In a physical theory, a field equation describes how the fundamental forces interact with matter. The four fundamental forces are gravitation, electromagnetism, strong interaction and weak interaction.
Before the theory of quantum mechanics was fully developed, there were two known field theories, namely gravitation and electromagnetism (these two are sometimes referred to as classical field theories, as they were formulated before the advent of quantum mechanics, and hence do not take into account quantum phenomena).
Modern field equations tend to be tensor equations.
Read more about Field Equation: Relativistic Fields, Unification Attempts
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