(a) A physical quantity that turns out to be useful in describing the evolution of magnetic fields in confinement devices is magnetic helicity. This is defined by H = ∫dV A · B, where A is the vector potential, and the integral should be performed over all the space visited by the field lines to remove a dependence on the electromagnetic gauge. Compute the partial derivative of A and B with respect to time to show that H is conserved if E · B = 0.

(b) The helicity H primarily measures the topological linkage of the magnetic field lines. Therefore, it should not be a surprise that H turns out to be relatively well-preserved, even when the plasma is losing energy through resistivity and radiation. To discuss magnetic helicity properly would take us too far into the domain of classical electromagnetic theory, but some indication of its value follows from computing it for the case of two rings of magnetic field containing fluxes Φ₁ and Φ₂. Start with the rings quite separate, and show that H = 0. Then allow the rings to be linked while not sharing any magnetic field lines. Show that now H = 2Φ₁Φ₂ and that dH/dt =−2∫dV E · B.