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    B What if an exercise wasn't uniquely defined?

    I am inclined to be in favour with the reply of "etotheipi', though I totally missed the perfectly logical reply of 'Vanadium 50'! Anyway, thanks for all contributions.
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    B What if an exercise wasn't uniquely defined?

    A math exam question asked for the ratio of two wheels given the required information about the number of revolutions made along a certain distance by the two wheels of a penny-farthing bike. Some students gave the ratio of the radii while others gave the ratio of the areas. What should an...
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    B Real numbers and complex numbers

    Now the math is more clear in my mind. Thanks Mark44.
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    B Real numbers and complex numbers

    I thank all those who posted their replies. But am I correct to argue as follows? Using only real numbers, √( (-2)(-3) ) = √6 is correct but √(-2)√(-3) does not exist as a real number. Hence Method 1 and Method 2 in my first post cannot be compared.
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    B Real numbers and complex numbers

    To find √(-2)√(-3). Method 1. √(-2)√(-3) = √( (-2)(-3) ) = √(6). Method 2. √(-2)√(-3) = √( (-1)(2) )√( (-1)(3) ) = √((-1)√(2)√(-1)√(3) = i√(2)i√(3) = (i)(i)√(2)√(3) = -1√( (2)(3) ) =-√6. Why don't the two methods give the same answer? Thanks for any help.
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    I Alternative formula for the Hamiltonian

    Thank you both, 'nrqed' and 'kimbyd'! Yes, my mistake was assuming that ##W_a## includes the rest mass energy. I derived it again and it turned out as the one given by Fermi, as 'nrqed' said.
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    I Alternative formula for the Hamiltonian

    Thanks 'Kimbyd' for your interest. I am using the usual ##E^2 = p^2 c^2 + {m_o}^2c^4,## and replacing E by ##W_a##, assuming ##W_a## stands for the total energy of the particle. I am also putting in the scalar potential V and vector potential U since the charged particle is in an...
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    I Alternative formula for the Hamiltonian

    I am still wondering from where did Fermi got his equation (13).
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    I Alternative formula for the Hamiltonian

    I tried the following. ## \left[ {W_a - ~eV} \right ]^2 = \left[ p -\frac {eU} c \right ] ^2 c^2 + m^2c^4 ## ## \left[ \frac{W_a - ~eV} c \right ]^2 = \left[ p -\frac {eU} c \right ] ^2 + m^2c^2 ## ## 0 = -\frac 1 {2m}\left(\left[ \frac{W_a - ~eV} c \right ]^2 -...
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    I Alternative formula for the Hamiltonian

    In his article 'Quantum theory of radiation', Reviews of Modern Physics, Jan 1932, volume 4, Fermi gives the relativistic hamiltonian function ##W_a## for a point charge by equation (13), ## 0 = - \frac 1 {2m} \left( \left[ mc +\frac {W_a - ~eV} c \right ]^2 - \left[ p -\frac {eU} c...
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    I Help required regarding deriving E-L equations for EM

    I think that it all depends on with whom I share my work. Perhaps if I share it with a beginner like me, it is good to show all steps. Thank you for your help, including your last hint.
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    I Help required regarding deriving E-L equations for EM

    Thanks 'Orodruin". I am not so fluent with tensors and hence, $$ \partial F_{\alpha\beta}F^{\alpha\beta} = 2F_{\alpha\beta} \partial F^{\alpha\beta} = 2F^{\alpha\beta} \partial F_{\alpha\beta}, $$ is not so obvious to me and I have to work it out. ##\begin {align} ∂(F_{αβ}F^{αβ}) & =...
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    I Help required regarding deriving E-L equations for EM

    I am following the suggestion by 'jedishrfu' and repeating my original post but removing the pdf attachment and trying LaTex instead. The langragian for an electromagnetic field with sources is, $$\mathcal{L}= - \frac {1}{4} F_{αβ}F^{αβ} + j^μA_μ.$$ Hence, ##\mathcal{L} = - \frac {1}{4}...
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    I Help required regarding deriving E-L equations for EM

    Thank you 'jedishrfu'. First I will have some practice using LaTex. Then I intend to repeat my post in a better format! Thanks again.
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    I Help required regarding deriving E-L equations for EM

    I am a retired High School teacher trying to use tensors in getting the Euler-Lagrange equations from the em lagrangian density. I attached a document in my post since I am not fluent in writing LaTex. Can anyone, please check my work. Thanks.
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    Finding the gravitational force over a flat infinite sheet

    Note also that only the vertical component of the force due to the ring is to be considered.
  17. G

    Finding the gravitational force over a flat infinite sheet

    The dr is included by putting the correct mass of the infinitesimal ring which is σ2πrdr.
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    Two capacitors with opposite terminals connected together

    Note that the POSITIVE plate of one capacitor is connected to the NEGATIVE plate of the other.
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    B Cyclic quadrilateral and alternate segment

    The post included all I had to say. Thanks.
  20. G

    B Cyclic quadrilateral and alternate segment

    As the secant AE is moved downwards, the exterior angle remains equal to the same interior angle, with the result that as the secant becomes a tangent, the cyclic quadrilateral disappears and the exterior angle becomes equal to the angle in the alternate segment. pdf is attached. It is...
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    B Does a tangent to a curve touch at 2 identical points?

    The geometrical description is a good answer to me. Thanks
  22. G

    B Does a tangent to a curve touch at 2 identical points?

    The limit of the above is 1 and IS attained. But although one knows the limit of 1, 1 + 0.5, 1 + 0.5 + 0.25, ... yet is it ever attained?
  23. G

    B Does a tangent to a curve touch at 2 identical points?

    Thanks 'fresh_42'. That's the idea that crossed my mind. But then then when one talks about 'limits' one may say that limits are approached as far as required ... but not attained.
  24. G

    B Does a tangent to a curve touch at 2 identical points?

    I am very sorry. I meant comparing the graphs of y = x - 1 and y = (x - 1)2. Both equations have the solution, i.e. x = 1 but that of the tangent is x = 1 (twice). I am asking if there is some deeper meaning for that 'twice'.
  25. G

    B Does a tangent to a curve touch at 2 identical points?

    The graph of y = x - 1 CUTS the x-axis at x = 1 while the graph of y = x2- 1 TOUCHES the x-axis at x = 1. The point at which the tangent touches the curve is shown mathematically by having two solutions of x, i.e. x = 1 (twice). Is there some deeper meaning to these two identical solutions for x?
  26. G

    Advantage/disadvantage of notation

    Thanks so much to you all. It seems that the first method is the better. As for the post of DarkoDornel, I am happy to say that I do not have to worry about "they always ask why that, and why not this" because I am a retired high school teacher who does not intend to work towards any thesis.
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    Advantage/disadvantage of notation

    When discussing Lorentz transformations some books use xμ/ = Λμ/αxα while some other books use x'μ= Λμαxα. Perhaps someone can help me understand which is the better notation. Thanks.
  28. G

    Help on Momentum problem

    A simpler notation may help. One can call the final velociies u and w and substitute for the masses right away. Then conservation of momentum and conservation of KE will give two equations in u and w which can be solved.
  29. G

    Free Response Momentum Problem

    You imagined correctly. I think that as for (h) and (i) you do not have to substitute any numbers. Just consider KEf = ( M1/(M1 + M2) )KEi. Then ignore the smaller mass when compared to the bigger one in (h) and in (i) and make the obvious deductions as regards what happens to the KEf and KEi.
  30. G

    Free Response Momentum Problem

    For part (h), imagine a very large heavy truck moving and colliding with a very small car. Do you think the initial KE of the truck will decrease much?
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