Consider the following magnetic potential expressed in spherical coordinates. k VM = (3 cos²0-1) (known as a quadrupolar potential). (a) If the magnetic field B=-VVM, then show that 3k B (3 cos²0-1)+ 6k sin cos 00. [Look up an expression for VV in spherical coordinates.] (b) Derive a simple value for the divergence of this magnetic field at all points in space [look up an expression for divergence in spherical coordinates]. (c) Show that the left-hand and right-hand sides of Gauss's theorem are equal for the case of vector field B and a sphere of radius 1 m, centred on the origin. [Hints: Calculate the total magnetic flux (B-dS) in/out of the sphere. The derivative of cos³ is -3 cos² 0 sin 0.]
Consider the following magnetic potential expressed in spherical coordinates. k VM = (3 cos²0-1) (known as a quadrupolar potential). (a) If the magnetic field B=-VVM, then show that 3k B (3 cos²0-1)+ 6k sin cos 00. [Look up an expression for VV in spherical coordinates.] (b) Derive a simple value for the divergence of this magnetic field at all points in space [look up an expression for divergence in spherical coordinates]. (c) Show that the left-hand and right-hand sides of Gauss's theorem are equal for the case of vector field B and a sphere of radius 1 m, centred on the origin. [Hints: Calculate the total magnetic flux (B-dS) in/out of the sphere. The derivative of cos³ is -3 cos² 0 sin 0.]
Modern Physics
3rd Edition
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Chapter13: Nuclear Structure
Section: Chapter Questions
Problem 2Q: A proton precesses with a frequency p in the presence of a magnetic field. If the intensity of the...
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could you please solve part a)
![Consider the following magnetic potential expressed in spherical coordinates.
k
VM = (3 cos²0-1)
(known as a quadrupolar potential).
(a) If the magnetic field B=-VVM, then show that
3k
B
(3 cos²0-1)+
6k
sin cos 00.
[Look up an expression for VV in spherical coordinates.]
(b) Derive a simple value for the divergence of this magnetic field at all points in space
[look up an expression for divergence in spherical coordinates].
(c) Show that the left-hand and right-hand sides of Gauss's theorem are equal for the
case of vector field B and a sphere of radius 1 m, centred on the origin.
[Hints: Calculate the total magnetic flux (B-dS) in/out of the sphere. The
derivative of cos³ is -3 cos² 0 sin 0.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9dea7f9-b5df-44eb-b722-5a6f233189b7%2F3ffcd125-896c-4f0b-bfc2-a504718794b8%2Fae6u5fk_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following magnetic potential expressed in spherical coordinates.
k
VM = (3 cos²0-1)
(known as a quadrupolar potential).
(a) If the magnetic field B=-VVM, then show that
3k
B
(3 cos²0-1)+
6k
sin cos 00.
[Look up an expression for VV in spherical coordinates.]
(b) Derive a simple value for the divergence of this magnetic field at all points in space
[look up an expression for divergence in spherical coordinates].
(c) Show that the left-hand and right-hand sides of Gauss's theorem are equal for the
case of vector field B and a sphere of radius 1 m, centred on the origin.
[Hints: Calculate the total magnetic flux (B-dS) in/out of the sphere. The
derivative of cos³ is -3 cos² 0 sin 0.]
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