Let (r) be an arbitrary scalar function. A magnetic field which satisfies B = B is called force-free because the Lorentz force density

Let α(r) be an arbitrary scalar function. A magnetic field which satisfies ∇ × B = αB is called force-free because the Lorentz force density j(r) × B(r) vanishes everywhere. There is some evidence that fields of this sort exist in the Sun’s magnetic environment.(a) Under what conditions is the sum of two force-free fields itself force-free?(b) Let α(r) = α. Find and sketch the force-free magnetic field where B(z) = B_x (z)x̂ + B_y (z) ŷ and B(0) = B₀ ŷ.(c) Let α(r) = α. Find and sketch the force-free magnetic field where B(ρ) = is finite.

(d) Suppose that B_z(R) = 0 in part (c). Find a simple magnetic field B^out(ρ) in the current-free volume ρ > R which matches onto the force-free magnetic field in the ρ

B(p)= B(p) + B (p)2 and B(0)