Consider an uncharged, conducting hollow spherical shell. The electric field on the inside surface of the spherical shell is measured to be directed toward the inner shell wall and varies in magnitude over the spherical shell's inside surface. It is known that the only charge in the hollow cavity inside the spherical shell is a point charge with magnitude q. What is the sign of the charge q?

The sign of charge q is negative.

Explanation:

This is because electric field lines usually emanate from positive charges and terminate at negative charges. The negative charge in the hollow cavity induces a positive charge on the inner shell wall of the hollow shell which in turn induces the negative charge q on the hollow spherical shell.

Since the induced electric charge on the inner wall is positive, the electric field lines from that charge terminates at the inner charge q. Since electric field lines terminate only at negative charges, q is negative.