What is the test for finding a prison official guilty of cruel and unusual punishment for failing to protect an inmate from physical abuse?

The Eighth Amendment prohibits the imposition of cruel and unusual punishment and "embodies 'broad and idealistic concepts of dignity, civilized standards, humanity, and decency.'" Estelle v. Gamble, 429 U.S. 97, 102 (1976). Prison officials have a duty to take reasonable steps to protect inmates from physical abuse. Farmer v. Brennan, 511 U.S. 825, 832-33 (1994) (quotations omitted). To establish a violation of this duty, the prisoner must show first, that he was incarcerated under conditions posing a substantial risk of serious harm; and second, that a prison official knew of and was deliberately indifferent to this risk. Id. at 834.