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nissanztt90
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Wave Mechanics help--for optics course
Consider a stretch string of length L along the x-axis in a stationary vibration mode of the form
z(x, t) = z0sin((2*pi*n/L)x) cos(omega*t)
where n is greater than or equal to 2 and is a given integer number and y0 and [tex]\omega[/tex] are constant (i think they meant z0). The left endpoint of the string is located at the origin x = y = z = 0 and the right endpoint is located at x = L, y = z= 0
An electrically charged particle with charge q>0 is glued to the string at a distance a from the left endpoint.
Write down the radiation fields ERAD(r, t) and BRAD(r, t) of the electric charge using spherical coordinates (r, [tex]\theta[/tex],[tex]\phi[/tex]) centered at x = a, y = z = 0, assuming that the vibration amplitude z0 is much smaller than |r|. Express your answer as a vector field written in terms of the unit vectors er, e[tex]\theta[/tex], e[tex]\phi[/tex].
There is more to it, but this is all i need help with for now.
[tex]\vec{E}[/tex]RAD([tex]\vec{r}[/tex], t) = (-q/(4*pi*epsilon0c2))(1/r)([tex]\vec{a}[/tex](t-r/c) X [tex]\vec{e}[/tex]r) X [tex]\vec{e}[/tex]r
By X i mean vector cross product, not multiplication, just so there's no confusion.
[tex]\vec{B}[/tex]RAD([tex]\vec{r}[/tex], t) = (1/c)[tex]\vec{e}[/tex]r X [tex]\vec{E}[/tex]RAD([tex]\vec{r}[/tex], t)
So my first guess was to convert the form of the vibration mode to spherical coordinates and i got...
z(x, t) = rcos[tex]\theta[/tex]((2*pi*n/L)(rsin[tex]\theta[/tex]cos[tex]\phi[/tex]))cos(omega*t)
So now, i sub my z(x, t) equation in spherical coordinates in for er in my ERAD equation, correct? I believe the ERAD equation will only be expressed in the unit vector er since this is only along the x-axis where [tex]\theta[/tex] and [tex]\phi[/tex] = 0?
Homework Statement
Consider a stretch string of length L along the x-axis in a stationary vibration mode of the form
z(x, t) = z0sin((2*pi*n/L)x) cos(omega*t)
where n is greater than or equal to 2 and is a given integer number and y0 and [tex]\omega[/tex] are constant (i think they meant z0). The left endpoint of the string is located at the origin x = y = z = 0 and the right endpoint is located at x = L, y = z= 0
An electrically charged particle with charge q>0 is glued to the string at a distance a from the left endpoint.
Write down the radiation fields ERAD(r, t) and BRAD(r, t) of the electric charge using spherical coordinates (r, [tex]\theta[/tex],[tex]\phi[/tex]) centered at x = a, y = z = 0, assuming that the vibration amplitude z0 is much smaller than |r|. Express your answer as a vector field written in terms of the unit vectors er, e[tex]\theta[/tex], e[tex]\phi[/tex].
There is more to it, but this is all i need help with for now.
Homework Equations
[tex]\vec{E}[/tex]RAD([tex]\vec{r}[/tex], t) = (-q/(4*pi*epsilon0c2))(1/r)([tex]\vec{a}[/tex](t-r/c) X [tex]\vec{e}[/tex]r) X [tex]\vec{e}[/tex]r
By X i mean vector cross product, not multiplication, just so there's no confusion.
[tex]\vec{B}[/tex]RAD([tex]\vec{r}[/tex], t) = (1/c)[tex]\vec{e}[/tex]r X [tex]\vec{E}[/tex]RAD([tex]\vec{r}[/tex], t)
The Attempt at a Solution
So my first guess was to convert the form of the vibration mode to spherical coordinates and i got...
z(x, t) = rcos[tex]\theta[/tex]((2*pi*n/L)(rsin[tex]\theta[/tex]cos[tex]\phi[/tex]))cos(omega*t)
So now, i sub my z(x, t) equation in spherical coordinates in for er in my ERAD equation, correct? I believe the ERAD equation will only be expressed in the unit vector er since this is only along the x-axis where [tex]\theta[/tex] and [tex]\phi[/tex] = 0?