
This paradox is not so knotty, but it can be a bit confusing when you first come across it (I was confused). It is related to a discussion of the theory of the Aharonov–Bohm effect that took place in the late 1970s and early into the 1980s [1–4].
We often write Maxwell's equations in terms of electric and magnetic fields, and ,
(1) 
However, the second and third of these equations can be reduced to mathematical identities if we work in terms of the scalar and vector potentials, and , defined by
(2) 
Maxwell's equations (1) can then be reduced to two equations for the potentials,
(3) 
which can often simplify calculations (Landau and Lifshitz [5] were good at this). We might therefore be tempted to immediately abandon the six quantities that define the electromagnetic field in favour of using the four quantities that determine the potentials. However this is not so straightforward as and are not uniquely determined by (2). The and fields stay the same after the following transformation,
(4) 
where is an arbitrary function of space and time. The transformation, (4) is known as a gauge transformation. Classically we don't ascribe any physical meaning to the potentials precisely because of this symmetry  you can't measure something that isn't uniquely defined. Yet when you write the theory of electromagnetism interacting with matter in terms of a Lagrangian or Hamiltonian, then using the potentials rather than the fields becomes unavoidable [5]. As a consequence (4) is a very important symmetry in the quantum theory of charged particles interacting via the electromagnetic field. Generalizations of this symmetry form the basis of our theories of the strong and weak forces.
Let's now consider a specific situation. A line of coils carrying electric current is placed along the axis, as shown in the figure below,
Now we'll determine the vector potential that describes this situation. In the Coulomb gauge, which is where is chosen such that , the second line of equation (3) reduces to
(5) 
which has the general solution (equation (5) is just a vector version of Poisson's equation),
(6) 
We now apply (6) to the situation shown in the figure, where the current density is given by
(7) 
where there are turns of the coil (radius ) per unit length along the axis, with a current, flowing through each. Inserting (7) into (6) and expanding the resulting expression to leading order in (the first order term integrates to zero), one obtains,
(8) 
where we have assumed a point of observation at , , and assumed that the line of coils has an infinite length. The final integral can be performed, with the result,
(9) 
So that,
(10) 
where the symmetry of the situation determines the direction of the vector in the second step, and, . Equation (10) is the vector potential that describes the field of a long, infinitesimally thin line of coils. However, (10) can also be rewritten as the gradient of a scalar function,
(11) 
[1]  Aharonov, Y. and Bohm, D. "Significance of electromagnetic potentials in quantum theory", Phys. Rev. 115 485 (1959) 
[2]  Bocchieri, P. and Loinger, A. "Nonexistence of the Aharonov–Bohm Effect", Il Nuovo Cimento 47 475 (1978) 
[3]  Berry, M. V. "Exact Aharonov–Bohm wavefunction obtained by applying Dirac's magnetic phase factor", Eur. J. Phys. 1 240 (1980) 
[4]  Tonomura, A., Osakabe, N., Matsuda, T., Kawasaki, T., Endo, J., Yano, S. and Yamada, H. "Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave", Phys. Rev. Lett. 56 792 (1986) 
[5]  Landau, L. D. and Lifshitz, E. M. "The Classical Theory of Fields", Butterworth–Heinemann (2003) 