INDUCTANCE OF A STRAIGHT WIRE

Tim Healy, Santa Clara University

This note concerns the inductance of a straight wire. Surprisingly, it is not easy to find an expression for the inductance of a straight piece of wire, and it is far more difficult to find out how that expression was obtained. And yet this result is very useful as a building block for more complex structures, and also because the inductance of a wire is important in high frequency or high speed electrical circuits.

The first really convenient derivation of the inductance of a wire which I have been able to find is due to Rosa(1). The first dilemma which Rosa faced is that there is no unambiguous definition of how to define the inductance of a straight wire. If we consider the wire in isolation we ignore the question of how the current gets to the wire. But that current, however it is delivered, will affect the flux which is developed in the vicinity of the wire. But this flux is a part of the definition. Rosa resolved the dilemma arbitrarily (there is no other way) by defining the inductance as the ratio of the flux developed in the region bounded by lines perpendicular to the beginning and end of the wire. Let's look at Rosa's own words in which he recognizes the dilemma in the definition.

"I have derived the formulae in the simplest possible manner, using the law of Biot and Savart in the differential form instead of Neumann's equation, as it gives a better physical view of the various problems considered. This law has not, of course, been experimentally verified for unclosed circuits; but the self inductance of an unclosed circuit means simply its self-inductance as a part of a closed circuit, the total inductance of which cannot be determined until the entire circuit is specified. In this sense the use of the law of Biot and Savart to obtain the self-inductance of an unclosed circuit is perfectly legitimate."

One might question here what the word 'legitimate'. Rosa has made an assumption in order to obtain a result. That is fine, but I don't think you should say it is legitimate, in the sense of being according to law. There is no law to guide this assumption. In any event let us proceed to the solution. The approach is the following.

1. Set up the geometry of a point at some distance from a straight line.

2. Write dH from Biot Savart at that point. Let the current equal one for simplicity.

3. Find the total H due to all of the current in the line by integrating over the line.

4. Set B = uH.

5. Find the magnetic flux phi in a differential area which is parallel to the wire by integrating B over this area which is a fixed distance from the wire.

6. Find the total flux phi over all of the area from the edge of the wire to infinity by integrating over the distance from the wire to infinity.

7. Since the current was set equal to one, the total flux equals the inductance.

I am not going to put in the full derivation until I find a more math-friendly language than HTML. You can get the full derivation by contacting me (thealy@scu.edu), or by looking up the original Rosa reference below.

The total low frequency inductance (internal plus external) of a straight wire is:

Ldc = 2L[ln(2L/r) - 0.75]nH

where Ldc is the "low-frequency" or DC inductance in nanohenries (nH), L is the length of the wire in cm, and r is the radius of the wire in cm.

This result is based on the assumption that the radius r is much less than the length L, which is commonly true.

For sufficiently high frequencies skin effects cause the internal inductance to go to zero and the inductance becomes:

Lac = 2L[ln(2L/r) - 1.00]

E.B. Rosa, "The Self and Mutual Inductances of Linear Conductors", Bulletin of the Bureau of Standards, Vol.4, No.2, 1908, Page 301ff.