Electric field inside conductor with a cavity

Suppose I have a neutral spherical conductor with a cavity inside. Suppose there's a $$+q$$ point charge inside the cavity.

I know that the electric field $$\vec{E}$$ is zero within the conductor, also know that there will be a $$-q$$ net charge in the inner surface of conductor with the cavity.

But I still don't understand if there can be an non-zero electric field inside the cavity due to $$+q$$.

Gauss law to a gaussian surface inside the cavity says it does:

$$\oint \vec{E} \cdot d\vec{A} = \frac{q}{\epsilon_0} \qquad \longrightarrow \qquad E \, A = \frac{1}{4\pi\epsilon_0} \, \frac{+q}{r^2},$$

where $$r$$ is the distance from $$+q$$ to a point inside the cavity.

Is my reasoning right? I mean, the electric field is always zero inside the conductor, but there can be a non-zero electric field inside the cavity?

Let the inner and outer radii of a spherical conductor with a cavity be $$R_1$$ and $$R_2$$ respectively. The electric scalar potential distribution is given by the following equation. $$$$\frac{4\pi\epsilon_0}{q}\phi(r) = \begin{cases} \frac{1}{r}+\left(-\frac{1}{R_1}+\frac{1}{R_2}\right)\;\;\;\text{(for }r Note that $$\phi(r)$$ is continuos at $$r=R_1$$ and $$r=R_2$$. Electric field for this case is given as $$$$\vec{E}=-\nabla\phi= \begin{cases} \frac{q}{4\pi\epsilon_0}\frac{1}{r^2}\vec{e}_r\;\;\;\text{(for }r where $$\vec{e}_r$$ is the unit vector. As seen in this solution, what you wrote "the electric field is always zero inside the conductor, but there can be a non-zero electric field inside the cavity" is correct.