What is the fundamental ethical dilemma of police work, as described by the police chaplain Folkhard Werth?

Question

Accepted Answer

Police officers must do harm to prevent harm: use coercion to prevent violence and, in the worst case, kill to protect life. They must legally and legitimately break their own values in order to protect the values of society as a whole.
The dilemma has three levels of severity:

officers must do evil to prevent evil,
they must use coercion to prevent violence,
in the worst case, they must kill to protect life.

Werth calls this an inner paradox that cannot leave a person untouched. Most people hold non-violence as a personal value, and the officer must act against that personal code, legally and legitimately, to protect society's code. The dilemma is never solved once and for all; it has to be endured (ausgehalten) in every use of force. This is the personal side of the utilitarian justification: society can count the harm prevented, but the person who applies the force carries the cost.
Go deeper:

Werth: Gewalt, ein ethisches Dilemma — the police chaplain's full text (German).