Construct field tensor an dual tensor

[leonne]

Homework Statement

straight wire along z axis carries charge density $$\lambda$$ traveling in +z direction at speed v. construct field tenor and dual at point (x,0,0_

Homework Equations

E=(2$$\lambda$$ /4$$\pi$$$$\epsilon$$o r)r^
B=($$\mu$$o I/2$$\pi$$r)$$\phi$$^

The Attempt at a Solution

I just don't get how they find the E and B they got
E=$$\lambda$$ /2$$\pi$$$$\epsilon$$o x)x^
B=($$\mu$$o $$\lambda$$V/2$$\pi$$x)y^
I know to find r it would be distance to the point, well i thought it would be r=(x²+z²)^1/2, but they just got x
also why is E in x^ and b in y^ does it have to do with the right hand rule?
thanks

 

[praharmitra]

Did you construct the field tensor $F^{\mu\nu}$. Then simply give it a boost in the +z direction, and read of E and B from there. If you've done that, you should get the right answer (i have not verified the result from the book that you've mentioned)

Please show your steps and I'll inspect them.

 

[leonne]

The field tensor formula is
f= |0 Ex/c Ey/c Ez/c
|-Ex/c 0 Bz -By
| -Ey/c -Bz 0 BX
...
So first I would need to find what Ex then Ey than Ez ect correct?
So the charge is moving on the Z and trying the find tensor at some point on x axis.
Not sure what to do. Don't really get what you mean by "Then simply give it a boost in the +z direction, and read of E and B from there" Didnt really study this chapter, but we only need to know this for the final. Going to try to figure it out later
the answer is F=$$\mu$$o$$\lambda$$/2$$\pi$$x(0 c 0 0)
(-c 0 0-v)
(0 0 00)
(0 v 0 0)

 

[praharmitra]

No, see this is what we do.

We imagine a system with a line charge in which the charge DOES NOT MOVE. Find its field strength tensor, which is easy.

Then we boost the system in the +z system, effecting a lorentz transformation on the tensor $$F^{\mu\nu}$$. We now have a new F. But since F is a tensor, it must be of the same form as before. From its components we simple read out $$E_x,E_y,E_z,B_x,B_y,B_z$$

 

[leonne]

hmm ok thanks for info going to try it later tonight, ill post what i get thanks

 

[leonne]

ok still kind of lost but here is what i found. so the components of B and E are Bij=(dAj/dXi)-dAi/dXj = diAj-djAi

Ei=-$$\phi$$/dxi- dAi/dT =-di$$\phi$$ -dtAi
than
A$$\mu$$=($$\phi$$-cA)
then the tensor =

F=dAv/dx$$\mu$$-dA$$\mu$$/dXv=d_$$\mu$$ Av-d_vA_$$\mu$$than from this they make a matrix
Is this corrent? well i am completely lost lol Final is tomorrow and i have a feeling this will be one of the problems. Can you show me how to do it or know any websites? I looked and could not find any.
btw is the A the Retarded potential?
thanks