Ampère's Law Proved Not to Be Compatible with Grassmann's Force Law

Efforts have frequently been made to create links between different approaches within electromagnetism. Names like Ampere, Coulomb, Lorentz, Grassmann, Maxwell et.al. are all linked to efforts to create a comprehensive understanding of electromagnetism. In this paper the very focus is on breaking the alleged links between Ampere’s law and the Grassmann-Lorentz force. Thanks to extensive mathematical efforts it appears to be possible to disprove earlier assumed links. That will tend to lead the further investigation of the subject effectively forwards


Introduction
Efforts have frequently been made to create links between different approaches within electromagnetism. Names like Ampère, Coulomb, Lorentz, Grassmann, Maxwell et.al. are all linked to efforts to create a comprehensive understanding of electromagnetism. In this paper the very focus is on breaking the alleged links between Ampère's law and the Grassmann-Lorentz force. Thanks to extensive mathematical efforts it appears to be possible to disprove earlier assumed links. That will tend to lead the further investigation of the subject effectively forwards

Finding the conceptual roots of the Lorentz force
Graneau discusses the assumption that the Ampère and the Lorentz forces are mathematically equal, claiming that this is not true [1]. He further makes the statement that the magnetic component of the Lorentz force was first proposed by Grassmann [2], [3]. This author shows that when Grassmann makes the derivation beginning with Ampére's law, he commits faults, which finally results in a term that is similar to the Lorentz force.

Grassmann on the electromagnetic force between currents
Grassmann himself discusses the conceptual problems that arise when studying Ampère's law [2], [3]. He says that the complicated form of Ampère's law arouses suspicion.
[4] Among others he complains over the fact that the formula in no way resembles that for gravitational attraction, indicating thereby the lack of analogy between the two kinds of forces 2 (2cos cos .cos ). / ra b r It ought to be mentioned that in the time of Grassmann and Ampère the electron has not yet been discovered and hence, the concept of what constituted a current must have been rather vague. It is therefore understandable that neither of them were able to apply Coulomb's law on the problem a law that by form fulfills the requirement to resemble the gravitational force. This author has been successful in doing so, beginning in his first paper on the subject in 1997 [5].

www.intechopen.com
Applying Ampère's law on a configuration [6] consisting of two current carrying conductors, with the elements a and b respectively, an angle  between the currents,  and  are the angles formed by the elements a and b respectively with the line drawn between the two mid-points, the current from the attracting element being i and its length ds , l being the perpendicular from the midpoints of the attracted element on the circuit element b on to the line of the attracting one, or sin lr   (2) gives the force between the two elements 2 ( . / )cos ( After he has arbitrarily chosen to put d at the end of the second term, due to the statement (6) above.
He thereafter integrates over the whole attracting line, thereby getting nothing from the second term. This is not mathematics. If there were originally undertaken an incremental step d attached to both terms, the integration along  can of course not be avoided by stating that dd    and choose to change d into d on the second term.
He continues the treatment of the two currents by deriving the force perpendicular to the attracted element b due to the attracting one 2 ( / )(cos .cos cos .sin ) Thereafter he neglects the second term of eq. (11), while leaving no indication of the reason for doing so and attains after using eq. (2) In the following a passage is cited from his paper in order to make it easier to understand his way of thinking [8] "From this expression he now at once attains the mutual interaction of two current elements, as he prefers to express it. It happens, since he is regarding the attracting current element .
idsas the combination of the two lines through which the current is passing, "these possessing the direction and intensity (i) of this element, and one of them having its current flowing in the same direction as that in the element, and the other in the opposite direction, while the first of them has its starting point in the initial point of the element , and the second has its starting point in the end of the element. We then obtain 2 (/ ) . s i n l ab r  as the effect exerted by a current element a on another b, distant r from it" This term proportional to sin is apparently that which has impelled people to name him as the first to define "Biôt-Savart's law".

Some comments on the method of Grassmann
If following the derivation Grassmann has made, at several occasions there are clear cases of confusion concerning what he is doing and why. The mathematical steps recall more an 'adhoc' way of using mathematical terminology than a real logical way of working. Hence, his results seem to be of no value with respect to factual electrodynamics.
www.intechopen.com It seems to be totally irrelevant to use the theoretical treatment by Grassmann in order to construct any links between Ampère's law and Biôt-Savart's law and related Lorentz force. They must be kept apart from each other.

Analysis of the results by Assis and Bueno in comparing Lorentz's force law and Ampère's law
Assis and Bueno have written a paper [9] in which he claims Ampère's law to be consistent with Grassmann's force law. They derive expressions for the force between the support and the bridge within a set of Ampère's bridge, using both laws. Conceding that the laws are not equal at every point of the circuit, they claim that the result for a whole closed circuit is equal. One special point of observation that they make is that Biôt-Savart's law and the related Lorentz force do not obey Newton's third law, whereas Ampère's law does [10] The set of Ampére's bridge is described by detail in two consecutive figures, with respect to the definitions of the integration domains [11]. The width of the conductor is small relative to the lengths of the branches. The bridge is described by two variables, the laminar thickness is being ignored. The shape is rectangular, the branches being of length 3 l (along the x direction) and 2 l (along the y direction). The branches along the y axis are cut off at the distance 1 l the segments thus attained have been numerated from one to six counterclockwise, beginning with the part of the support being situated along the x axis, where presumably the supporting battery is practically applied. Thus, the cuts appear between segment 2-3 and 5-6 respectively. In this paper it is especially being focused on the force that appears from segment 5 onto segment 4, since the Lorentz force from segment 6 to segment 5 (and similarly between segment 2 and 3) is zero, whereas Ampère's law is not. Interestingly, the Lorentz force from segment 5 onto segment 4 differs from zero, contrary to the force in the opposite direction. The authors claim that this net force within this branch (the bridge) is able to account for the force that otherwise Ampère's law produces from segment 6 to 5. Normally, according to Newtons's third law, there cannot be any net force, if there is no acceleration, since during conditions of balance, no net force is active. Biôt-Svart's law and the Lorentz force law implies a singledirected force, that if it were to be real would immediately blow up the circuit, which does not happen. The force that Ampère's law produces between the two segments corresponds to an internal tension that is made visible when cutting of the bridge, thereby creating two equal forces, but of opposite direction, that exactly cancel.
To conclude, this discussion seems to verify that the Lorentz force is by nature unphysical, whereas Ampère's law is physical. Furtheron, the derivation that will follow in the next chapter, gains momentum to this conclusion, since it will appear that the derivation by Assis and Bueno that constitutes a fundamental basis for their claim, has been fallaciously performed. It must be mentioned that the Assis and Bueno mention the law as "Grassmann's", without showing the reference. The reference that this author has found is that given by Peter Graneau [13], [14], [15]. It would be preferred to be used the name "Lorentz' force law (based on Biôt-Savart's law)". For the reader's convenience, also Ampère's law is given here:

Result of the calculations with respect to segment 5-4
The integrations can usually be performed straightforwardly, thereby using the normal rules for integrations. At some points, however, it appears necessary to make approximations, when a term is extremely small as compared to the others, for example terms w  in the numerator. That has been dome at every actual occasion in this work. This is the precondition for attaining a closed expression at the end.

The result according to Assis and Bueno
Assis and Bueno have been using two ways to approximate the integrals that have to be performed. In the first case they assume the circuit to be divided into rectangles (21), in the second case they let a diagonal line at the corners define the border between two segments (22). In the first case they claim that the result of the calculation of Grassmann's force on segment 4 due to segment 5 is: In the second case their result is a result that they identify as equal to that which Ampère's law gives rise to.

The result according to the analysis of this author, first approach
However, the intention with this paper is to judge the claims by Assis and Bueno. In order to attain that goal, the Grassmann force they have been using will be used in this paper in an independent derivation, by this author.
The first step in the calculation procedure is to give the problem a strict formulation in the shape of an integral, thereby identifying as well the variables of the integrations as the boarders. Applying Eq.(13) above to the segments 5-4 will give rise to the following integral equation: www.intechopen.com

The second step: Integration with respect to 5 y
The subsequent integration with respect to 5 y gives rise to the following expression: For convenience, the four terms may be named (19a), (19b), (19c) and (19d) in consecutive order.

The third step:
Integration with respect to 4 y and 5 x

The first term of Eq.(19) above treated, (19a)
In order to solve the integration, some integration formulas must be used: This expression contains two terms: The first term becomes after integration: The second term Solving with respect to z, thereby using formula [10] gives: Straightforward integration accordingly gives, provided also the approximation 35 wlx  is used in the final stage, both terms of the integral equal zero in the limit. The second term of those also requires the usage of an integration formula, Eq. [20], before the null result can be achieved. Hence, Here it appears to be evident that this expression might simply be divided into two, according to the logarithm product law (Eq. (61) later in the text): These terms will hereafter be denoted Solving straightforwardly, and using series expansions of the ln function, thereby neglecting terms of higher order of w, gives for the first term of the expression (34) above: The second part will demand some more computational work, as will appear below: Evaluating now the integral with respect to z , leads directly to the following, rather complicated expression: The three terms on the right hand expression will be called (36a), (36b) and (36c) respectively. Now, instead of evaluating the expressions straightforwardly, it appears to be favorable to find suitable combinations of terms that would be able to make the solution simpler.

The sum of the first two terms of Eq. (36)
In order to solve the integral (36) it will be favorable to make the following variable substitution: Applying this on the first term above (36a) gives: The treatment of the third term (36c) will be postponed, until the first two have been developed and simplified. Doing so, leads to the result with respect to the first term (36a )   2  22  0  213  21  21  3  2  33   22  2  3  21  32  1  2  1  3  2  1  32 1  3   2  3  21  21  21  3 In the next step the two results above, (40) and (41) will be added. The result is: www.intechopen. com  2  22  03 21  21  21  3  2  33  3  3   22 3  21  21  21  3  33  3  3   2  21  21  21 3  33   2  21  21  21 3 33 It has been used the series expansion of ln (1 ) xx Since the 'x' is very small, only the first term in the expansion has been taken into account.
The next step will be to simplify the expression (42).This must be done in a deliberate way in order to succeed. A practical method is to separate out the dominant terms first, thereafter put the terms of first order in w thereafter those of second order, neglecting the terms of even higher order. The reason for this is that the numerator in the very first term of the expression contains 2 w in the numerator.
In the following this integral formula will again be usable: Following this procedure gives for the terms of expression (29)  The sum of these eight expressions is equal to expression (42). Now, it will be easier to perform the addition of the terms of Eq. (42), by adding the eight separate terms above to each other. An important key to success has been to approximate every ln and root expression with a series expansion for every case when a 'small' term is appearing besides the big ones. By using this method it has been possible to restrict the complicated ln and root expression to the 'standard'ones, without 'small' terms. In doing so it is possible to gather similar terms from the different expression, even when there have been 'small' terms added inside the ln and root expressions.
The result is:

The third term of Eq.(36)
The third term of Eq. (36) was denoted (36c) is repeated here for convenience. These three terms will now be denoted (51a), (51b) and (51c) respectively. In order to solve these integrals, it appears practical to make the variable substitution The same procedure is repeated with the next term, which leads to: Term (53b) becomes

The fourth term of Eq.(19) above treated, (19d)
Straightforward integration leads again to a result. Many partial integration to be done, but in a rather straightforward way. In order to solve the fourth term, Eq. (19c), it is at first reasonable to use the substitution This expression may be dissolved into two terms, using the logarithmic product formula: The first term simply becomes 2 0 (l n 1 ) 4 The second term can be rewritten: In order to perform the evaluation successfully, the integral formulas [20] and [25] will be needed. Further series expansions have to be done with respect to root and logarithmic expressions that will arise during the work, namely [28], [29] and [30]. It is the hope that these advices will lead the reader successfully to the result, which is

The sum of all contributions, simplified expression
After having performed the summation, some simplification occurs and gives rise to the following result:

The result according to the analysis of this author, second approach
The first step in the calculation procedure is to give the problem a strict formulation in the shape of an integral, thereby identifying as well the variables of the integrations as the boarders. Applying Eq.(13) above to the segments 5-4 will give rise to the following integral equation, now in this case with a diagonal line at the corner, defining the border between two segments. The change appears at the upper border in the integral over 5 y : 25 lx  instead of 2 lw  of Eq. (4) above.
This corresponds to the procedure in section 4.5 and the integration result here (Eq.(68)) is equal to that (Eq.(18)). The border line at the corner has namely not yet been involved.

The second step: Integration with respect to 5 y
The subsequent integration with respect to 5 y gives rise to the following expression: This expression must be separated into several separate terms in order to be solved.
The first term will be: www.intechopen.com The second term will accordingly be: The second terms appears to give a zero result, hence the total result may be written  2  2  0  21  21  54  3  33   22  2  21  21  3  21  3  2  22  32  1  2  1  2  1   33  3   2  3  22  22  33  3  3   This integral apparently consists of three separate terms, which may be treated separately from each other.
The first term is only a constant term, which has to be integrated with 5 x .over the variable 5 x . Hence, the result is easily written The second term of Eq. (80) causes a real difficulty when trying to integrate the expression, since no approximations in using seres expansions are allowed. That is so, since the terms are all of the same order. Instead, it is possible to estimate the limits, between which the result must lay. The method is given by the expression 1 () ( 6 9 ) { l n l n ( ) l n 2 } 42 Hence, Assis and Bueno are wrong in their claim that the Grassmann force gives the same result as Ampère's law [1]. To conclude, it seemed to be rather wise to reject the claim of coincidence between these two laws, as they did not coincide before integrating. To be stated again,

Discussion and conclusions
From the rigorous analysis that has been performed above it is evident that Grassmann's law is not equal to Ampère's law. It is also evident that the very roots of the idea of their equality by Grassmann is false, too. However, it does not exist any need to regard them as equal. It was only one idea among many invented by science in its search for a better understanding of physics. In this respect it constitutes a progress for physics that it has been possible to reject one of the speculative ideas that must inevitably arise in the search for the truth. More seriously, it is of course grave for the proponents of the Lorentz force (identified as the Grassmann force), if it cannot be justified by referring to preceding established ideas but must stand as a mere ad-hoc invention.