Thursday, April 12, 2012

Proof of the remaining resultant Lorentz force internal to a triangular coil



PengKuan 彭宽, titang78@gmail.com
12 April 2012


The Lorentz force respects the third Newton's law. Is the Lorentz force internal to a coil consistent with the third Newton's law ? Let us analyze the triangular coil ABC in the Figure 1; the current is I . Each side feels a Lorentz force from the magnetic field of the coil itself. The resultant force of all the Lorentz forces on the 3 sides is the double integrated F.

I have done a numerical computation for a triangular coil with base length of 1 and height of 10. The calculated force is dimensionless and the value of the overall resultant force is (see the Figure 2):
S= 35.21 ey

This force is not 0, violating the third Newton's law. In general, this value suffices to prove that the Lorentz force law is flawed, because only one counter example is sufficient to topple a general law. However, to exclude any doubt about the accuracy of this numerical calculation, I have done a rigorous analytical proof, which gives the expression of the dimensionless resultant force for a isosceles triangular coil, Fres  .

Thus, the analytical method proves without a doubt that the Lorentz force law is flawed. The mathematical derivation of the proof is given in the Mathematical Proof (see pdf link or below).
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4 comments:

  1. AnonymousJanuary 19, 2016 at 7:05 PM

    Very interesting. Ok lets see if I understand it OK.. From your equation as Theta approaches 0deg, the triangle folds in upon itself, and the net force becomes zero. This make sense to me.

    As the angle Theta approaches 90deg, the triangle open up toward infinity, and the force is a maximum. This also seems reasonable to me.

    I believe there also should also be a zero force at 60 deg, where the triangle becomes and Equilateral triangle. Here, by symmetry, the force at the top should exactly cancel out the forces due to the other two points of the triangle. Maybe the zero force at Theta=60 is bound up in the following integrals?, or maybe you did not include the contribution of forces from the other corners? At any rate it is very interesting.

    -Dr Jaynes

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    Replies
    1. PengKuan EMJanuary 20, 2016 at 11:43 PM

      You are absolutely right. I have explained the Equilateral triangle coil case here: http://pengkuanem.blogspot.com/2012/03/lorentz-forces-internal-to-equilateral.html

      When the triangle is split into 2 parts, there is a self force. If the field on one side is from the other 2, then the force will be balanced. This is because the summit of 2 sides creates a self force.

      Delete
  2. AnonymousMarch 15, 2016 at 5:55 PM

    I have spent the last few months finding an analytical solution to the self force of an isosceles triangle loop of current.
    I did this after I realized that numerical calculations were a loosing battle since you ended up subtracting the difference between values
    of singularities approaching infinities, so the smaller you make 'dx' the larger the error will get.

    The analytical double integral solution gave both finite and diverging
    terms as a function of angle of the triangle. In order for the solution to be meaningful, all diverging singularities must cancel out, and for all angles.
    Unfortunately this did not seem to be the case.
    I was able to cancel out all terms except one. This diverging term cancels only at the angle of 60 deg, but seems to be divergent for
    all other angles?!... I can live with a self force, if proven out by experiment, but diverging terms mean that the theory gives non-sense answers.

    The diverging term is of the form:




    "f1(theta)* ln( [2b/( eps sin(2 theta) )] / [F2(theta)*b/eps]^[2*sin(pi/2-theta/2)] ).... lim eps ->0 "



    Where b is the length of the base of the triangle, and eps is the integration distance from the singular corners.

    For theta = 60 deg, the exponent in the denominator becomes 1, exactly canceling the terms in the numerator, only at 60 deg.
    otherwise [ b/eps ] -> infinity, lim [eps] -> 0 , of function 'ln' -> infinity,... 'ln' must diverge to +/- infinity, as far as I can see.
    (although it does seem to diverge very slowly)

    This is definitely a problem, a theory that predicts nonsense answers.

    -Dr Jaynes

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    Replies
    1. PengKuan EMMarch 15, 2016 at 11:14 PM

      You are correct that for 60° the self-force is zero. But in fact, if you compute using the balance of force between the base side and the two upper sides as a whole, you will end up with a non zero self force. Here is why
      Equilateral triangle coil case
      http://pengkuanem.blogspot.com/2012/03/lorentz-forces-internal-to-equilateral.html

      In fact, when you have 60° at each angle, the three forces cancel out because they are the same in directions of 0° 120° 240°.

      The singularity of infinite force at an angle is a proof that the Lorentz force law is wrong. Infinite force does not occur in nature.

      For avoiding singularity, I have done a computation of self-force using non symmetrical coils in semicircles here:
      Numerical computation of the Lorentz force internal to an asymmetric coil
      http://pengkuanem.blogspot.com/2013/04/numerical-computation-of-lorentz-force.html

      The result is the same, there is a self force even there is no angular point.

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