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New analytic expressions for the magnetic and electric fields generated by circular current loops (magnetic dipoles) are derived. They apply to near-zone, Fresnel-zone, and far-zone conditions; they asymptotically reduce to the popular small-loop equations in the far-zone. Analytical expressions for the fields produced by current-carrying conductors are valuable because of their rarity. They offer physical insight and a way to test the numerical methods needed for solving more complex problems. Areas for application include medicine, weapons, and nondestructive testing.