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Maximum Horizontal Force of Relativistic Point Charge

  1. Feb 14, 2019 at 2:17 AM #1
    1. The problem statement, all variables and given/known data
    A charge q1 is at rest at the origin, and a charge q2 moves with speed βc in the x-direction, along the line z = b. For what angle θ shown in the figure will the horizontal component of the force on q1 be maximum? What is θ in the β ≈ 1 and β ≈ 0 limits? (see image)

    2. Relevant equations
    Equation for the electric field of a stationary point charge: Q/(4*pi*ε*R^2)
    Lorentz transformations

    3. The attempt at a solution
    Starting out with the equation for the electric field of a stationary point charge, I used the Lorentz transformations to transform the electric field expression to the reference frame moving with βc and multiplied that expression by z / ((γx)^2 + z^2)^(1/2) (equivalent to cos θ) to account for the horizontal component of the electric field. I took the derivative of this electric field expression to obtain the x-value for which the horizontal electric field was at a maximum, which was at x = b/(sqrt(2) * γ), and I rewrote this value in terms of sin θ (since the problem is asking for it in terms of θ). My final answer was sin θ = sqrt(2γ/(2γ + 1)), but that does not match up with the actual solution, which is sin θ = sqrt(2 / (3 - β^2), although for the limiting cases of β = 1 and β = 0 they are identical.
     

    Attached Files:

  2. jcsd
  3. Feb 14, 2019 at 12:36 PM #2

    TSny

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    Homework Helper
    Gold Member

    OK. Note that γ is not inside the square root.

    This isn't correct. I think you might have made a careless error here. Check your work.
     
  4. Feb 14, 2019 at 7:40 PM #3
    Found the error. Thank you.
     
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