Resistive electric flow -- Ohm's law

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

$$F^{resistive} = -bv$$

The minus sign says that the force is in the opposite direction from the velocity.

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):

$$F^{net} = qE - bv = 0$$

or

$$qE = bv.$$

Ohm's law

Now let's 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 of $n$. 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 the page, quantifying electric current) the amount of charge crossing an area per second, or

$$I = \frac{\mathrm{amount\;of\;charge\;crossing\;area\;in\;a\;time} Δt}{Δt}$$

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

$$I = \frac{qn(AvΔt)}{Δt} = qnvA$$

We can therefore solve for $v$ in terms of $I$ as

$$v = \frac{I}{qnA}$$

Putting this into our balance of forces equation gives

$$\frac{qΔV}{L} = \frac{bI}{qnA}$$

Solving for $\Delta V$ gives

$$ΔV = \bigg(\frac{bL}{q^2nA}\bigg) I$$

The combination $bL/q^2nA$ 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, $R$:

$$R = \frac{bL}{q^2nA}.$$

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. It's a standard gradient-driven flow equation, where a change in some scalar field in space leads to something moving. Other examples include the H-P equation, heat flow by conduction, and Fick's law of diffusion.

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, the system often adjusts its resistance in order to actively manipulate the current flow. In these situations, it is sometimes more convenient to think of the current as a result of not just a voltage drop, $\Delta V$, but of the inverse of the resistance as well. We therefore sometimes define the inverse of the resistance, $1/R$, as the conductance, $G$. So our equations become:

$$G = 1/R = \frac{q^2nA}{bL}$$

$$I = GΔV$$

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.) 

Resistivity: A density of resistance

When we have had properties of matter that depend both on what the matter is and what its shape is, we've found it useful to create a density. The obvious example is mass, where an oblect's mass is equal to its (mass) density times its volume: $m = \rho V$ (here, volume, not potential). Another place we had this was in creating a spring constant for a block of matter: $k = YA/L$ where $Y$ is Young's modulus. You can find a number of others.

Here, the electrical resistance can also be separated into a density-like quantity that only depends on the material, and the properties of the material:   

$$R = \bigg(\frac{b}{q^2n}\bigg) \frac{L}{A}.$$

The combination $b/q^2n$ only depends on the substance and not on the particular shape of the resistance. The combination is called the electrical resistivity  and is often written $\rho$. (Sorry about that.) 

Note that the HP equation for fluid flow in a pipe has a form very similar to Ohm's law: a pressure difference produces a flow through a resistance:

$$\Delta p = R Q$$

where $p$ is the pressure differential, $Q$ is the volumetric rate of flow, and the fluid resistance, $R$, is

$$R = (8 \pi \mu) \frac{L}{A^2},$$

a similar form to Ohm's law, but the resistance in the HP equation depends on $1/A^2$ rather than on $1/A$. Can you see from the derivations what the source of this difference is?

Where does the voltage difference come from?

Since we knew we had a steady current, and since we knew we had some resistance, Newton 2 told us we had to have a pushing force to balance the resistive force. We assumed this was produced by an E field associated with a potential difference. But where does that potential difference come from?

Let's consider an idealized model where we have a battery (that creates a voltage difference and hence an E field) pushing some charges down a wire. If the wire is essentially resistanceless (and most wires have a very low resistance), any charges that start moving don't slow down. They keep moving at a constant speed. But suppose now the hit a resistive region as shown in the figure below.

When the moving charges (blue, or + in this diagram) hit the resistor they feel the resistance and start to slow down, building up some excess + charge at the front end of the resistor. These excess charges create an E field in the resistor which drives positive charges out, leaving a deficit of + which is an excess of - charges.

Once this stabilizes (in about a nanosecond in a typical macroscopic circuit), we have a sheet of + on one side of the resistor and a sheet of - on the other side. This is like a capacitor, establishing an E field between them that is just the right amount to keep the charges moving through at a constant rate. (There's no magic to this. If there weren't enough charges to keep them from slowing down, more charges would build up increasing the E field until is was just enough. Then it wouldn't increase any more and a steady state would be established.)

We can measure the voltage difference across the ends of the resistor. If the E field were constant inside, then its magnitude would be given by

$$E = \frac{\Delta V}{\Delta x} = \frac{\Delta V}{L}$$

Though $E$ probably isn't constant, this will still be the average E field, $\langle E \rangle$.

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 the unit of resistance is

• 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 detailed 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".**

*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.

** Alas, this is no longer so. The official term has been redefined so that 1 inverse Ohm is officially called a Siemens.

Joe Redish 2/27/12