= Solution
A field assigns a physical quantity to each point. <Gravitational field> is <force> per unit test <mass>; <electric field> is <force> per unit test charge. In a conservative, time-independent field,
$$
\boxed{\mathbf F=-\nabla U,\qquad \Delta U=-\int\mathbf F\cdot d\mathbf r.}
$$
Write $U=m\Phi$ for gravitational potential $\Phi$, or $U=q\phi$ for <electric potential> $\phi$, giving $\mathbf g=-\nabla\Phi$ and $\mathbf E=-\nabla\phi$. A <point mass> has $U=-GMm/r$ with zero at infinity, while two point charges have $U=Qq/(4\pi\epsilon_0r)$. Only potential-energy differences matter; a constant can be added freely. The <mechanical energy> $K+U$ is conserved if no nonconservative <forces> do work. A magnetic <Lorentz force> $q\mathbf v\times\mathbf B$ does no work on a point charge and generally cannot be represented by a position-only scalar <potential energy>.
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