Update: I believe I have now found a solution, which I will post below.
Define the four vectors
##\begin{align}
\vec{X}_1 & = (ct_1,\vec{r}_1)^T \\
\vec{X}_2 & = (ct_2,\vec{r}_2)^T
\end{align}##
that correspond to the displacements of the ends of the interval ##\vec{w}## in the wave-front, so...
Well, you could define two four vectors
##\begin{align}
\vec{X}_1 & = (ct, \vec{x}_1) \\
\vec{X}_2 & = (c(t+\Delta t), \vec{x}_2)
\end{align}##
such that their difference gives
##\begin{align}
\vec{R}= \vec{X}_2 - \vec{X}_1 = (c\Delta t, \Delta \vec{x})
\end{align}##
where we have defined...
Yeah, sorry for not being specific; it is indeed a plane wavefront corresponding to an EM wave.
I might be getting myself in a pickle here, but wouldn't ##\vec{U}## be a good option to consider, seeing as the direction of the rays remains unchanged? Or ##\vec{K}## due to the nature of the...
Homework Statement
For a plane, monochromatic wave, define the width of a wavefront to be the distance between two points on a given wavefront at a given instant in time in some reference frame. Show that this width is the same in all frames using 4-vectors and
in-variants.
Homework...
Hi tburke2,
I believe that you can solve the system by considering the displacement x of the particle as the displacement from some support that oscillates in the same way as your forcing.
This would lead to a lagrangian of:
L =1/2mx'^2 - 1/2kx^2
were x = x_o - z, where x_o = F_o cos(wt) and z...
Hey there Eric. If you are still having trouble with this problem, you could consider the total energy of the system. In polar coordinates (where r is the distance of the alpha particle from the lead nucleus), the kinetic energy of the system can be written as, noting that the mass is...
Hey there Calpalned. I have worked the problem through, and I am not sure that your answer is correct. Are you sure you have evaluated the cross product correctly?