Approximations for the natural logarithm from solenoid-toroid correspondence
AApproximations for the natural logarithm fromsolenoid-toroid correspondence ˙Ibrahim Semiz ∗ Physics Department, Bo˘gazi¸ci University,Bebek, ˙Istanbul, Turkey
Abstract
It seems reasonable that a toroid can be thought of approximatelyas a solenoid bent into a circle. The correspondence of the inductancesof these two objects gives an approximation for the natural logarithmin terms of the average of two numbers. Different ways of averaginggive different approximants. They are expressions simpler than Taylorpolynomials, and are meaningful over a wider domain.
The calculation of the inductance of an ideal solenoid, L sol = µ N l A, (1)is a standard part of any introductory level college physics course. Here, L denotes inductance, µ is the permeability of free space, N the number ofturns of the solenoid, l its length and A its cross-sectional area. An ideal solenoid is one that is infinitely long, i.e. we are assuming that l >> a ,where a is any length characterizing the cross-sectional area.If the said course is calculus-based, the calculation of the inductance of atoroid with rectangular cross-section (Fig.1) will often be among the end-of-chapter problems, see e.g. [1]. The result of that calculation is L tor = µ N h π ln (cid:18) ba (cid:19) , (2) ∗ e-mail: [email protected] a r X i v : . [ phy s i c s . pop - ph ] M a r igure 1: Geometry of a rectangular toroid. Adapted from ref. [1].where µ and N are as above, a and b are the inner and outer radii, respec-tively, of the toroid, and h its height.One might wonder, in fact, an occasional student will ask; if a toroidcannot be considered as a solenoid bent into a circle. Of course, this mustbe a good approximation at least in some limit; and comparing (1) and (2)tells us that putting L sol ≈ L tor will give us an approximation for the naturallogarithm. We will get h π ln (cid:18) ba (cid:19) ≈ Al (3)Now, the area of the solenoid bent into the circle is ( b − a ) h , and its lengthis 2 πr , where r is some kind of average of a and b , which we denote by < a, b > . Defining b = ax and using the necessary property of linearity ofany definition of average, we haveln x ≈ x − < x, > . (4)This is our main result. Any reasonable definition of average will give aparticular approximation. In the next section, we will give three such specialcases. 2 Particular Approximations
We start with the most familiar concept of average, the arithmetic one, < a, b > A = ( a + b ) /
2. This givesln x A ≈ x − x + 1 . (5)The geometric average < a, b > G = √ ab givesln x G ≈ ( x − √ x , (6)and the harmonic average < a, b > H = 2 aba + b (7)results in ln x H ≈ ( x − x . (8)While the approximation (6) is not a rational function due to the presenceof the factor √ x , the approximations (5) and (8) are; i.e. they are fractionsof polynomials. In fact, when aproximating a function, such an expression iscalled a Pad´e approximant [2].Another common type of average is the weighted average, but no clearmotivation exists for weighting the inner and outer radius of the toroid differ-ently, nor are there any guidelines for what the weighting factors would be;so we do not use this average at this point. On the other hand, the harmonicaverage (7) can be seen as a special weighted average, where each number isweighted by the other.
If an approximation is needed for a function, the immediate impulse, almostreflex, of a physicist is to use a Taylor series. However, when we construct theTaylor series of the function we are interested in here, f ( x ) = ln x , around x = 1 (the point x = 1 is dictated by our problem: x >
1, since b > a ), andmake plots of the Taylor polynomials (truncated Taylor series, henceforthoccasionally abbreviated as TP’s), we see that at x -values beyond 2.5, thepolynomials are totally useless (Fig.2); in fact, the higher the number ofterms taken, the worse a representation of the function the series is. On3igure 2: Graphs of the function ln x (red), the first 13 Taylor polynomialsof that function around x = 1, and the three approximants (blue) of thiswork. The x -range is about five units. ln(x)HGA13 5 3 16 4 2 (cid:45) (cid:45) the other hand, our approximations (5), (6) and (8) are also shown on thatfigure, and they perform much better in that range.Contrast this behavior of the Taylor polynomials with the correspondingcase for the function f ( x ) = sin x (Fig.3): Here, the higher the order of thepolynomial, the later it peels off from the curve of f ( x ), i.e. by increasingthe order of the polynomial, we can find one that will be a good approxima-tion in any desired range around the origin. The difference comes from theconvergence properties of the respective Taylor series: The series for sin x converges for all x , while for ln x , around x = 1, the radius of convergence R is 1, as can be shown with standard techniques (e.g. [3]). In fact, an upperlimit of 1 on R could have been guessed without calculation, by noting thesingularity of the function at x = 0.This tells us that for this particular function, the Taylor series expandedaround x = 1 is meaningless for x ≥
2, hence, other approximations areneeded. While Taylor series around other x values can be constructed, theynecessitate calculation of ln x first; the approximation scheme (4) provides aneat alternative. Comparing to the TP’s around x = 1; already at x = 1 . x (red), the first 7 Taylor polynomialsof that function around x = 0. The Taylor polynomials are labeled by theirorders.
15 9 13 sin(x)3 7 11 (cid:45) (cid:45) the geometric approximant is better than the first 13 TP’s, and at x = 2 . x , since the solenoid( ≡ constant magnetic field inside) approximation for the toroid becomesmonotonically worse with x ; the error of the geometric approximant is %2at x = 2, %5 at x = 3, %11 at x = 5 and %24 at x = 10, still usable forsome purposes. At large x values, the approximants tend to 2, √ x and x/ x > x = 0 and x = 1. For most of thisregion, the fifth and higher order Taylor polynomials are better than all threeof our approximants, the geometric approximant is of comparable accuracyto the fourth order TP (near x = 1 /
2, TP4 is better, near x = 1 /
3, theapproximant), followed by the arithmetic one, the third order TP and theharmonic approximant, in this order. The error of the geometric approximantincreases to %5 near x = 1 / x = 1, that is, for | x − | <<
1, the Taylor poynomials are goodapproximations, as our reflexes tell us; in fact, the closer to x = 1 we are,5igure 4: Graphs of the function ln x (red), the first 13 Taylor polynomialsof that function around x = 1, and the three approximants (blue) of thiswork, shown near x = 2. ln(x)AGH1 23 the shorter the Taylor polynomial adequate for a given level of accuracy.However, the accuracy of all three of our approximants also increases towardsperfection as we near x = 1 from either direction, so in this region also, theyare eminently usable. On the other hand, the singularity at x = 0 cannot beexhibited by any of the Taylor polynomials, while the geometric and harmonicapproximants do a visually good job of it, even if the errors are large.One final note is that, having seen the behavior of the approximants (Figs.2 and 4), some kind of average of the geometric and arithmetic averages willgive a more accurate closed-expression result. For example, their geometricaverage, ln x A ≈ ( x − (cid:115) x + 1) √ x (9)has only %6 error at x = 10, and the error could be decreased even more byweighting the factors. But the higher accuracy of expression (9) and anal-ogous ones comes at the cost of losing some of the simplicity of expressions(4), (5), (6) and (8). 6igure 5: Graphs of the function ln x (red), the first 13 Taylor polynomialsof that function around x = 1, and the three approximants (blue) of thiswork, shown near x = 0 . (cid:45) (cid:45) (cid:45) (cid:45) Starting from the reasonable assumption that a toroid can in some limitbe thought of as a solenoid bent into a circle, we derived the simple andneat approximation scheme (4) for the natural logarithm function. Everyreasonable definition of the average of two numbers will give a particularrealization of the scheme, and we exhibited three such approximants, usingthe most common types of average. Among the three, the approximantbased on the geometric average, (6), is the best performer, with %5 error at x = 1 / x = 3, the error decreasing monotonically down to zero as x = 1is approached from either side.These approximants are much simpler expressions than multi-term Taylorpolynomials around x = 1, which do not make sense for x (cid:39) . 2, as well. They could be combined for more accurate approximations,at the cost of losing some of the simplicity.7 cknowledgements The author thanks A. Kazım C¸ amlıbel for help with the figures. References [1] RA Serway and JW Jewett Jr, Physics for Scientist and Engineers withModern Physics , Brooks & Cole, USA, 2014.[2] GA Baker Jr and P Graves-Morris, Pad´e approximants , Cambridge Uni-versity Press, New York, USA, 1996.[3] GB Thomas Jr, MD Weir and JR Hass,