Let \(\mathbf{A}\) be the vector potential and \(\mathbf{B}\) the magnetic field of the infinite solenoid of radius \(R\) in Example 4 . Use Stokes' Theorem to compute:
(a) The flux of \(\mathbf{B}\) through a circle in the \(x y\)-plane of radius \(r(b) The circulation of \(\mathbf{A}\) around the boundary \(C\) of a surface lying outside the solenoid

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EXAMPLE 4 Vector Potential for a Solenoid An electric current I flowing through a solenoid (a tightly wound spiral of wire; see Figure 11) creates a magnetic field B. If we assume that the solenoid is infinitely long, with radius R and the z-axis as the central axis, then B(r) = A(r) = 0 Bk ifr < R where r = = (x + y2)/2 is the distance to the z-axis, and B is a constant that depends on the current strength I and the spacing of the turns of wire. (a) Show that a vector potential for B is B if r > R 21 B (-y.x,0) X ifr > R if r < R