Question

how can you use an electroscope to determine an unknown charge using a known charge?

Answer 1

The electroscope is generally used to determine unknown charges.

Let us consider we have a known positive charge and an uncharged electroscope. If we touch the known charge with the electroscope it will transfer some of its charges thereby making the electroscope positively charged.

Now if we bring an unknown charge near the electroscope, depending on the nature of the unknown charge it will move towards or away from the charge. If it moves away the unknown is a positive charge as like charges repel. If it moves towards the charge it is negatively charged as opposite charges attract.

We could derive the same conclusion with a known negative charge.