If a boy’s hand has a negative charge, what is the charge on the area of a metal doorknob closest to his hand?

What this question is really asking

You are being asked about electrostatic induction, how charges in a nearby object can rearrange charges in a conductor without touching it.

What a negative charge does to a nearby metal object

A metal doorknob is a conductor, so its electrons can move. If a negatively charged hand is brought near it, the excess electrons in the hand repel the doorknob’s electrons.

Charge separation on the doorknob (induced charges)

• Electrons in the doorknob shift to the side farther from the hand.
• The surface closest to the hand loses some electrons.
• Losing electrons means that near surface has a net positive charge.

So the charge on the area closest to the hand is positive.

Extra detail (what happens to the rest of the knob)

If the doorknob is initially neutral overall, the far side becomes relatively negative, while the near side becomes positive. The doorknob’s total charge can still be $0$ overall, it is just redistributed.