Which Egyptian god is portrayed as a falcon, with a large circle representing the sun hovering over his head?

The Egyptian god you are looking for is Ra, the sun-god. The Egyptians depicted Ra in several ways, but he was most often depicted as a man with the head of a hawk or falcon. Over Ra's head hovered a yellow disk (representing the sun) with a snake coiled around it.

According to Egyptian mythology, Ra was responsible for creating all life. According to one myth, Ra created humans from his tears. As the god of the sun, the Egyptians believed he was responsible for light, heat, and the growth of crops. Thus he was intricately connected to most aspects of Egyptians' lives.

The city of Heliopolis served as the center for cult-worship of Ra. From this cult developed the idea that pharaohs were the "sons of Ra."