Lorentz boost, electric field along x-axis, maths confusion?

  • Thread starter JesseC
  • Start date
  • #1
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Homework Statement



Given that (φ/c,A) is a 4-vector, show that the electric field component Ex for a
Lorentz boost along the x-axis transforms according to Ex' = Ex.

Homework Equations



[tex]E_x = -\frac{\partial \phi}{\partial x} - \frac{\partial A_x}{\partial t}[/tex]

[tex]A_x[/tex] being the x component of the vector potential

The Attempt at a Solution



So I don't have a problem getting φ'/c or Ax'

Obviously:
[tex]E_x^{\prime} = -\frac{\partial \phi^{\prime}}{\partial x^{\prime}} - \frac{\partial A_x^{\prime}}{\partial t^{\prime}}[/tex]

But I don't understand how to get the partial derivative w.r.t. x' in terms of x and t. Likewise for the partial derivative w.r.t. t'.

In the solutions it seems:

[tex]\frac{\partial}{\partial x^{\prime}} = - \frac{\gamma \beta}{c}\frac{\partial}{\partial t} - \gamma \frac{\partial}{\partial x}[/tex]

?? Does it come from the lorentz co-ord transformation?
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
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Hi JesseC! :smile:
But I don't understand how to get the partial derivative w.r.t. x' in terms of x and t. Likewise for the partial derivative w.r.t. t'.
Chain rule: ∂/∂x' = ∂x/∂x' ∂/∂x + ∂t/∂x' ∂/∂t :wink:
 

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