Equilateral triangle coil case

Analyze of the Lorentz forces internal to an equilateral triangle coil

Take an equilateral triangle coil shown in the Figure 1, which is rigid to obtain the resultant of the forces on all sides; a current I flows in it. Each sides would feel a Lorentz force from the magnetic field of the other sides. Due to the magnetic field of the base side S_b , the force on the left side S_l is F_bl; the force on the right side S_r, is F_br. So, the force that the base exerts on the 2 upper sides is the sum:

R_up= F_bl + F_br

Due to the magnetic field of S_l , the force on S_b is F_lb; due to the magnetic field of S_r , the force on S_b is F_rb. So, the force that the upper ensemble S_l + S_r exerts on the base is the sum:

R_b= F_lb + F_rb

R_b and R_up are the action forces between the 2 parts (S_b and S_l + S_r) that form the triangle. Their sum is the resultant internal force that the 2 parts exert on each other:

R_net= R_b + R_up = F_lb + F_rb + F_bl + F_br

Can we find the value of R_net? Let us examine the right and base sides and their forces. The force F_br is perpendicular to S_r , F_rb is perpendicular to S_b . Because S_r and S_b have the same length, F_br and F_rb have the same magnitude and their resultant R_br will lay on the bisector of the angle. In the same way, we find that the resultant R_bl of the forces F_bl and F_lb has the same magnitude than R_br but with an opposite x component.

R_br =F_rb + F_br ,   R_bl = F_lb + F_bl

So, the sum of R_br and R_bl will be vertical:

R_br + R_bl = R_y e_y≠0

We notice that this sum is equal to R_net:

R_net= (F_rb + F_br ) + (F_lb + F_bl ) = R_br + R_bl = R_y e_y≠0

So, R_net is vertical and non null. The total force internal to the coil, R_net, must be 0 according to the principle of dynamic. But it is not, violating the third Newton's law. R_net is predicted by the Lorentz force Law, thus, this law is in contradiction with the third Newton's law, and is flawed.

 Pengkuan

Polygon coil

Lorentz forces internal to a polygon  coil, analyze and computation

Take a coil of the shape shown in the Figure 1, which is made rigid by a wooden plate (yellow in the Figure 1); a current I flows in it. Each of the 5 sides would feel a Lorentz force from the magnetic field of the other sides. The forces on the left and right low sides, S_ll and S_rl , are F_ll and F_rl , which are horizontal and symmetrical. The forces on the left and right high sides, S_lh and S_rh , are F_lh and F_rh , which are perpendicular to their sides and make an angle between them. The force on the base side, S_b, is F_b, which is vertical.

These forces are internal to the coil. What is the sum of F_ll , F_rl , F_lh , F_rh and F_b? F_ll and F_rl cancel because of symmetry. The x components of F_lh and F_rh cancel because of symmetry but their y components make a vertical resultant force F_top. So, the sum of these forces is:

R=F_ll + F_rl+ F_lh + F_rh + F_b = 0 + F_top + F_b

As R is the sum of all internal forces, it must be 0. However, this requires that F_top and F_b have the same magnitude. Is this condition fulfilled? Let us analyze a coil having long vertical sides S_ll and S_rl. For this coil, the top and base sides are distant from each other. For sufficiently long vertical sides, the intensity of magnetic field being inversely proportional to the square of the distance, the magnetic field from the base becomes negligible at the top and vice versa. In this case, the Lorentz force on the base and the top due to the opposite sides are very weak. In fact, from a certain length of S_ll and S_rl, F_b and F_top become independent to the opposite sides.

F_top depends on the angle of the top. When this angle varies, F_top varies strongly. But F_b will stay unchanged since the distance is large. Because of the variability of F_top and the constancy of F_b, they do not have the same intensity. Hence, the resultant force R is not constantly 0.

R is the sum of all internal forces, but is not 0. This is a violation of the third Newton's law. As R is predicted by the Lorentz force law, the latter is not consistent with the third Newton's law.

Above we have used distance to separate the top and the base in terms of magnetic field. In reality, this trick is not necessary. The resultant of internal Lorentz forces is non null even for ordinary triangle. The Figure 2 gives the result of a computation for the shown triangle coil. The base line is 1 and the height is 10. The values of the forces on all sides are given in the figure, they are dimensionless. The resultant force is:

R=35.21

This is not permitted by the fundamental laws of dynamic. The analyze and the numerical example have shown that the Lorentz force law does not predict correct internal forces. Thus it is flawed. 

Paradoxes and Solutions

Lorentz torque experiment

The Lorentz torque experiment

Peng Kuan 彭宽, titang78@gmail.com

Sunday, April 15, 2012

The paradoxes have shown the deficiency of the Lorentz force law; the differential Ampere's force law has successfully solved all these paradoxes. It is however necessary to confirm by new experiments never carried out before. A success will show the flaw of the Lorentz force law and prove the new law experimentally. Below is the design of the experiment.

Paradoxical Lorentz force internal to a triangle coil

Take a rigid triangle coil (Fixed on a wooden plate, yellow in the Figure 1), a current I flows in it. The 3 sides would feel a Lorentz force from the magnetic field of the other sides. These force are internal to the coil.

Now, put the side s₃ inside a ideal magnetic shield. So, the sides s₁ and s₂ would not feel the magnetic field from the side s₃ and the latter does not feel that from s₁ and s₂. What will be the total internal force on the triangle in this case?

The side s₃ would feel no force. The side s₁ would feel the Lorentz force F₁ from the side s₂ and the side s₂ the Lorentz force F₂ from s₁ respectively. So, the total internal force of the triangle coil is F₃= F₁+F₂. As the forces F₁ and F₂ are Lorentz forces, they are perpendicular to their sides and they make an angle between them. Their resultant force F₃ is non null.

If the total internal force is non null, any movement in the direction of the resultant force would do a work and create a energy. This is impossible because energy cannot be created.

So, the fact that the Lorentz force is perpendicular to the current violates the third Newton's law and the energy conservation law. This is the paradox of the internal Lorentz force.

This paradox is explained and solved in the document behind the link Paradoxes and Solutions.

The Lorentz force law presents some deficiencies. A study is in this document..
Paradoxes and Solutions