is 20 nm. (1 nm = 1 nanometer
= 10-9 m). We then deposit nanowires from solution
onto these electrodes. We load the sample into our
measurement chamber, where we use spring-loaded
“pogo pins” to make electrical contact to the AuPd
electrodes; the pogo pins connect to our measuring
apparatus. Finally, we apply a voltage difference
between the two pogo pins, and measure the current
that flows through the nanowire.
Our nanowires only act as conductors when
they’re illuminated with a laser. For the test described
here, we don’t have the laser on. Therefore, for this
preliminary test, the nanowires are not electrically
important, and so are not shown in the circuit model at
the bottom of the figure. However, the capacitors
formed by each of the AuPd electrodes and the Si
substrate below it are important. For each of these capacitors, the SiO2 acts as a dielectric, with  = 3.9. As
shown, these two capacitors are in series, because the part of the substrate below the left AuPd electrode is
connected to the part of the substrate below the right electrode.
(Problem continues on the next page.)
To insure that we have good electrical contact between the pogos and the AuPd electrodes, we run a
“capacitance test”, in which we ramp the voltage (i.e. change the voltage at a constant rate) that is applied
across the pogos from 0 V to 0.25 V, hold it for several seconds, then ramp it back to 0 V. While doing this, we
measure the current. The results are shown here; ignore the spikes at 20 s and 30 s; these are due to other tests
we perform while the voltage is being held constant at 0.25 V. (The current scale is given in “pA”, short for
“picoamps”; 1 pA = 10-12 A.) The increasing voltage ramp begins at 7.46 s and ends at 12.38 s. The decreasing
voltage ramp begins at 41.11 s and ends at 46.03 s. As you can see from the graphs, the current is
approximately constant during each of these voltage ramps. The level of current tells us that we have good
electrical contact between the pogos and the AuPd electrodes.
a. Explain why there is a constant positive current while the voltage is being increased, and a constant negative
current while the voltage is being decreased.
b. From the information given in this problem, estimate the thickness of the SiO2. Explain your reasoning
clearly.