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Physics E-1bx: Pre-reading for Lecture 3
February 24, 2015
Lecture 3: Pre-reading
Today, we’ll talk more about circuits in which batteries, resistors, capacitors, and other
circuit elements are connected together by wires (conductors). A battery creates a potential
difference (voltage), which can cause current to flow.
Last week we examined resistors in parallel and resistors in series, and we found that we
could replace a bunch of resistors with a single equivalent resistor. It turns out that you can do
the same thing with capacitors, although the rules end up being (mathematically) exactly the
opposite. We will derive the results for capacitors in class, but here is a simple table to
summarize the key points:
Resistors
Requiv = R1 + R2 +
in series:
1
in parallel:
Requiv
=
1
1
+
+
R1 R2
Capacitors
1
Cequiv
=
1
1
+
+
C1 C2
Cequiv = C1 + C2 +
An important use for capacitors is to store electrical potential energy U. If a capacitor
with capacitance C has charge q and voltage ΔV, the stored potential energy is:
U elec
1 q2 1
1
=
= qΔV = C(ΔV )2
2C 2
2
We write the energy in these three equivalent forms using the basic relationship C = q/ΔV.
(Check yourself that this is true!)
For any circuit element, if a current i has a potential drop ΔV, the power dissipated by
that circuit element is given by:
P = iΔV
In the case of a resistor you can use Ohm’s law to derive:
P = i2R
P=
(ΔV )2
R
(Again, check the math for yourself to see how we’re using Ohm’s Law here…)
With more complicated circuits, it’s not always possible to simply combine circuit
elements in series or in parallel—we need more powerful tools to analyze circuits. The basic
Physics E-1bx: Pre-reading for Lecture 3
February 24, 2015
tools are known as Kirchhoff’s Laws. They are not really new ideas, but it will take some
practice to learn how to use them.
The first law is the junction rule. At any point in a circuit where two or more wires
come together at a junction, the total current flowing in to the junction must equal the total
current flowing out of the junction. This rule arises because charge is conserved, and no charge
can actually build up at a junction. (It may be helpful to think by analogy about pipes with water
flowing through them—if three pipes come together, the water flowing in must equal the water
flowing out.)
The second law is the loop rule. If you add up the potential differences ΔV for every
circuit element around a complete loop, the sum must be zero. This is another way of saying that
the potential V is defined at every point in a circuit—so if the potential at a point is, say, +3V,
then if you go around a loop you must get back to a potential of +3V. (For this rule, a good
analogy is to think about altitude and hiking: if you hike around a complete loop, the height that
you climb must equal the height that you descend. In other words, it’s impossible for your walk
to class and back to be uphill both ways.) We will often use the term voltage to refer to potential
differences, so the loop rule says that the sum of the voltages for all elements in a loop must be
zero. You’ll have to pay close attention to the sign of ΔV here—we’ll explain that in class.
Finally, in this lecture we introduce time-dependent circuits: these are circuits in which
some parameter (like the charge, or voltage, or current) changes with time. The most important
example for us are RC circuits, which contain (as their name suggests) a Resistor and a
Capacitor. The circuit shown at right is the simplest possible example of an RC circuit.
Suppose the capacitor starts out with an initial charge Q0, before the circuit is connected.
There will be a corresponding voltage across the capacitor, ΔV = Q/C. At the moment that the
circuit is connected, there will also be the same voltage across the resistor. By Ohm’s Law, this
will create a current through the resistor—and this current will tend to discharge the capacitor.
The current will flow from the (+) plate of the capacitor, through the resistor, to the (–) plate of
the capacitor. Over time, then, the charge on the capacitor will decrease to zero.
As we’ll see in class, the charge Q on the capacitor, as a function of time, is given by the
expression:
Q(t) = Q0 e−t RC
Physics E-1bx: Pre-reading for Lecture 3
February 24, 2015
This expression says that there is an exponential decrease in the charge, starting with the initial
charge Q0 when t = 0. The decay of the charge is characterized by the time constant, which is
given the symbol τ = RC. (Yes, that’s just the product of the resistance R times the capacitance
C.) The time constant tells you how long it takes for the charge to decay to about 37% of its
initial value. (It’s kinda like a half-life, but instead it’s a “37%–life.”) As we’ll see in class, the
number 37% arises because it’s just 1/e—and as you can see, when t = RC, the exponent is just
equal to –1.
You can also have an RC circuit that charges the capacitor—
but in this case you need a battery. The circuit at right will charge the
capacitor, and as we’ll see in class, this circuit can start off with zero
charge on the capacitor and will end up with the capacitor fully
charged to the same potential as the battery. The expression for
charge as a function of time is now: Q(t) = Qmax (1 − e−t RC ) . You should convince yourself that
Q in this case will start at zero when t = 0 and end up at Qmax as t approaches infinity.
The biological significance of these RC circuits is that they model the electrical behavior
of the cell membrane. The cell membrane is an insulator (the lipid bilayer) surrounded on both
sides by conductors (aqueous solutions)—and so it acts just like a capacitor. Embedded in the
membrane are ion channels, which allow ions (current) to flow—like a resistor—and ion pumps,
which are like batteries. So the electrical properties of a membrane are very much like an RC
circuit.
But the potential of a cell membrane isn’t just static—we know that electrical signals can
propagate along the membrane. To understand signal propagation, we need to consider how the
membrane potential can change with distance. A single capacitor can’t have several different
potentials: by definition, it can have only one potential. So we must model the membrane as a
whole bunch of capacitors connected in a row, each of which can have a (slightly) different
potential from its neighbors. Although it may look silly, the simplest circuit model for the axon
would look something like this:
Physics E-1bx: Pre-reading for Lecture 3
February 24, 2015
(Imagine that it continues like that for a long way in both directions.) The wire along the top
represents the fluid outside the membrane, which we assume is a perfect conductor, so all along
the top the potential is the same everywhere. Each capacitor, though, can have a different
voltage—so the potentials along the bottom can all be different. The resistors along the bottom
represent the resistance to current flow inside the cell (note that the cytosol is quite viscous and
will hinder the flow of ions). In this model, there is no flow of charge across the membrane (no
ion channels), and there are no batteries (no ion pumps). Yet an electrical potential will
propagate along the length of this membrane.
As we’ll see in class, we can consider this model to be a whole bunch of RC circuits
linked together. As one of these circuits discharges, its neighbor becomes charged—and vice
versa. It’s kinda like doing “the wave”: when your neighbor stands up, you stand up too; when
your neighbor sits down, you sit down—but there’s a bit of a delay as the wave propagates from
one person to the next. The speed of “the wave” depends on the effective RC time constants of
the membrane, which in turn depends on the materials and the geometry of the cell. We will
construct more elaborate models later involving the ion channels and ion pumps, but this simple
model—known as the cable model—can explain a great deal about the membrane biophysics of
nerve signals. For instance, we can use this model to understand how myelination will
dramatically enhance the speed of nerve propagation along the axon, and why the famous squid
giant axon is so huge (as much as 1 mm in diameter—the size of a strand of spaghetti!)
If you’ve never heard about neurons, or axons, or myelination, don’t worry—we’ll
introduce the basic biological concepts as needed. But there’s a very brief introduction to
neurons that I found online: http://webspace.ship.edu/cgboer/theneuron.html
•
Learning objectives: After this lecture, you will be able to…
1.
Explain what we mean by electrical circuit and construct simple circuits involving
resistors or capacitors in series or in parallel.
2.
Explain the concept of an equivalent resistor or equivalent capacitor.
3.
Replace several capacitors or resistors (either in series or in parallel) with a single
equivalent circuit element.
4.
Explain how a capacitor stores energy, and calculate the energy stored in a capacitor.
Physics E-1bx: Pre-reading for Lecture 3
February 24, 2015
5.
Discuss how circuit elements dissipate power, and calculate the power dissipated by a
circuit element (e.g. for a resistor).
6.
Apply Kirchhoff’s Laws to one- or two-loop circuits to obtain equations for the current
flowing through any element of the circuit.
7.
Solve the equations obtained from Kirchhoff’s Laws to calculate some unknown
quantities (such as currents, voltages, resistances, etc.) for a circuit.
8.
Describe how we can model the cell membrane as an electrical circuit containing
resistors, capacitors, and batteries. Explain the relationship between the biological
structures (membranes, ion channels, and ion pumps) and the analogous circuit elements.
9.
Discuss the qualitative behavior of a simple circuit model of a cell membrane.
10.
Calculate the voltage, current, and charge as a function of time for a circuit containing a
resistor, a capacitor, and (optionally) a battery. This is an RC circuit.
11.
Describe the qualitative behavior of an RC circuit when it is being charged, or
discharged.
12.
Construct a model of the cell membrane as a passive carrier of charge, including the fact
that the potential can vary along the length of the membrane (e.g. along the axon).
13.
Describe quantitatively how the physical and electrical properties of the cell membrane
affect the speed of propagation of an electrical signal, and use this model to explain the
effect of myelination on the speed of nerve signals.