A negative charge -Q is placed inside the cavity of a hollow metal solid. The outside of the solid is grounded.

Q: Is there any excess charge induced on the inner surface of the piece of metal?
A: Yes, +Q

Q: Is there any excess charge on the outer surface of the metal?
A: No

Q: Is there an electric field in the cavity?
A: Yes

I understand the first two. The +Q induced charge to match the -Q charge, and the outer part of the metal is neutralized. However, I don't understand the third. Wouldn't the electric field from the -Q and the field fromt the +Q induced charge cancel each other out?

Forget it... Inside the cavity of a charged metal object (such as a sphere or cylinder), there is no net field. The net field exists only on the outside.

That explains it.

Ah, not quite that simple.

There is no field inside a charged hollow ball that is only charged on the periphery.
However in this case there is a -Q at the center I and a +Q on the inner surface. There is an electric field between these two. You can apply Gauss Law around the charge at the center and get an E vector surrounding the charge at the center.
In fact that field continues outside the sphere with only a gap with No E field in the interior of the metal material of the shell itself. That is because in that metal shell material the Gauss surface surrounds the -Q at the center and the +Q on the interior surface of the metal, for a net charge inside of zero.
Outside the sphere, your Gauss surface encloses the -Q at the center, the + Q at the inner surface, and the -Q at the outer surface for a net charge of -Q enclosed for any point totally outside the sphere.

You bring up an interesting point. At first glance, it might seem that the electric fields from the -Q charge and the +Q induced charge would cancel each other out in the cavity. However, that is not the case.

To understand why there is still an electric field in the cavity, consider the following:

1. When the negative charge -Q is placed inside the cavity, it induces a redistribution of charges on the inner surface of the metal. The charges on the inner surface rearrange themselves to neutralize the electric field inside the metal. This happens because in a conductor, charges can freely move and redistribute in response to an external electric field.

2. The grounding of the outer surface of the metal solid ensures that the excess negative charge on the inner surface is neutralized. The grounding allows for the flow of electrons between the solid and the Earth, balancing out the charges. As a result, the outer surface of the metal remains neutral.

3. Now, although the electric field due to the -Q charge is cancelled by the induced charge on the inner surface, there is still a residual electric field inside the cavity. This residual field is solely due to the induced +Q charge on the inner surface.

The reason for this is that the induced charge on the inner surface does not completely cancel out the effect of the -Q charge. While the induced charge can redistribute to neutralize the electric field inside the metal, it cannot fully counteract the electric field of the -Q charge in the cavity itself. As a result, there is still an electric field present in the cavity, even though the exterior remains neutral.

To summarize, the presence of a negative charge -Q in the cavity induces a positive charge +Q on the inner surface of the metal solid. This induced charge, while neutralizing the electric field inside the metal, still produces a residual electric field inside the cavity.