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4-Vector Transformation

  1. Nov 6, 2013 #1
    Hi all,

    I got a 3 part Qs: γ=1/√1-v^2-c^2

    Part A

    1. The problem statement, all variables and given/known data

    Consider the Lorentz transformation tensor

    Matrix
    Row 1: [ γ 0 0 -vγ/c]
    Row 2: [ 0 1 0 0 ]
    Row 3: [ 0 0 1 0 ]
    Row 4:-[vγ/c 0 0 γ ]

    for transforming 4-vectors from frame S to [itex]\overline{S}[/itex] according to[itex]\overline{A}[/itex][itex]^{\mu}[/itex] = L[itex]^{\mu}[/itex] [itex]_{v}[/itex] A[itex]^{v}[/itex] . The coordinate system is x[itex]^{0}[/itex] =ct, x[itex]^{1}[/itex] = x, x[itex]^{2}[/itex] = y, x[itex]^{3}[/itex] = z .

    3. The attempt at a solution

    Doing the transformation and then solving for it gives the answer:

    d/d[itex]\overline{t}[/itex]=γ(d/dt-vd/dx), d/d[itex]\overline{x}[/itex]=γ(v/c^2 d/dt - d/dx), d/d[itex]\overline{y}[/itex] = d/dy, d/d[itex]\overline{z}[/itex]=d/dz

    That's the answer I get but I am not sure about if I have the addition and substraction signs correct.

    Part B

    1. The problem statement, all variables and given/known data

    In above question, if the 4-vector potential is given by [itex]\underline{A}[/itex]=([itex]\phi[/itex]/c, Ax, Ay, Az) in frame S what are its components in frame [itex]\overline{S}[/itex]?

    3. The attempt at a solution

    Again solving for and getting the answer, I am confused on the addition and subtraction signs:

    [itex]\overline{A}[/itex]=(γ[itex]\varphi[/itex]/c + γv/c Ax, γAx+ γv[itex]\varphi[/itex]/c^2, Ay, Az)

    Part C

    1. The problem statement, all variables and given/known data

    In Part B, the electric and magnetic fields are defined in frames S and [itex]\overline{S}[/itex] by

    E[itex]^{(3)}[/itex]=-∇[itex]\varphi[/itex]-dA[itex]^{(3)}[/itex]/dt, [itex]\overline{E}[/itex][itex]^{(3)}[/itex]=-∇[itex]\overline{\varphi}[/itex]-d[itex]\overline{A}^{(3)}[/itex]/d[itex]\overline{t}[/itex], B[itex]^{(3)}[/itex]=∇xA[itex]^{3}[/itex], [itex]\overline{B}[/itex][itex]^{(3)}[/itex]=[itex]\overline{∇}[/itex]x[itex]\overline{A}^{(3)}[/itex],
    [itex]\overline{A}[/itex]=([itex]\overline{\varphi}[/itex]/c, [itex]\overline{A}[/itex]x,

    If

    [itex]\overline{A}[/itex]y, [itex]\overline{A}[/itex]z)=([itex]\overline{\varphi}[/itex]/c, [itex]\overline{A}^{(3)}[/itex])

    what is value of [itex]\overline{E}[/itex]x?

    3. The attempt at a solution

    Again solving for it I get my answer in which I am unsure of the addition and subtraction signs.

    [itex]\overline{E}[/itex]x=Ex, [itex]\overline{E}[/itex]y=γ(Ey+vBz), [itex]\overline{E}[/itex]z=γ(Ez-vBy)

    I am also not sure if the have the vector components assigned to the correct axis.

    Help would be appreciated.
     
    Last edited: Nov 6, 2013
  2. jcsd
  3. Nov 6, 2013 #2
    Hi,

    no reply?

    Help?
     
  4. Nov 7, 2013 #3
    I'm not even sure what the questions are.
     
  5. Nov 7, 2013 #4
    In Part A - I am supposed to find the transformation of the L matrix using that tensor equation. Is my transformation correct? It was my attempt at the question.

    In Part B - Again, are the components of [itex]\overline{S}[/itex] correct (ie. is [itex]\overline{A}[/itex] correct)? It was my attempt at the question.

    In Part C - It is a bit crowded (the formulae) but essentially they are the electric and magnetic field equations E, E (dashed), B and B (dashed) of the S and S (dashed) frames.

    A (dashed, the 'if' was supposed to start before the A dashed equation and not in the middle)

    I am supposed to find the E (dashed, the 'x' is a typo, sorry) components of this system (from the A dashed equation of part B). If the above is wrong then so is my following working. Are the + and - signs in the answer? It was my attempt.

    Thanks for brings that up.
     
    Last edited: Nov 7, 2013
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