The capacity of a channel with an energy-harvesting (EH) encoder and a finite battery remains an open problem, even in the noiseless case. A key instance of this scenario is the binary EH channel (BEHC), where the encoder has a unit-sized battery and binary inputs. Existing capacity expressions for the BEHC are not computable, motivating this work, which determines the capacity to any desired precision via convex optimization. By modeling the system as a finite-state channel with state information known causally at the encoder, we derive single-letter lower and upper bounds using auxiliary directed graphs, termed $Q$-graphs. These $Q$-graphs exhibit a special structure with a finite number of nodes, $N$, enabling the formulation of the bounds as convex optimization problems. As $N$ increases, the bounds tighten and converge to the capacity with a vanishing gap of $O(N)$. For any EH probability parameter $η\in \{0.1,0.2, \dots, 0.9\}$, we compute the capacity with a precision of ${1e-6}$, outperforming the best-known bounds in the literature. Finally, we extend this framework to noisy EH channels with feedback, and present numerical achievable rates for the binary symmetric channel using a Markov decision process.