Magnetic Field: Spinning, Charged Spherical Shell
A spherical shell of radius R, carrying a uniform surface charge density, is set spinning about a fixed axis at a constant angular velocity. What is the magnetic field at a point P outside the sphere?
QUESTIONS
(Score is number right minus number wrong.)
There is no loss in generality in putting the point P on the z-axis and selecting a Cartesian coordinate system in which the angular velocity vector is in the x-z plane.
True
False
The surface current K generated by the spinning shell is given by
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
The vector potential can be expected to follow the direction of
Equation (8a)
Equation (8b)
Equation (8c)
According to Equation (5), we expect the vector potential at point P to point in the direction of
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
Equation (9f)
The SOURCE POINT is given correctly by
Equation (10a)
Equation (10b)
Equation (10c)
Equation (10d)
Equation (10e)
Equation (10f)
Equation (10g)
Equation (10h)
The FIELD POINT is given correctly by
Equation (10a)
Equation (10b)
Equation (10c)
Equation (10d)
Equation (10e)
Equation (10f)
Equation (10g)
Equation (10h)
The ANGULAR VELOCITY is given correctly by
Equation (10a)
Equation (10b)
Equation (10c)
Equation (10d)
Equation (10e)
Equation (10f)
Equation (10g)
Equation (10h)
The length of WAVY r is given correctly by
Equation (10a)
Equation (10b)
Equation (10c)
Equation (10d)
Equation (10e)
Equation (10f)
Equation (10g)
Equation (10h)
The DIFFERENTIAL AREA is given correctly by
Equation (10a)
Equation (10b)
Equation (10c)
Equation (10d)
Equation (10e)
Equation (10f)
Equation (10g)
Equation (10h)
In calculating Equation (5), one can safely discard any terms that are not proportional to
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
Equation (9f)
The terms that we discard are all proportional to a single factor of either sine or cosine of phi and integrate to zero.
True
False
The integral for the vector potential is correctly given by Equation (11).
True
False
The integral can be transformed into Equation (6) by the substitution,
Equation (12a)
Equation (12b)
Equation (12c)
To evaluate the expression for the integral of Equation (6), we must use in our specific problem,
Equation (13a)
Equation (13b)
For a point outside the spinning, charged sphere, the vector potential is given by
Equation (14a)
Equation (14b)
For a point outside the spinning, charged sphere, the vector potential is given by
Equation (15a)
Equation (15b)
Unlike Equation (14), Equation (15) is written in "coordinate free form", which means that there is no remaining reference (except an implied origin from which r is measured) to either the original Cartesian or spherical coordinate systems used in the derivation. We are free to choose a new coordinate system (same origin) with the angular velocity along the z-axis (which seems more "natural").
In this new spherical coordinate system, for a point outside the spinning, charged sphere, the vector potential is given by
Equation (16a)
Equation (16b)
Equation (17b) correctly gives the magnetic field at an arbitrary point outside the spinning charged shell. See Equation (4).
True
False
EQUATIONS
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