Electric Field of Plate to a Point

[miniissacnewton]

Homework Statement

[/B]
A square plate of side-length L, charged with uniform surface charge density η.
It is centred at x = y = z = 0 and orientated in the z = 0 plane.
The task is to determine the z-component of the electric field at the point r(x, y, z) = r(x, 0, d), offset along the x-axis and a height d above the plane of the plate.
A strategy is to break the plate into a set of thin strips, with the i th strip having a thickness ∆xi and position xi along the x-axis (see figure).
The electric fields of the strips can then be individually determined, and summed up via an integral to obtain the total electric field.

(a) Draw two sketches of the system in the y = 0 plane (side view), the first showing the electric field lines from the plate, and the second showing the total electric field vector at point r as well as its z-component.

(b) Using your field vector sketch, find the electric field in the z direction (the z component)

Homework Equations

[/B]
In the y = 0 plane bisecting the plate, the electric field magnitude of strip i is
E(i) = ηLΔx(i)/(4πεr(i)√(r(i)²+L²/4))
where η=surface charge density
r(i) = distance from the centre of the strip to the point r
Δx(i) = distance of strip along x-axis

The Attempt at a Solution

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I have drawn multiple diagrams but I'm not confident in my calculations. I'm pretty sure that the field lines from the plate act perpendicular to the surface. However, from this I'm not sure how to get to the z component of the electric field...

 

[BvU]

Hello mini,

You want to show your calculation in detail. You only want the z-component at $(x,0,d)$ but the field lines aren't just in the z direction (imagine x >> L).
Posting a drawing might help ...

 

[BvU]

Any progress ?

 

[miniissacnewton]

Any progress ?

Am i correct in saying that the field lines will be perpendicular to the surface of the square plate?

 

[BvU]

No. Why should they ?
Again, imagine positions with x > L

 

[miniissacnewton]

No. Why should they ?
Again, imagine positions with x > L

So where the distance from the centre of the square plate is much bigger than the side length of plate?
Will the electric field go off from the plate at all angles?

 

[haruspex]

So where the distance from the centre of the square plate is much bigger than the side length of plate?
Will the electric field go off from the plate at all angles?

Clearly this plate is not a conductor. The field will not be normal to the plate even close to it. At the point (x, 0, d) (x>0), the net field from the plate in the rectangular region (L/2, *, 0) to (2x-L/2, *, 0) will be parallel to the Z axis. When we include the field from the rest of the plate, (2x-L/2, *, 0) to (-L/2, *, 0), the net field can no longer be parallel to the ZY plane.

Edit: ... and it's Isaac, not Issac.

 

[physicstotesrox]

And so how would you apply the equation E(i) = ηLΔx(i)/(4πεr(i)√(r(i)2+L2/4)) to go on to solve the magnitude of the electric field?