Problem in finding the electric field()

[vchris5]

Consider a, practically,infinite metallic conductor in which inside there is a spherical cavity with radial R. At the paries there is a surface allocation of electric brunt with a surface density: σ(θ)=σ_οcosθ .
θ: polar angle at the system of spherical coordinates with the center at the center of cavity.
Find the electric field at each point of the cavity.

Thanks in advance!

 

[malawi_glenn]

attempt to solution?

 

[vchris5]

attempt to solution?

Inside the conductor, the electric field it's zero.
I think inside the cavity the field is zero
And at the surface of the cavity how can I work?