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