## 1 Introduction

[Version 4.2.1] It is now common to find that whenever special relativity is discussed, it is accompanied by the assertion that it is one of the best tested and verified physics theory to date. The Kaufmann(1901)[3], Bucherer(1908)[4], Neumann[5] and Rogers et.al(1940)[6] experiments have always been represented as conclusive experimental verification of a relativistic mass varying with speed, thereby indirectly also repudiating Newtonian mechanics and verifying special relativity. It will be shown that despite the excellent agreement the relativistic mass of special relativity has with the Bucherer experiment, the experiment is not a verification of special relativity; instead, it is a clear experimental repudiation of the Lorentz force law of electrodynamics at relativistic speeds.

## 2 The Bucherer Experiment, 1908

The purpose of this paper is not to critique the Bucherer experiment in its details, but only its theoretical basis. Professor A.K.T. Assis gives a simplified description of the Bucherer experiment in one of his papers[2], but with a clear description of the theory behind the experiment. We reproduce it here.
The Bucherer apparatus may be considered as a capacitor with a linear dimension $L$ much greater then the separation of of the two oppositely charge plates with surface charge distribution of $±\sigma$. The $x$-axis is perpendicular to the plates from $-\sigma$ to $+\sigma$. Classical electrodynamics shows that there is a uniform electric field ${\stackrel{\to }{E}}_{x}=-\left(\sigma /{\epsilon }_{0}\right)\stackrel{^}{x}$ between the capacitor plates. The axes origin is a radium $\beta$-particle(electron) source at the center of the capacitor between the plates. The $y$-axis is the path an electron would leave the capacitor after traversing the distance $L$ leaving the capacitor with a velocity ${\stackrel{\to }{v}}_{y}$. A uniform magnetic field ${\stackrel{\to }{B}}_{z}$ in the $z$-axis direction is superimposed on the capacitor. Only those electrons in the $y$-direction could leave the capacitor when the electric deflection and the magnetic deflection in the $x$-direction are in balance. Furthermore, the initial electrons has to have no velocity component in the $x$-direction else they would collide with the capacitor plates. For electrons that leaves the capacitor moving along the $y$-axis, the only forces acting on the electrons are from the electric and magnetic deflections that act only in the $x$-direction; the velocity ${\stackrel{\to }{v}}_{y}$ is the natural electron ejection velocity. The force acting on the electrons is: $\begin{array}{cc}\stackrel{\to }{{F}_{x}}=-e\left({\stackrel{\to }{E}}_{x}+\stackrel{\to }{{v}_{y}}×\stackrel{\to }{{B}_{z}}\right)& \left(2.1\right)\end{array}$Equating the force with zero, we have: $\begin{array}{cc}{v}_{y}=\sigma /{\epsilon }_{0}{B}_{z}& \left(2.2\right)\end{array}$The Bucherer apparatus is also a velocity selector as changing the magnitude of the voltage across the capacitor and the magnetic field would allow electrons of varying speed to leave the capacitors. Five runs of the experiment were made giving data points for speed from about 0.3c to 0.7c. After the electrons leave the capacitor it would only be under the deflection of the magnetic field and it would travel in a circular path with a constant speed as in (2.2) until it strikes a photographic plate at some known distance away. From the coordinate of the points the electrons make on the photographic plate and the other dimensions, the radius $r$ of the circular path could be computed. Applying the Lorentz magnetic force as the centripetal force for circular motion, we have: $\begin{array}{cc}|e\left(\stackrel{\to }{v}×\stackrel{\to }{B}\right)|=ma=m{v}^{2}/r& \left(2.3\right)\end{array}$$a$ being the centripetal acceleration and $v$ is the constant speed equal to the speed in (2.2). Combining equations (2.2) and (2.3) gives: $\begin{array}{cc}e/m=\sigma /r{\epsilon }_{0}{B}^{2}& \left(2.4\right)\end{array}$The RHS of (2.4) could be evaluated as the terms are known physical constants or measured variables of the experiment. The ratio $e/m$ for the data points was found to vary with velocity, decreasing with velocity increase. As the electron charge was accepted to be constant, the varying charge-mass ratio of the electron was interpreted to mean that mass increases with velocity. The ratio $e/m$ was found to have a strong correlation with $e/\left({m}_{0}/\sqrt{1-{v}^{2}/{c}^{2}}\right)$. This showed that the experiment was in agreement with the Lorentz-Einstein model where the electromagnetic mass was: $\begin{array}{cc}{m}_{r}=\frac{{m}_{0}}{\sqrt{1-{v}^{2}/{c}^{2}}}& \left(2.5\right)\end{array}$${m}_{0}$ being the invariant rest mass of the electron. The textbook of Professor Robert Resnik [1] gives a table of the data for the experiment.The Bucherer experiment was viewed as evidence that inertia mass of matter has an electromagnetic origin and that it varies with velocity, not invariant.

## 3 Interpretation of the Bucherer Experiment

The result of the experiment did have profound implications. Prima facie, it repudiated invariant mass and verified the relativistic mass of special relativity. It was neither. The physicists then had electron models that predicted mass increasing with speed. For whatever reasons, they were unwilling to forego their models and consider alternative interpretations of the experiment. Some were quick to accept the Bucherer experiment as a conclusive repudiation of the invariant mass. None cast any suspicion on the equation (2.3) which was the basis of experiments such as that of Bucherer's. Electrons were deflected in a circular path and the Lorentz magnetic force of $e\left(\stackrel{\to }{v}×\stackrel{\to }{B}\right)$ was the only force acting on the electrons. It was the application of Newton's second law that gave rise to the equation.
We would re-examine the Bucherer experiment in a manner which has not been considered in the past. The novelty of this treatment is to attempt three different interpretations of Newton's second law to be used in equation (2.3) of the Bucherer experiment and to analyze the implications of each.
1. $Force\propto \frac{dp}{dt}$. This is an attempt to go back to the original statement as in the Principia. Momentum would be the relativistic definition : $\begin{array}{cc}Force\propto \frac{d}{dt}\left(\frac{mv}{\sqrt{1-{v}^{2}/{c}^{2}}}\right)& \left(3.1\right)\end{array}$This interpretation of Newton's second law as in relativistic mechanics fails - it leads to a force that is fictitious.
2. $Force=relativistic_mass×acceleration$. Relativistic mass may be defined as $\phi \left(v\right)m$ where $\phi \left(v\right)$ is a scalar function dependent on velocity $v$, $m$ being the invariant mass. This interpretation takes the form of a definition of a force as the relation here is an identity, not a proportionality as in the first case. As the dimension of the RHS is $\left[M\right]\left[L\right]\left[{T}^{-2}\right]$, the same dimension of force as with classical Newtonian mechanics, we first assume that the force here is defined and has a real unit the same as that for Newtonian mechanics. This definition of force would be consistent with the actual Bucherer experiment as it would accommodate a mass that varies with speed if it is found to be the case in the result. If we take $\phi \left(v\right)$ to be $\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}$, the relativistic mass would be that of special relativity. As we have seen, such a mass agrees with the result of the Bucherer experiment. The case here seems to give a formulation of relativistic mechanics that has a real unit of force and leads to a valid mechanics which agrees with the Bucherer experiment.
We now use this definition of force in the work-energy theorem to get the formula for kinetic energy. $K=W={\int }_{0}^{v}\left(\frac{m}{\sqrt{1-{v}^{2}/{c}^{2}}}\right)\frac{dv}{dt}dx={\int }_{0}^{v}\frac{mv}{\sqrt{1-{v}^{2}/{c}^{2}}}dv$ $\begin{array}{cc}K=m{c}^{2}\left(1-\sqrt{1-{v}^{2}/{c}^{2}}\right)& \left(3.2\right)\end{array}$The formula (3.2) is not the same as the kinetic energy formula of special relativity which is: $K=m{c}^{2}\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)$If the definition of force in this case formulates a valid relativistic mechanics, it is not the relativistic mechanics of special relativity. In fact, the definition of force in this case is also invalid as a variable mass dependent on speed could not be use to define a consistent standard unit of force. This case too is dismissed.
3. $Force=mass×acceleration$, mass being invariant. This is the definition of force in classical Newtonian mechanics. It interprets Newton's second law as an axiom of truth defining a force. This interpretation has been the only one since the time of Newton and there never was any other. Here, mass is an invariant as an axiom of Newton's laws of motion. This force definition is used in the circular motion force equation (2.3) of the Bucherer experiment and it leads to a result which shows a mass that is not invariant, but increases with speed - there is a contradiction between the physics theory and the experimental result. If the physics underlying the experiment is correct, such a contradiction should not occur. The force law behind the Bucherer experiment is based on an invariant mass and yet, the result shows a mass that increases with speed. This contradiction indicates that the physics on which the experiment is based are not all correct - it includes physics that are incorrect.
The physics behind the Bucherer experiment are Newton's force law, electromagnetism including the Lorentz force law. One of them is invalid giving rise to the contradiction.The classical Newton's force law is one of the best tested laws in physics since the time of Newton, rigorously tested for three centuries without any instant of failure where it is applied - it cannot be incorrect. The conclusion cannot be other than that electromagnetism and the Lorentz force law contain fundamental errors in some manner. As others have noted [7], the Lorentz law has to date not been directly tested under relativistic speeds; that it is the cause of the contradiction is very probable. To assume that only the Lorentz force law alone is invalid and the rest of electromagnetism is all clean and correct is illogical. As Lorentz force law itself involves both fields $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$, its failure may well have its origin in the very formulation of electromagnetism itself.
The first two cases above have to be ignored. Only the third case need to be considered and the conclusion could only be the failure of the Lorentz force law and the theory of electromagnetism.
The Bucherer experiment was experimental proof that electromagnetism and the Lorentz force law fail under relativistic speed conditions.

## 4 A Mass Definition Is Not Testable

In the “Introduction to Special Relativity"[1], the well known author Robert Resnick shows the Bucherer experiment as “proof” that the idea of an invariant mass was contradicted by experiment - mass was verified to vary and even fits the $\gamma$-factor for the relativistic mass of special relativity. The invariance of mass in Newtonian mechanics is a definition - defined as an absolute “quantity of matter” in the Principia. Even the relativistic mass of special relativity is founded on this same mass, but as a “rest mass${m}_{0}$ with a $\gamma$-factor added, $\gamma =\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}$: $\begin{array}{cc}{m}_{r}=\frac{{m}_{0}}{\sqrt{1-{v}^{2}/{c}^{2}}}& \left(4.1\right)\end{array}$The formula (4.1) is just a new definition for mass (indirectly through relativistic momentum) giving rise to a new formulation of mechanics of special relativity.
Experiments in the scientific paradigm is meant only to verify or test predictions of a theory, not any of its defined concepts. As an example, the invariance of mass in Newtonian mechanics is not testable, but the prediction that planets orbits the sun in elliptical orbits is verifiable. So the claim of Professor Robert Resnick is logically untenable. Neither the invariant Newtonian mass nor the relativistic mass of special relativity is testable.

## 5 A New Coulomb's Law and Lorentz Force Law

1. Case $f=\phi$: ${\phi }_{e}={\phi }^{2n+1};\phantom{\rule{20px}{0ex}}{\phi }_{b}={\phi }^{n+1}$ where $n>=0$.
2. Case $f=1/\phi$: This case is equivalent to case (1).
3. Case $f\ne \phi$: ${\phi }_{e}={\phi }^{2n-1}{f}^{2};\phantom{\rule{20px}{0ex}}{\phi }_{b}={\phi }^{n}f$ where $n>=1$
This shows that there are infinite forms the Lorentz force could take which would have the Bucherer experiment to not lead to any contradictions. The only way to determine what the actual form would be is through experimental verification of the velocity in (5.1) by a direct time-of-flight measurements of the ejected electron speeds. Furthermore, examination of empirical observations may help to deduce what its actual form should take. If the velocities of electrons of the Bucherer experiment had been verified through direct time-of-flight measurements, then the Bucherer experiment would have been a verification of the relativistic form of the Lorentz force law.
The Bucherer experiment could have been an experiment to verify the Lorentz force law for relativistic speed if the predicted speeds of the electrons had been verified through direct time-of-flight measurements.
There is empirical phenomenon on which we could rely on to determine what the probable form $f\left(v\right)$ is to take. It is shown in the section below that if $f\left(v\right)=\sqrt{1+{v}^{2}/{c}^{2}}$, the force equation (5.9) between two long parallel current-carrying conductors may be derived using only Coulomb forces only without any use need magnetism. The fact that this parallel-force equation has been well tested may be taken to mean the form $f\left(v\right)=\sqrt{1+{v}^{2}/{c}^{2}}$ may most likely be correct.
With the new Lorentz force law, there would not be any contradiction of the Bucherer experiment with the underlying physics on which the experiment is based on. The electric charge would be an invariant. Mass would be speed invariant, consistent with its definition in Newton's Principia. The relativistic Lorentz force law would explain why protons within particle accelerators cannot exceed the speed of light. This is due to the electric force having a necessary factor of $\psi \left(v\right)$ which approaches zero as velocity of the protons near that of light speed.
Contemporary physics has incorporated special relativity into electromagnetism. It shows how $\stackrel{\to }{F}=q\stackrel{\to }{E}$ to be correct for any electrostatic field and for any charge, whether at rest or moving at any speed, including relativistic speed. This derivation relies on the transformation of relativistic force between inertial reference frames based on the Lorentz transformation.[8, 5.8] Our analysis earlier has shown special relativity to be invalidated; the relativistic force based on $\stackrel{\to }{F}=\frac{d}{dt}\left(\gamma m\stackrel{\to }{v}\right)$ is fictitious. Therefore, $\stackrel{\to }{F}=q\stackrel{\to }{E}$ at best may only be an approximation for a charge moving at small speed as with the speed of electrons in currents in conductors.

### 5.1 Force Between Parallel Current-carrying Conductors

From classical electromagnetism, two long parallel current-carrying conductors will have forces acting between them; the formula for the force acting on a small element of a conductor of length $l$ by the other long conductor is: $\begin{array}{cc}F=\frac{{\mu }_{0}}{2\pi R}\left({I}_{1}{I}_{2}l\right)& \left(5.9\right)\end{array}$It will be shown here that the exact same equation (5.9) may be derived based only on the relativistic Coulomb forces between the $+q$ and $-q$ charges within the two conductors. This is in contrast to current electromagnetism which derives the equation through the mediation of the Lorentz magnetic force $\stackrel{\to }{F}=q\left(\stackrel{\to }{v}×\stackrel{\to }{B}\right)$ where $\stackrel{\to }{B}$ is based on the Biot-Savart law for the magnetic field.
The relativistic Coulomb's law has a factor $\psi \left(v\right)=\sqrt{1-{v}^{2}/{c}^{2}}\left(1+{v}^{2}/{c}^{2}\right)$ and is valid for any speed $v<=c$. The drift speed of current electrons in conductors is in the order of $1{0}^{-5}m/s$, very much smaller than $c$. For currents in conductors, $\psi \left(v\right)$ may be approximated by series expansion:$\sqrt{1-{v}^{2}/{c}^{2}}=1-\frac{{v}^{2}}{2{c}^{2}}+...$ giving $\psi \left(v\right)=1+\frac{{v}^{2}}{2{c}^{2}}+...$. The Coulomb's law would then be: $\stackrel{\to }{F}=k\left(1+\frac{{v}^{2}}{2{c}^{2}}\right)\frac{{q}_{1}{q}_{2}\stackrel{^}{r}}{{r}^{2}}$
Figure (5.1) shows two typical elements of the same length $dl$. The aim is to find the force that acts on the single top element ${q}_{1}$ due the the long conductor 2. Element 1 will have a $+{q}_{1}$ charge due to the fixed proton lattice ions; it will be balanced by an equal amount $-{q}_{1}$ of drift electrons. It is similar for the element 2. If the classical Coulomb's law is used, the forces between the charges between the two elements will exactly balance - the Coulomb forces will not give rise to any net force between the parallel conductors. As we are now using a relativistic Coulomb's law with a scalar factor dependent on the relative speed of the interacting charges - electrons have a drift speed - the Coulomb forces between the charges will not exactly balance as before; it will give rise to a net force between the current elements, either attractive or repulsive.
The method is straightforward. The force between the charges ${q}_{1}$ and ${q}_{2}$ will be along the direction of $\theta$. The total force on element ${q}_{1}$ is found by integrating the forces for the conductor 2 from $-\infty$ to $+\infty$. From symmetry, the force between the elements need only the transverse components as the longitudinal components will cancel out when integrated. The forces between elements ${q}_{1}$ and ${q}_{2}$ are:
${f}_{12}=A{\rho }_{2}{a}_{2}{v}_{2}\frac{dl}{{r}^{2}}$
As $\frac{1}{r}$=$\frac{cos\theta }{R}$,we have: $\begin{array}{cc}{f}_{12}=A{\rho }_{2}{a}_{2}{v}_{2}\frac{dlco{s}^{2}\theta }{{R}^{2}}& \left(5.13\right)\end{array}$Refering to Figure (5.1), we see that $l=Rtan\theta$ giving $dl=Rse{c}^{2}\theta d\theta =\frac{Rd\theta }{co{s}^{2}\theta }$. For integration over the whole length of conductor 2, we need only the transverse component of ${f}_{12}$, i.e multiplying (5.13) by $cos\theta$. Substituting for $dl$:
As ${q}_{1}{v}_{1}={I}_{1}dl$, ${\mu }_{0}=\frac{1}{{\epsilon }_{0}{c}^{2}}$, ${I}_{2}={\rho }_{2}{a}_{2}{v}_{2}$, we finally have: $\begin{array}{cc}{F}_{dl}=\frac{{\mu }_{0}}{2\pi R}{I}_{1}{I}_{2}dl& \left(5.14\right)\end{array}$The formula is valid for any element of length $l$ of the conductor where $l$ is much smaller than the lengths of the parallel conductors. For parallel currents, the force would be attractive. The formula is also valid for anti-parallel currents where the force would be repulsive. This can be seen in equation (5.10). The term $\left(v2-v1{\right)}^{2}$ becomes $\left(v2+v1{\right)}^{2}$ and would cause a change in sign in ${f}_{12}$. This sign change would cause a sign change in the final equation (5.14).
A curious observation may be made of equation (5.14). It is used as the basis to define the SI unit of Ampere for electric currents. This fact may be taken to mean that the equation has been rigorously verified experimentally by all the standard's laboratories around the world to be reliable and consistent; any inconsistency of the equation, if any, should by now be discovered. So far, no inconsistency of the equation has ever been observed. As the equation (5.14) is also derived based on the new relativistic Lorentz force law and the Coulomb's law, the implication here is that it is a verification of the new force laws for speed much smaller than the light speed.
The fact that the formula for the forces between long parallel conductors has been rigorously tested is a verification of the relativistic Lorentz force law and the Coulomb's law for speed much smaller than that of the light speed.

## 8 1908 Bucherer 实验

A.K.T.Assis [
2] 在他的一篇论文中简要介绍了Bucherer实验，并对实验背后的理论进行了清晰的描述。 我们在这里重现。
Bucherer 装置也是速度选择器，用于改变电容器两端的电压的大小，并且磁场将允许变化速度的电子离开电容器。进行了5次实验，数据点的速度从 $0.3c$$0.7c$。 在电子离开电容器之后，它将仅在磁场的偏转之下，并且将以（8.2）中的恒定速度在圆形路径中行进，直到在某个已知距离处撞击照相板。 从电子在照相板上形成的点的坐标和其他尺寸，可以计算圆形路径的半径 $r$。 应用洛伦兹磁力作为圆周运动的向心力，得： $\begin{array}{cc}|e\left(\stackrel{\to }{v}×\stackrel{\to }{B}\right)|=ma=m{v}^{2}/r& \left(8.3\right)\end{array}$$a$ 是向心加速度，$v$ 是等于（8.2) 中速度的恒定速度。 组合方程（(8.2) 和 (8.3)给出： $\begin{array}{cc}e/m=\sigma /r{\epsilon }_{0}{B}^{2}& \left(8.4\right)\end{array}$方程 (8.4） 的右手边可以被评估为术语是实验的已知物理常数或测量变量。 发现数据点的 $e/m$ 的比率随速度而变化，随着速度的增加而减小。 随着电子电荷被认为是恒定的，电子的电荷-质量比变化被解释为质量随着速度而增加。 发现 $e/m$ 的比率与 $e/{m}_{0}/\sqrt{1-{v}^{2}/{c}^{2}}$ 有很强的相关性。 这表明实验与洛伦兹爱因斯坦(Lorentz-Einstein)模型一致，其中电磁质量为： $\begin{array}{cc}{m}_{r}=\frac{{m}_{0}}{\sqrt{1-{v}^{2}/{c}^{2}}}& \left(8.5\right)\end{array}$${m}_{0}$ 被电子的静止质量。 Robert Resnik教授的教科书 [1] 给出了实验数据表.Bucherer 实验被认为是物质的惯性质量具有电磁起源，并且随速度变化而不是不变的证据。

## 9 Bucherer实验的解释

1. $Force\propto \frac{dp}{dt}$。 这是试图回到原来 “Newton‘s Principia” 的声明。动量将是相对论定义： $\begin{array}{cc}Force\propto \frac{d}{dt}\left(\frac{mv}{\sqrt{1-{v}^{2}/{c}^{2}}}\right)& \left(9.1\right)\end{array}$牛顿第二定律在相对论力学中的解释失败了-它导致了力的虚构。比例关系只有当关系的双方都有定义值时才有意义。右侧被定义，量纲为 $\left[M\right]\left[L\right]\left[{T}^{-2}\right]$。 左侧的力尚未定义; 因此左侧未定义。在牛顿力学中习惯的力的定义不能在这里假设，因为牛顿第二定律的这种解释有效地定义了一种新的力学。如果这种新的机制是有效的，狭义相对 论的这种力的定义必须先出现在相对论力学中。由于没有相对论力的确定单位，这种相对论力学只能是虚构的。虽然REF中所示的相对论质量满足了 Bucherer实验中所要求的质量，但是在这种情况下定义的力就会失效。此案被驳回。
2. $Force=relativistic_mass×acceleration$: 相对论质量可以定义为 $\phi \left(v\right)m$，其中 $\phi \left(v\right)$ 是取决于速度 $v$ 的标量函数，$m$ 是不变质量。 这种解释是以力的定义的形式，因为这里的关系是个相等值,而不是第一种情况下的比例。由于右手边的量纲为 $\left[M\right]\left[L\right]\left[{T}^{-2}\right]$，与经典的牛顿力学相同的力量维度，我们首先假设这里的力被定义，并且具有与牛顿力学相同的真实单位。力的这种定义将与实际的 Bucherer 实验一致，因为它将适应随速度变化的质量，如果在结果中被发现是这样。如果我们将 $\phi \left(v\right)$ 作为 $\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}$，质量将是狭义相对论的质量。正如我们所看到的，这样的质量与 Bucherer 实验的结果一致。这里的情况似乎是给出了一个相对论力学的表达式，它具有真实的力量单位，并导致与 Bucherer 实验一致的有效力学。
我们现在在工作能量定理中使用力的定义来得到动能公式。 $K=W={\int }_{0}^{v}\left(\frac{m}{\sqrt{1-{v}^{2}/{c}^{2}}}\right)\frac{dv}{dt}dx={\int }_{0}^{v}\frac{mv}{\sqrt{1-{v}^{2}/{c}^{2}}}dv$ $\begin{array}{cc}K=m{c}^{2}\left(1-\sqrt{1-{v}^{2}/{c}^{2}}\right)& \left(9.2\right)\end{array}$公式（9.2）与狭义相对论的动能公式不一样，即： $K=m{c}^{2}\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)$如果这种情况下的力量定义形成了一种有效的相对论力学，那就不是狭义相对论的相对论力学。 事实上，在这种情况下，力的定义也是无效的，因为取决于速度的变量质量不能用于定义一致的标准力单位。 此案也被驳回。
3. $Force=mass×acceleration$, 质量是不变的。这是古典牛顿力学中力的定义。它将牛顿的第二定律解释为定义力量的真理公理。这个解释一直是牛顿以来唯一的公理解释。在这里，质量是牛顿运动定律的一个不变量。该力定义用于 Bucherer 实验的圆形运动力方程 (8.3)， 它使显示质量不变，但随着速度的增加。这代表了这实验的物理学理论与实验结果之间存在矛盾。如果实验所基于的物理学是正确的，那么这种矛盾就不应该发生。 Bucherer 实验背后的力法基于不变质量，但结果表明随着速度增加的质量。这个矛盾表明，实验所依据的物理学并不完全正确 - 它包括不正确的物理理论。 Bucherer 实验的理论是基于牛顿力法和洛伦兹力法在内的电磁学。其中一个是无效的，导致矛盾。牛顿时代以来，经典的牛顿定律是一个经过最严格检验的物理学规律。经过 三个世纪的严格测试，没有一次失败;问题不应该归结于牛顿定律。结论应该是电磁学和洛伦兹力定律包含一些根本误差。正如其他人指出的那样 [7]，洛伦兹法律迄今尚未以相对论速度直接测试; 这是矛盾的原因是非常有可能的。假设只有洛伦兹的力法是无效的而其余的电磁学是清洁的，正确的是不合逻辑的。 洛伦兹力法涉及电场 $E$ 和磁场 $B$，因此它的失败很可能源于电磁本身的制定。

Bucherer实验证明了电磁学和洛伦兹力律在相对论速度条件下失败.

## 11 新库仑法和洛伦兹力法

### 11.1 并联载流导体之间的力

${f}_{12}=A{\rho }_{2}{a}_{2}{v}_{2}\frac{dl}{{r}^{2}}$
$\frac{1}{r}$=$\frac{cos\theta }{R}$, 得: $\begin{array}{cc}{f}_{12}=A{\rho }_{2}{a}_{2}{v}_{2}\frac{dlco{s}^{2}\theta }{{R}^{2}}& \left(11.13\right)\end{array}$从参考图 (11.1),可看 $l=Rtan\theta$ 导致 $dl=Rse{c}^{2}\theta d\theta =\frac{Rd\theta }{co{s}^{2}\theta }$。 为了在导体 2 的整个长度上进行整合，我们只需要 ${f}_{12}$ 的横向分量，即将 (11.13) 乘以 $cos\theta$。 代替 $dl$