Problem. Let B denote the magnetic field generated by the solonoid, with flux . The field vanishes away from the origin (r, y, z) = (0,0,0) (where the solenoid is located). (a) Consider the vector potential Â= ,0 r2 + y2' x2 + y? defined for all (x, y, z) # (0,0, 0). Compute the curl of A. (Hint: is a constant number, not a variable.) (b) Compute the line integral

Problem. Let B denote the magnetic field generated by the solonoid, with flux . The field vanishes away from the origin (r, y, z) = (0,0,0) (where the solenoid is located). (a) Consider the vector potential Â= ,0 r2 + y2' x2 + y? defined for all (x, y, z) # (0,0, 0). Compute the curl of A. (Hint: is a constant number, not a variable.) (b) Compute the line integral

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

In class, we discussed the Aharonov-Bohm effect. The setup was as follows. Consider two beams of electrons, shot
around a tiny magnetic solenoid. Classically, the two beams of electrons do not feel the magnetic field at all, since
it vanishes away from the solenoid. However, in quantum mechanics you do see an effect. This effect is detected by
a wave interference pattern that appears on a screen. Let's predict this difference using calc III.
solonoid
II
Figure 1: Two beams of electrons y1 and yII shot around a magnetic solonoid.
Problem. Let B denote the magnetic field generated by the solonoid, with flux . The field vanishes away from
the origin (x, y, z) = (0,0,0) (where the solenoid is located).
(a) Consider the vector potential
().
Â=
x² + y2' x² + y²
defined for all (x, y, z) # (0,0, 0). Compute the curl of A. (Hint: & is a comstant number, not a variable.)
(b) Compute the line integral
A- dî
where C is the circle of radius 1 in the xy-plane, oriented counter clockwise. (Hint: Your answer should involve
Ф)
(c) Now let y = Y1 – YII be the closed curve in the figure (i.e. the path that goes along Y1 and goes back along Y1).
Using Green's theorem for a non-simply connected region, we conclude that
A · dî =
Â · dî.
The Aharanov-Bohm effect can be detected provided that
Cos
A · dî ) + 1.
For what values of & does this happen?

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