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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 A = 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 dî where C is the circle of radius 1 in the ry-plane, oriented counter clockwise. (Hint: Your answer should involve Ф) (c) Now let y = 1- 1 be the closed curve in the figure (i.e. the path that goes along n and goes back along 1)- 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 df) + 1. Cos For what values of does this happen?

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 A = 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 dî where C is the circle of radius 1 in the ry-plane, oriented counter clockwise. (Hint: Your answer should involve Ф) (c) Now let y = 1- 1 be the closed curve in the figure (i.e. the path that goes along n and goes back along 1)- 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 df) + 1. Cos For what values of does this happen?

Advanced Engineering Mathematics
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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This is not a physics problem. It's a calculus test problem 

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?
Transcribed Image Text: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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