Aplastic rod is charged up by rubbing a wool cloth, and brought to an initially neutral metallic sphere that is insulated from ground. it is allowed to touch the sphere for a few seconds, and then is separated from the sphere by a small distance. after the rod is separated, the rod: a) is repelled by the sphere. b) is attracted to the sphere. c) feels no force due to the sphere. i know the answer is a, but i don't understand why. how do we know that the sphere and the rod have the same charge? what does it mean to be insulated from the ground?

You are right. The answer to this question is option A. The metallic sphere is a conductor and can allow electric charges flow through it from one region to another. The reason the sphere is insulted from the ground is to block out the transfer of charges from the sphere to the ground or from the ground to the sphere. The ground itself can serve as infinitely large sink or source for electric charges.

The act of touching the sphere with the charged rod deposits some electric charge on the sphere. These charges in the absence of an insulator between the sphere and the ground will flow straight into the ground. But because there is an insulator connected to the sphere the charges remain on the sphere with no way of getting out of the sphere.

These charges are the same with that on the rod as they were conducted and not induced by polarization (in general if the shepre became charged by polarization, it would have a charge opposite that on the rod). Since like charges repeatedly and unlike charges attract. The sphere would always repel the rod.

Thank you for reading and I hope this was helpful to you.