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We self-consistently derive the magnetic energy and relative magnetichelicity budgets of a three-dimensional linear force-free magnetic structurerooted in a lower boundary plane. For the potential magnetic energy we derive ageneral expression that gives results practically equivalent to those of themagnetic Virial theorem. All magnetic energy and helicity budgets areformulated in terms of surface integrals applied to the lower boundary, thusavoiding computationally intensive three-dimensional magnetic fieldextrapolations. We analytically and numerically connect our derivations withclassical expressions for the magnetic energy and helicity, thus presenting aso-far lacking unified treatment of the energy/helicity budgets in theconstant-alpha approximation. Applying our derivations to photospheric vectormagnetograms of an eruptive and a noneruptive solar active regions, we findthat the most profound quantitative difference between these regions lies inthe estimated free magnetic energy and relative magnetic helicity budgets. Ifthis result is verified with a large number of active regions, it will advanceour understanding of solar eruptive phenomena. We also find that theconstant-alpha approximation gives rise to large uncertainties in thecalculation of the free magnetic energy and the relative magnetic helicity.Therefore, care must be exercised when this approximation is applied tophotospheric magnetic field observations. Despite its shortcomings, theconstant-alpha approximation is adopted here because this study will form thebasis of a comprehensive nonlinear force-free description of the energetics andhelicity in the active-region solar corona, which is our ultimate objective.

Magnetic Energy and Helicity Budgets in the Active-Region Solar Corona. I. Linear Force-Free Approximation