Electric Field at a Point Near a Charged Rectangular Plate

[Harrisonized]

Homework Statement

Given an arbitrary curve or surface with a total charge of Q, find the vector equation for the electric field at any point in space.

Homework Equations

dE = 1/(4πε₀) dq /r²

The Attempt at a Solution

Problem 1

Take the unit circle on the plane, for example. Find the vector equation for the electric field at a point (x₀,y₀).

The unit circle on the plane can be parametrized by
x = r cos θ
y = r sin θ

or alternatively,

r = √(x²+y²)

I know that the electric field lines generated are normal to such a curve. Therefore, if I take the gradient:

∇r(x,y) = 1/r*(x,y)

I get the unit normal vector. The only way I know how to do this problem is by letting Q be concentrated in the middle of the circle to obtain

E = 1/(4πε₀) Q /r² * 1/r*(x,y)

Unfortunately, this is wrong for the interior of the sphere.

I'm not really sure what to do. I've looked in so many textbooks, and none of them tell how to actually find the vector field E. All the sources I found only concern themselves with finding |E| at a point specified from the "object" of charge, and most of them just use symmetry to reduce the problems to pointlessness.

Any help is appreciated. If possible, please link me to some source that provides the method of finding such fields.

Problem 2

Find the electric field at a point a perpendicular distance a away from center a rectangular plate of uniformly distributed charge. The rectangular plate has dimensions m*n and no thickness.

2. Homework Equations

dE = 1/(4πε₀) dq /r²

The Attempt at a Solution

Let dq = σ dx dy, where σ = Q/(mn), Q is the total charge of the plate.

Let the center of the plate be the origin of x and y. Then:
r=√(x²+y²+a²)

By symmetry, the x and y components along the direction of the plane cancel out. Let θ represent the angle between a and r_x and φ represent the angle between a and r_y. Then:

dE = 1/(4πε₀) dq /r²
dE = 1/(4πε₀) σ dx dy /r² * cos θ * cos φ
dE = 1/(4πε₀) σ dx dy /(x²+y²+a²) * a/√(a²+x²） * a/√(a²+y²)
Integrate...
... and...
Wolfram Alpha: (No result found in standard mathematical functions.)

:(

Also, how do you do this for any point (without assuming an infinite plane)?

 

[Harrisonized]

I found the solution to Problem 1.

http://www.physics.buffalo.edu/~sen/documents/field_by_charged_ring.pdf

Anyone want to help me on Problem 2?