help me to clear the concepts in electric field

A capacitor consist of two parallel, circular conduction discs with radius b and separated by a small distance h. The space between the circular discs is supported by a concentric circular cylinder with permittivity ε and radius a.

I consider the electric filed is in z-direction, however, my friend argued with me that it should be in r-direction(he considered it as a circular cylinder), who is correct?

Moreover, the electric field outside the region where r>b should be zero if we assume the fringe effect is not significant. Usually, in what situation can we ignore the fringe effect ?

help you gays can help me to clear the concepts

The electric field will be directed in z-direction since it is due to the interaction between the negative and the positive charges that are located on the two discs (at a fixed time frame one disc will be positive and the other one negative). Inside the dielectric it will be more intense with respect to the outside air ring and almost zero outside the disc.
In real world there will be a fringe effect at both the boundary: dielectric-air and inside-outside the disc. Usually you can ignore these effects.
To calculate the capacitance you can consider your system as two capacitors in series, the cylinder with dielectric and the ring in air. So you can calculate separately the two capacitances and then sum them toghether.