Download The Physics of Electricity

The Physics of Electricity
Before we can understand the function of nerve cells, receptors, and muscle fibers, we need
to understand something about their bioelectric properties. And, before we can understand
bioelectrical phenomena, there are some terms and concepts from the physics of electricity with
which we need to be familiar. These are:
• Charge
• Electrical potential and potential difference
• Current
• Resistance and conductance
• Capacitance.
A basic understanding of those seven terms will make your life easier when we start talking
about the neurophysiology in a week or so.
I. Electric Charge
• Charge is a fundamental property of matter.
• Charges are either positive (+) or negative (−)
• Charge results from the presence of electrons (e−) or protons (p+), and a charged
region in space results from a net excess of electrons or protons.
• Charge is represented by q.
• Charge is measured in coulombs; one coulomb of charge equals the amount of
electrical charge in 6.241506 ×1018 electrons, protons, or other charges.
II. Electric Potential and Potential Difference
• Electric potential is the form of energy that makes charges tend to move.
o It thus serves as the driving force for movement of charges.
o Electric potential can also be thought of as the tendency for charges to move.
o We usually drop the “electric”, and just refer to potential.
• Potential is measured in units of volts (V) or millivolts (mV).
o Potential is variously represented by V, E, ε, or ψ.
o We will usually use V or ψ, but your physics buddies probably prefer E.
• A region of net negative (or positive) charge is surrounded by a potential field,
and the energy in this field will cause charges to move.
o The more charges in a region, the stronger the potential field surrounding the
charged region, which can be appreciated by studying the following diagrams:
++++
+++ ++
+++
−−
−−−−−−
−−−−
More separated charge
= greater tendency for opposite charges to
move toward one another and for like
charges to move away from one
another
= greater driving force for charge to move
= increased (electric) potential
+
+
++
− −
− −
Less separated charge
= decreased tendency for opposite
charges to move toward one another
and for like charges to move away
from one another
= less driving force for charge to move
= decreased (electric) potential
o The ‘strength’ of the potential field – and therefore the tendency of charges to
move – decreases exponentially with distance:
++++ − −
+++ ++ −−−−−
+++ −−−−−
−−−−
−−− −−
−−−
++++
+++ ++
+++
Opposite charges interact strongly
= greater tendency for opposite charges
to move toward one another
= greater driving force for charge to
move
= increased (electric) potential
Opposite charges interact weakly
= less tendency for opposite charges to
move toward one another
= less driving force for charge to move
= decreased (electric) potential
Another way of visualizing the same concept is presented in the diagram
below, in which
1. the ‘strength’ of the potential field indicated by the spacing of the
dashed lines (greater separation = weaker field), and
2. the tendency for the charges to move relative to the charged region at
the center is loosely indicated by the length/weight of the arrows:
+
−
+
++++
+++ ++
+
++ +
−
−
•
The more charges in a region
o the stronger the driving force for current flow, and
o the farther the field extends outward from the edge of the region containing
the charges.
•
•
•
•
The tendency for charges to move between the two regions of opposite charge – and
therefore the potential difference between the two regions – depends on two important
parameters:
1. the amount of net positive or negative charge in each of the regions.
a. more net charge = more separated charge = greater potential difference 
greater tendency for charges to flow between the two regions.
2. the distance separating the charged regions.
a. For a given amount of separated charge, increasing the distance between
the charged regions decreases the strength of the potential field between
them  decreased tendency for charges to flow between the two regions.
In biological systems, potential differences (across a plasma membrane, say) are
established by the separation of charge. There’s no other way. Based on this, we can
state:
o No separated charge = no potential difference  no tendency for charges to
move.
o The greater the amount of separated charge, the greater the potential
difference (i.e., the driving force), and therefore the greater the tendency for
charges to move.
Potential difference between two points is measured using a voltmeter, and the units
are volts.
Potential difference can be represented as ∆V or ∆Ψ, but we usually drop the ∆.
IV. Current
• Current is simply the movement, flow, or flux of charge
• In solids, the flowing charge is usually electrons; in liquids, it can be either electrons
or ions. In biological systems, current almost always involves ions. In this course
we’ll make frequent reference to sodium, potassium, calcium, and even chloride
currents.
• Current is usually represented as I, occasionally as A.
• The unit is the ampere; 1 ampere = 1 coulomb of charge moving past a point in one
second.
V. Resistance and Conductance
• The resistance of a medium (a copper wire, a plasma membrane, cytoplasm, etc.)
measures how difficult it is for charges to move through the medium. This is true
whether the charges are in the form of electrons or ions.
• Conductance measures how easy it is for the charges to move through the medium.
• Resistance is represented by R, conductance by g.
• Resistance and conductance are inversely related:
𝑔𝑔 = 1�𝑅𝑅 ⟺ 𝑅𝑅 = 1�𝑔𝑔
• The unit of resistance is the ohms, represented by the Greek letter omega (Ω).
• Because conductance is the inverse of resistance, the unit of conductance was
originally the mho (the inverse of ohm, you see…). Probably out of disgust with the
geeky mindset that led to that choice, the unit was changed to the Siemen – much to
the obvious amusement of Biology majors.
VI. Capacitance
• Capacitance is basically the ability to store (separated) charge.
• A simple capacitor consists of two parallel plates of a conducting substance, usually
metal, separated from each other by a gap that contains a non-conducting dielectric.
In a typical application, the two plates of the capacitor are connected by individual
wires to an electromotive force, such as a battery, in such a way that it can pull
electrons (e−) from one plate and ‘deposit’ them on the other:
Capacitor, C
−
+
−
+
−
− +
e
−
+
−
+
+
−
+
−
Plate
e−
Dielectric
Plate
Battery, 𝜀𝜀
•
•
•
•
e−
As the left-hand plate becomes more positively charged (through loss of electrons), it
becomes progressively more difficult to continue removing electrons from the plate.
Likewise, the accumulating negative charge on the right-hand plate makes it
progressively more difficult to add more electrons. Ultimately, the accumulating net
charges on the two plates stop the flow of current through the circuit, and the
capacitor is said to be fully charged.
Once developed, the charge separation will persist as long as there’s no path for
electrons to flow from one plate to the other, even if you disconnect the capacitor’s
plates from the battery. (this is why it’s dangerous to start poking around TV sets,
even after they’ve been unplugged; their capacitors contain enough charge to knock
you on your patootie, maybe even kill you.)
The amount of charge (𝑞𝑞𝐶𝐶 ) stored on the plates depends on ‘strength’ of the
electromotive force (𝜀𝜀, measured in volts) and the inherent ability of the plates to
store charge, which is termed the capacitance, 𝐶𝐶. The mathematical representation of
this relationship is:
𝑞𝑞𝐶𝐶 = 𝐶𝐶𝐶𝐶
The unit of capacitance is the farad = 1 coulomb/volt ( = amount of charge per unit
driving force).
VII. Ohm’s Law
Ohm’s law is a quantitative expression of the relation among potential, current, and
resistance:
𝑉𝑉 = 𝐼𝐼𝐼𝐼
or, in words, potential = current × resistance. Now, to a budding physiologist such as
yourself, who’s becoming accustomed to thinking in terms of fluxes (current, in this case)
being caused by driving forces (potential, in this case), that formulation of Ohm’s Law
probably looks rather strange and uninformative.
A much more useful formulation can be derived by substituting conductance for
resistance:
1
𝑉𝑉 = 𝐼𝐼𝐼𝐼 = 𝐼𝐼 � �
𝑔𝑔
which can be rearranged to yield:
𝐼𝐼 = 𝑔𝑔𝑔𝑔
Equation 1
In other words, we now have:
current = conductance×potential
which can be expressed verbally as “the current (i.e., the amount of charge moving past a
point) is equal to the driving force tending to cause current to flow (i.e., the potential)
multiplied by the ease with which charges can flow (i.e., the conductance).” Note the
analogy between our reformulated Ohm’s law and Newton’s Law of Cooling.
VIII. The RC Circuit
The RC circuit is a special type of electrical circuit that consists of a resistance, a capacitor,
an electromotive force (usually in the form of a battery), and a switch. The RC circuit exhibits
electrical properties and temporal dynamics that are useful in the study of the electrical
properties of excitable cells such as neurons and muscle fibers. Refer to the diagram below as
you work through the following discussion.
Initially, the switch is in the open position, meaning that there is no possibility of current
flow from one plate, through the battery, to the other plate. The capacitor thus will have no
separated charge stored on its plates (𝑞𝑞𝐶𝐶 = 0; left, below).
Capacitor, C
e−
Resistor, R
Close
Switch
+
+
+
+
+
+
+
Switch
Battery, 𝜀𝜀
e−
−
−
−
−
−
−
−
e−
When the switch is closed to complete the circuit between the capacitor and the battery, a
current in the form of electrons is driven through the circuit by the battery, just as we saw earlier.
This will result in one plate of the capacitor developing a net positive charge (as electrons are
removed), while the other becomes negatively charged as electrons accumulate on it.
However – and this is critically important for the functioning of neurons – the resistance (R)
slows electron flow through the circuit, and charging of the capacitor takes time, as evidenced in
this graph of the capacitor’s charge state vs. time, which is based on 𝑅𝑅 = 3 ohms, 𝐶𝐶 =
0.8 farads, and 𝜀𝜀 = 10 volts:
The equation for the line in the above graph is:
𝑡𝑡
𝑡𝑡
𝑞𝑞𝐶𝐶 (𝑡𝑡) = 𝐶𝐶𝐶𝐶 �1 − 𝑒𝑒 − �𝑅𝑅𝑅𝑅 � = 8 �1 − 𝑒𝑒 − �2.4 �
Equation 2
and we note that the plot ultimately stabilizes at 𝑞𝑞𝐶𝐶 = 𝐶𝐶𝐶𝐶 = 8.0 coulombs. The potential
difference between the two plates is given by 𝑉𝑉𝐶𝐶 = 𝑞𝑞𝐶𝐶 ⁄𝐶𝐶 , and is therefore a linear function of 𝑞𝑞𝐶𝐶 ,
so the graph of VC versus time exhibits the same qualitative temporal dynamics as 𝑞𝑞𝐶𝐶 :
The change in 𝑉𝑉𝐶𝐶 with time in the above figure described by the equation:
𝑡𝑡
𝑡𝑡
𝑉𝑉𝐶𝐶 (𝑡𝑡) = 𝜀𝜀 �1 − 𝑒𝑒 − �𝑅𝑅𝑅𝑅 � = 10 �1 − 𝑒𝑒 − �2.4 �
Equation 3
Note that the potential difference between the two plates, 𝑉𝑉𝐶𝐶 , stabilizes at the value of 𝜀𝜀, which
was given as 10 V. Also note the general similarity between the form of Equations 2 and 3 and
the mathematical expression for Newton’s Law of Cooling as presented in lecture.
Once the capacitor is fully charged, we flip the switch to its other pole, thereby removing the
battery from the circuit and giving the excess electrons stored on the capacitor’s right-hand plate
a path to the left-hand plate:
−
−
+
+
−
−
+
+
−
− −
+
+
−
e
−
− e
+
+
−
−
+
+
−
−
+
+
−
−
+
+
Flip the
Switch
e−
The electrons immediately start to flow to the left-hand plate, driven by the potential difference
between the two plates, just as we described earlier. Once again, however, this process, termed
discharging of the capacitor, requires time:
The equation describing the discharge of a capacitor through a resistance is, as you would
expect, closely related to the equation that describes the charging dynamics. I.e.,
𝑡𝑡
𝑡𝑡
𝑞𝑞𝐶𝐶 (𝑡𝑡) = 𝐶𝐶𝐶𝐶𝑒𝑒 − �𝑅𝑅𝑅𝑅 ⟺ 𝑉𝑉𝐶𝐶 (𝑡𝑡) = 𝑉𝑉0 𝑒𝑒 − �𝑅𝑅𝑅𝑅
Equation 4
where 𝑉𝑉0 , the initial potential difference between the plates of the fully-charged capacitor, is
determined by the ‘strength’ of the battery in the RC circuit (10 volts in this example).
As we’ll see, this temporal dynamics relates directly to one of the most important functions
of neurons in the brain – transmission of information from one neuron to another.
The Take-home Messages
OK. The preceding is a lot to digest and assimilate, but nonetheless very important to
understand if you’re going to get comfortable with how many physiological systems, including
the nervous, cardiovascular, gastrointestinal, respiratory, and reproductive systems, are able to
perform their very essential functions. For now, the key facts and concepts you should
remember from this essay are:
• Potential differences in biological systems result from separation of positive and
negative charges, and the greater the amount of separated charge, the greater the
potential difference.
• Current in biological systems is usually the result of moving ions such as Na+, K+,
Ca++, or Cl−, rather than moving electrons.
• Potential differences are the driving force for current, and the larger the potential
difference, the larger the current, all else being equal.
• Ohm’s Law can be rearranged to yield a very useful and informative form (Equation
1) that describes the dependence of current on conductance and potential:
•
𝐼𝐼 = 𝑔𝑔𝑔𝑔
The capacitor ‘experiments’ described above show that:
o resistance reduces the rate at which current moves in response to a potential
difference.
o potential differences cause current to flow, and
o current flow can create or destroy the charge separation that produces a
potential difference.
=====================================================================
Questions You Should Ponder
1. What are we really saying when we talk about a “1.5 volt flashlight battery” or a “12 volt
automotive battery”? Hint: note that the units are volts.
2. When we talk about a 120 volt wall outlet, what are we really saying?
3. Thinking about what produces potential differences, can you explain why potential can
be thought of as the tendency for charges to move? How does this relate to current?
4. What exactly did I mean when I referred to the “analogy between our reformulation of
Ohm’s law and Newton’s Law of Cooling”?
5. What is directly responsible for the potential difference between the plates of a charged
capacitor?
Extra Credit (5 points)
1. What is the effect of varying R and/or C on the time required for a capacitor to
completely charge or discharge? Hint: construct models of Equations 3 and 4 using
Excel or a programming language of your choice, and use the output of your models to
produce graphs of 𝑉𝑉𝐶𝐶 versus time for a few values of R and C. (you could also use the
simulation referred to below to accomplish this.) What is the effect of increasing R
and/or C on the time required to fully charge or discharge the capacitor? Use what
you’ve learned about the physics of electricity to explain your results.
More Stuff
1. These websites may be of some interest/assistance:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://ww2.slcc.edu/schools/hum_sci/physics/tutor/2220/index.html
2. If you want to learn more about RC circuits and their importance to neurophysiology, you
might work through the exercise and computer simulation that I’ve posted at:
http://biology.unm.edu/toolson/b435/rc_circuit.html.