Hopkins was born at Kingston-on-Soar, in Nottinghamshire. In his youth he learned practical agriculture in Norfolk and afterwards took an extensive farm in Suffolk. In this he was unsuccessful. At the age of thirty he entered St Peter's College, Cambridge, taking his degree of BA in 1827 as seventh wrangler and MA in 1830. In 1833 he published Elements of Trigonometry. He was distinguished for his mathematical knowledge, and became eminently successful as a private tutor, many of his pupils attaining high distinction.
About 1833, through meeting Adam Sedgwick at Barmouth and joining him in several excursions, he became intensely interested in geology. Thereafter, in papers published by the Cambridge Philosophical Society and the Geological Society of London, he entered largely into mathematical inquiries connected with geology, dealing with the effects which an elevatory force acting from below would produce on a portion of the earth's crust, in fissures and faults. In this way he discussed the elevation and denudation of the Lake District, the Wealden area, and the Bas Boulonnais. He wrote also on the motion of glaciers and the transport of erratic blocks.
So ably had he grappled with many difficult problems that in 1850 the Wollaston Medal was awarded to him by the Geological Society of London; and in the following year he was elected president. In his second address (1853) he criticized Elie de Beaumont's theory of the elevation of mountain-chains and showed the imperfect evidence on which it rested. He brought before the Geological Society in 1851 an important paper On the Causes which may have produced changes in the Earth's superficial Temperature. He was president of the British Association for 1853. His later researches included observations on the conductivity of various substances for heat, and on the effect of pressure on the temperature of fusion of different bodies. He died at Cambridge.