Electric Field of a Uniformly Charged Cylindrical Surface

[erisedk]

Homework Statement

An infinitely long cylindrical surface of circular cross-section is uniformly charged lengthwise with the surface density σ = A cosΦ where Φ is the polar angle of the cylindrical coordinate system whose z axis coincides with the axis of the the given surface. Find the magnitude and direction of the electric field strength vector on the z axis.

Source: Irodov

Homework Equations

The Attempt at a Solution

Since σ depends only on Φ, according to cylindrical coordinate system, σ is different on different planes that pass through z = 0. I just don't know where to begin, I looked up the answer and it didn't make sense to me very much.
Please just tell me how to start!

 

[theodoros.mihos]

On axes x,y make the expretion of electric field components E_x, E_y for the field on the x=0, y=0 and take integrals to whole circle and z. May polar coordinates be more useful.
Also may Gauss's law be useful.

 

[haruspex]

Since σ depends only on Φ, according to cylindrical coordinate system, σ is different on different planes that pass through z = 0.

Don't you mean through r=0?
Consider a section through it at some value of z. What does the charge distribution look like? Does it vary with z?