Problem 5: Recall that in electrostatics, we defined a scalar electrostatic potential V such that →E = −∇V. Let's see if we can find something similar with →B as well.
(a) Let us first try to define a scalar magnetostatic potential A such that →B = −∇A. In general, does such a definition for →B comply with the known laws for →B? If not, under what conditions does a scalar potential for →B work?
(b) The "no monopole" law for magnetism (∇ ∙ →B = 0) prompts that our desired potential can be a vector potential →A such that →B = ∇ !! →A. Explicitly show in Cartesian co-ordinates (do not invoke vector calculus identities as of now) that this vector potential always satisfies the "no monopole" law for magnetism.
(c) OPTIONAL – FOR EXTRA CREDIT. Notice how the definition of →A is in terms of its curl, not divergence. Let's explore what we can say about it. (You may invoke vector calculus identities here.)
i) Consider we have a vector potential →A₀ that produces a given →B. Show what happens to →B if we replace →A₀ with →A = →A₀+∇λ, where λ is some scalar function. (Some fancy words: setting a function for λ in this substitution is called a gauge transformation.)
ii) The divergence of a vector is a scalar, so let's say ∇ ∙ →A₀ = a₀ and ∇ ∙ →A = a, where a, a₀ are some scalar functions. Find an expression for ∇²λ in terms of a and a₀. Based on your expression and your result in i), argue that, as the equivalent of a gauge transformation, we are free to set the divergence of a vector potential to be an arbitrary scalar quantity and we will still retrieve the same given →B. (More fancy words: this invariance of →B under gauge transformations is called gauge invariance.)