Finding the Longitudinal Force: A Scientific Approach

[cdummie]

Homework Statement

Very long thread, with constant longitudinal charge Q' is placed in a vacuum parallel to a very long conductive strip, whose width is a. thread is placed in the middle of the strip and it's a/2 away from it, if the surface density of charge of the strip is σ, find the longitudinal force to a thread.

Homework Equations

The Attempt at a Solution

since F=Q*E it means that F'=Q'*E so i have to find E to find F, i could think of strip as a very big number of thin lines of charge, then, knowing the value of E for every single line i could sum up all of the contributions from one to another end of the strip. Since the E of the single line is (it's dE for the whole system):

where d is distance from one charged line to a thread.

Because of the symmetry, x component of the vector E will be zero, which means there's only y component and it's
dE_y=(σ*dl*cosθ)/(2πε₀d)

Since dl, d, and cosθ are unknowns i need to express them with some values i know, since cosθ*dl=r*dθ it means that cosθ/r=dθ/dl i have:

dE=(σ*dl*dθ)/(2πε₀dl)

dE=(σ*dθ)/(2πε₀)

which means i have to integrate over angle,
i have the distance form middle of strip to a thread and it's a/2 and it's in the middle so i have the distance to both ends form middle equal a/2, which means i could find distance form first (and last) thin line using Pythagoras theorem and it's sqrt(2)*a/2, now i can easy find the angle I'm looking for since sinθ equals opposite over hypotenuse it's sqrt(2)/2 and it means that angle is π/3 so i have integral form -π/3 to π/3 for this expression
dE=(σ*dθ)/(2πε₀)

when i solve it i get

E=σ/3ε₀
and F'=Q'*σ/3ε₀

Now, my question is. Is this good approach, because I'm not sure is this correct?

 

[haruspex]

A number of things I'm not following here.
Wouldn't longitudinal force mean force parallel to the thread? But I see no reason why there would be such a force. On the other hand, if it means the force normal to the strip then that will be proportional to the length, which is not given. Maybe I have the wrong picture.

It says the strip is conducting, but then indicates it has a uniform charge density. That's contradictory.

The equation you show for the field has it inversely proportional to the distance. Not inverse square?

I can't tell if your integral is right without more detail. I believe it should involve a $\cos^3$ term.