Prospero has separated and isolated the king because he has plans for the king.
In Act 1, Scene 2, the relationship between Prospero and his servant Ariel is explored. Prospero uses magic to free Ariel from his imprisonment in his tree but keeps him bonded in servitude to him because he needs Ariel's help. When Ariel reminds Prospero of his promise to liberate him, Prospero reminds Ariel that he has done a great job of freeing him from his prison and does not ask too much of him with his requests. Prospero tells Ariel that his former mistress, the witch Sycorax, asked him to perform tasks so wicked that Ariel himself could not comply. Prospero uses his intellect to raise arguments to persuade Ariel to remember to be grateful for his semi-liberty and relatively easy tasks assigned to him. Prospero does not use magic to persuade Ariel here. He succeeds and Ariel agrees to serve him still. Prospero has to enlist the help of Ariel and he could have exaggerated the extent of his magical powers to incite fear and obedience from Ariel. He reminds Ariel of the good he has done for him. Prospero's magical hold over Ariel is more of a persuasive and influential one than that of mysterious manipulation through magic.
It is not elaborated on exactly how Caliban is confined to the rock. He could have been physically chained and not kept restricted by magic. Prospero uses magic and influence to colonize Caliban because he thinks that the dark skinned native is incomparable to the whites. Prospero is correct because when Caliban rebels against him, he voluntarily changes his old master for new one in Stephano and Trinculo. This happens in Act 2, Scene 2, when Caliban is infatuated with two men whom he has never seen before; Trinculo and Stephano. He wants to follow them. Caliban seems simple minded and irrational. He does not understand that if he follows a new master, he is free from Prospero but does not get his own liberty. So he wants to follow Trinculo and Stephano for the wrong reasons. Sometimes, it seems that Prospero has no real magical hold over Caliban other than his exhortations to get Caliban's loyalty and service. Prospero uses threats to cause bodily pain to Caliban. Caliban is