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Resistive electric flow: Ohm's law

Page history last edited by Joe Redish 8 years, 3 months ago

Class Content > Electric Currents

 

Prerequisites:

 

Moving charge feels resistance

When electric charge is moving through a material, it typically feels a resistance that tends to oppose the motion, coming from their interaction with other charges in the material moving thermally.* From our experience with resistive forces, we can't be sure whether the resistance felt by the moving charge is independent of velocity (like friction), proportional to velocity (like viscosity), or proportional to the square of the velocity (like drag). For many situations, it appears that the resistive force a moving charge feels is more viscosity like -- proportional to velocity. We will see that the assumption that this is what the drag is like is equivalent to Ohm's law -- a relation that holds very well for many systems. Since we don't know exactly what the mechanism for the drag is (it's complicated), we'll describe it phenomenologically, saying just that it's proportional to the velocity times some constant that is characteristic of the resisting medium:

 

Fresistive = - bv

 

What keeps it going?

To keep a charge moving through a resistive medium we need a force to balance the drag. Since  we want to move charges, it's most natural to think about the electric force as the force pushing them  through the resistive medium. 

 

To keep a charge, q, moving at a constant velocity through a resistive medium, we need an electric force, qE. If our charge is moving with velocity v, to keep it moving at constant velocity we need the two forces to balance (by Newton's second law):

 

Fnet = qE - bv = 0

or

qE = bv.

 

Ohm's law

Now lets consider a cylinder consisting of, say, ions and electrons, on which we place an electric field. The ions will respond perhaps 120,000 times less than the electrons (the ratio of the mass of a copper ion to the mass of an electron) so we can ignore the motion of the ions.

 

Let's consider a cylinder of charge of cross sectional area A and length L with charge carriers q having a density n. To get an E field across the volume we'll impose a potential difference ΔV. This will produce an average E field 

 

E = ΔV/L

 

Balancing our forces gives

qE = bv

qΔV/L = bv

 

Now we want to get rid of the v in favor of the current, I. Recall that current is given by (See Quantifying electric current) the amount of charge crossing an area per second, or

 

I =(amount of charge crossing area in a time Δt)/Δt

 

I = (charge on a single carrier)(number of carriers per unit volume) x
(volume crossing area in time Δt)/Δt

 

I = qn(AvΔt)/Δt = qnvA

 

We can therefore solve for v in terms of I as

 

v = I/qnA

 

Putting this into our balance of forces equation gives

 

qΔV/L = bI/qnA

 

or 

ΔV = (bL/q2nA) I

 

The combination bL/q2nA is a property of the particular cylinder we are looking at -- its material (which determines what are q, n, and b) and its shape (which determines L and A). It's called the resistance of the cylinder, R:

 

R = bL/q2nA.

 

The result is the powerful equation, Ohm's law,

 

ΔV = IR.

 

What does it mean?

Basically, we can see from the derivation where Ohm's law comes from. It all starts with the statement that the push (coming from the E field) is balanced by the drag (proportional to v) so we maintain a constant velocity (according to Newton's 2nd law). Since we can't easily create E fields quantitatively but can easily manipulate potential, we express this in terms of the potential difference across the cylinder (resistor). Since we can't easily measure the speed of our current carriers, but do have devices (ammeters) to measure currents directly, it's convenient to express the velocity in terms of the current.

 

The result makes a kind of intuitive sense: more push means more flow; more resistance for the same push results in less flow. 

 

To think about what the implications of this are, we will have to consider a variety of models and establish some principles for the use of this law to help figure out what flows where.

 

Biologist vs Electrical Engineer's Ohm's Law

Since electrical resistors are basically passive, electrical engineers are very comfortable with the idea of resistance -- that matter resists a current flow. But in biological systems, there system often adjusts its resistance in order to actively manipulate the current flow. In these situations, it is sometimes more convenient to think about not just a voltage drop, ΔV, as what creates a current, but the inverse of the resistance as contributing. We therefore sometimes define the inverse of the resistance, 1/R, as the conductance, G. So our equations become:

 

G = 1/R = q2nA/bL

I = GΔV

 

These are equally valid forms of Ohm's law. (It's no different from describing a motion in terms of velocity -- miles/hour -- or its reciprocal, pace -- minutes/mile. Which is easier to use depends on what you are calculating and they are fully equivalent, formally.) 

 

Units 

From Ohm's law it is clear that an appropriate unit for resistance is the "volt/Ampere". This combination can be unpacked -- volt = Joule/Coulomb, Ampere = Coulomb/sec, so volt/Ampere = Joule*sec/Coulomb2 = kg-m2/C2-s. Since "b" has to have units of kg/s in order for bv to come out a force (kg-m/s2), this matches with our formula for R.

 

This messy combination is given the designation "Ohm" and is written with a Greek capital omega (Ω). The unit for conductance is (of course -- what else could it be?) the "Mho".

 

Follow-ons:

 

*Except in very special circumstances -- such as metals and some other materials at very, very low temperatures -- at which point the resistance to flow can vanish. This is called superconductivity.

 

Joe Redish 2/27/12

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