# Derivation of electric field intensity for Line charge

These snapshot is taken from Principle of Electromagnetics by Matthew N.O. Sadiku.

In this derivation I don't understand the highlighted step. In previous step we are defining a position vector R from element dl to point in space (x,y,z) which is equal to xax + yay + (z-z')az , this position vector is defined for a unique point.

However in the next step we are defining same position vector in cylindrical coordinate but we are defining only two cylindrical coordinate and $\phi$ coordinate is missing. I know that Electric field intensity will be constant for all values of $\phi$ for a given value of z. But I think position vector should be directed toward a unique point in space. Can anyone justify why we are not defining $\phi$ in the highlighted step?

In the highlighted area vector R is the place translation from a charge element dl in z axis to the observation point where the total E is wanted.

The first term of R is the placement of the xy projection of the observation point (a constant vector in xy plane when the integration is done), the second term is the z component of R, it's the z-difference times z-unit vector. The second term contains the integration variable (=the z coordinate of the charge element).

ϕ is constant in integration, it is included to xy-place of the observation point (ϕ=the direction angle of unit vector ap), but it isn't needed as variable in the integration, where only a zone in z axis is walked through to collect total E