An infineltly long cylindrical surface density `sigma = sigma_(0) cos varphi`. Where `varphi` is the polar angle of the cylindrical coordinate system whose `z` axis coincides with the axis of the given surface. Find the magnitude and direction of the electric fiels strength vector on the `z` axis.

Explanation

Take a section fo the cylinder perpendicualr to its axis through the point where the electric field is to be calculated. (All points on axis are equivalent.) Consider an element `S` with azimuthal angle `varphi`. The length of the element is `R d varphi, R` being teh radius of cross section of the cylinder. The element iteself is a section of an infinite strip. The electric field `O` due to this strip is
`(sigma_(0) cos varphi (R dvarphi))/(2pi epsilon_(0) R)` along `SO`
This can be resolved into
`(sigma_(0) cos varphi d varphi)/(2pi epsilon_(0)){{:(cos varphi" along OX towards O"),(sin varphi" along YO"):}`
On intergation the component along `YO` vanishes. What remain is
`int_(0)^(2pi) (sigma_(0) cos^(2) varphi d varphi)/(2pi epsilon_(0)) = (sigma_(0))/(2 epsilon_(0))` alonng `XO` i.e, along the direction `varphi = pi`.