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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 -y x2 + 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.)

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 -y x2 + 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.)

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.4: Hyperbolas
Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
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This not a physics question! It's a calculus / advance math 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?
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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