Why the Electric field outside 3 infinite boards is 0?

[axcelenator]

1. Three boards are in the x axis. the middle one is in x=0 and charged in a positive sign density is: σ. the other 2 boards are in : x=d and x=-2d. This 2 boards are grounded.

The question is why outside the boards the ElecField is 0?
I though that i don't waste energy to bring an electric charge from x=∞ to x=d because the potential in ∞ is 0 (from my theoretical assumption - but in the question nobody told me that).

So, what is the best answer for that? Thanks!

 

[ehild]

I think, yours is the best answer! Those plates are grounded, that is, they are at the same potential as infinity. No work is done when a charge moves from infinity to those plates, so the electric force (-negative gradient of the potential energy) has to be zero.

Of course, it would not be true if there were charges outside the grounded plates.

ehild

 

[axcelenator]

Another question about that: I found that on the middle plate the charge is divided in 2:1 ratio: The electric field from X=0 to X=d is:E1 = (8/3)*pi*k*σ
and from X=0 to X=-2d is: E2 = (-4/3)*pi*k*σ.
If I want to find the total force on the middle plat should I do: F=σ*(E1-E2) ??
Thanks

 

[ehild]

It is all right but you can simplify the result you got for E1, E2, and F.

ehild

 

[ehild]

Sorry, it is half of that: F=σ*(E1-E2)/2

See for example: http://web.mit.edu/8.02t/www/materials/StudyGuide/guide04.pdf
The charge does not act on itself, and half the electric field between the plates is due to the charges on the middle plate.
You can also derive it from the energy of the electric field between the plates. The energy density is U=1/2ε₀E². If the middle plate is moved by dx, the electric field does work, and that is equal to Fdx. At the same time, this work of the field will decrease the energy. The change of energy is equal to the work done.

ehild