Question

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?

Answer 1

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.

Answer 2

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.