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