Lorentz transforming θ-dependent magnetic field with longitudinal distribution

[matt_crouch]

Basically I am trying to lorentz transform the magnetic field along θ of a bunch particles which have a gaussian distribution to the radial electric field. However the magnetic field in θ is dependent on the longitiudinal distribution.
Now initially i thought we would just use the standard LT,

x=x'
y=y'
s'=/gamma (s-βct).
Now someone suggested to me that infact the transform will be non trivial when a longitudinal dependent radial field is perpendicular to the boost axis.
Can someone suggest some literature that would point me in the right direction?

Just for reference the field follows as,

B_{\theta}=Const \times r^(-1/2)e^{-r^{2}/2*sigma_{r}^{2}} e^{-s^{2}/2\sigma_{s}^{2}}

Sorry i am not sure how to make it latex

 

[Meir Achuz]

You have to Lorentz transform the B vector also. This is described in advanced EM books.

 

[DrGreg]

B_{\theta}=Const \times r^(-1/2)e^{-r^{2}/2*sigma_{r}^{2}} e^{-s^{2}/2\sigma_{s}^{2}}

Sorry i am not sure how to make it latex

Put a double-dollar before and after
$$B_{\theta}=Const \times r^(-1/2)e^{-r^{2}/2*sigma_{r}^{2}} e^{-s^{2}/2\sigma_{s}^{2}}$$
Then correct the errors
$$B_{\theta}=\mbox{Const} \times r^{-1/2}e^{-r^{2}/2 \sigma_{r}^{2}} e^{-s^{2}/2\sigma_{s}^{2}}$$
...if that's what you meant.

 

[jtbell]

Now someone suggested to me that infact the transform will be non trivial when a longitudinal dependent radial field is perpendicular to the boost axis.
Can someone suggest some literature that would point me in the right direction?

Maybe this will help:

http://farside.ph.utexas.edu/teaching/em/lectures/node123.html

which presents the Lorentz transformation for E and B fields. To see the derivation, you have to work backwards through the preceding pages.

 

[matt_crouch]

Ok thanks.. I'll have a look through