The velocity of electron near speed of light?

  1. This isn't a homework question, simply one I found in a book that I'm trying to do:

    momentum p, of electron at speed v near speed of light increases according to formula

    p = [tex]\frac{mv}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}[/tex]

    if an electron is subject to constant force F, Newton's second law of describing motion is

    [tex]\frac{dp}{dt}[/tex] = [tex]\frac{d}{dt}[/tex] [tex]\frac{mv}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}[/tex] = F

    This all makes sense to me. It then says, find v(t) and show that v --> c as t --> infinity. Find the distance travelled by the electron in time t if it starts from rest.

    Now I could get an expression for v by using the first formula, but I don't understand how I can show that v -->c as t --> infinity as t isn't in the equation. I haven't even attempted the second part, but I'm assuming some integration is involved

    Can anyone help?
  2. jcsd
  3. What you want to do to find v(t) is to solve the differential equation:
    F = \frac{d}{dt} \left( m v(t) \gamma (t)\right)
    for v(t).
    Now, you can differentiate the product to get
    F = m a(t) \gamma (t) + m v(t) \left( \frac{v(t)a(t)}{c^2}\gamma^3(t)\right)=m a(t) (\gamma + \beta ^2(t)\gamma ^3(t)), ~~~~ \beta=\frac{v(t)}{c}
    Now, solving this (I used Maple) and imposing the condition v(0)=0, one gets the expression
    v(t)=\frac{Fct}{\sqrt{F^2 t^2 + c^2 m^2}}
    As you can see, v approaches c a time goes to infinity.
    The expression is easily (with some computer) integrated to give you an expression for the distance travelled over time:
    d(t)=\frac{c}{F} \sqrt{t^2F^2+c^2m^2}
    Last edited: Apr 8, 2010
  4. F=m*d/dt(v/sqrt(1-(v/c)^2)



    Since F/m is constant, the above becomes a very simple differential equation with the solution:


    For at<<c, you recover the Newtonian equation v=at

    If you integrate one more time, you will get x as a function of a and t. Indeed:



    Again, for at<<c, you recover the Newtonian formula x(t)=at^2/2
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