Resistance

The concept of resistance is most likely familiar to readers via Ohm’s Law for Devices; i.e.,

where

is the potential difference associated with a current

. This is correct, but it is not the whole story. Let’s begin with a statement of intent:

Resistance is a property of devices such as resistors, which are intended to provide resistance, as well as being a property of most practical electronic devices, whether it is desired or not.

Resistance is a manifestation of the conductivity of the materials comprising the device, which subsequently leads to the “

” relationship. This brings us to a very important point and a common source of confusion. Resistance is not necessarily the real part of impedance. Let’s take a moment to elaborate. Impedance (

) is defined as the ratio of voltage to current; i.e.,

; or equivalently in the phasor domain as

. Most devices – not just devices exhibiting resistance – can be characterized in terms of this ratio. Consider for example the input impedance of a terminated transmission line (Section 3.15). This impedance may have a non-zero real-valued component even when the transmission line and the terminating load are comprised of perfect conductors.

Restating the main point in yet other words: Resistance pertains to limited conductivity, not simply to voltage-current ratio.

Also important to realize is that whereas conductivity

(units of S/m) is a property of materials, resistance depends on both conductivity and the geometry of the device. In this section, we address the question of how the resistance of a device can be is determined. The following example serves this purpose.

Figure 6.4.1 shows a straight wire of length

centered on the

axis, forming a cylinder of material having conductivity

and cross-sectional radius

. The ends of the cylinder are covered by perfectly-conducting plates to which the terminals are attached. A battery is connected to the terminals, resulting in a uniform internal electric field

between the plates.

To calculate

for this device, let us first calculate

, then

, and finally

. First, we compute the potential difference using the following result from Section 5.8:

Here we can view

as a “given” (being the voltage of the battery), but wish to evaluate the right hand side so as to learn something about the effect of the conductivity and geometry of the wire. The appropriate choice of

begins at

and ends at

, following the axis of the cylinder. (Remember:

defines the reference direction for increasing potential, so the resulting potential difference will be the node voltage at the end point minus the node voltage at the start point.) Thus,

and we have

We do not yet know

; however, we know it is constant throughout the device and points in the

direction since this is the direction of current flow and Ohm’s Law for Electromagnetics (Section 6.3) requires the electric field to point in the same direction. Thus, we may write

where

is a constant. We now find:

The current

is given by (Section 6.2)

We choose the surface

to be the cross-section perpendicular to the axis of the wire. Also, we choose

to point such that

is positive with respect to the sign convention shown in Figure 6.4.1. With these choices, we have

From Ohm’s Law for Electromagnetics, we have

So now

Finally:

This is a good-enough answer for the problem posed, but it is easily generalized a bit further. Noting that

in the denominator is the cross-sectional area

of the wire, so we find:

i.e., the resistance of a wire having cross-sectional area

– regardless of the shape of the cross-section, is given by the above equation.

It is important to remember that Equation 6.4.3 presumes that the volume current density

is uniform over the cross-section of the wire. This is an excellent approximation for thin wires at “low” frequencies including, of course, DC. At higher frequencies it may not be a good assumption that

is uniformly-distributed over the cross-section of the wire, and at sufficiently high frequencies one finds instead that the current is effectively limited to the exterior surface of the wire. In the “high frequency” case,

in Equation 6.4.3 is reduced from the physical area to a smaller value corresponding to the reduced area through which most of the current flows. Therefore,

increases with increasing frequency. To quantify the high frequency behavior of

(including determination of what constitutes “high frequency” in this context) one requires concepts beyond the theory of electrostatics, so this is addressed elsewhere.

• “Resistor” on Wikipedia.
• “Ohm's Law” on Wikipedia.
• “American wire gauge” on Wikipedia.