Maxwell Stress Tensor

How do we calculate forces using the Maxwell Stress Tensor? Consider a spinning sphere of radius R with a uniform surface charge. What is the magnetic force (ignoring any purely electrical forces) on the northern hemisphere? Is the northern hemisphere attracted or repulsed by the force? Figure 1



QUESTIONS

(Score is number right minus number wrong.)

Spinning or not, there is a purely electric force from the uniformly distributed charge that causes the southern hemisphere to exert a force on the northern hemisphere.
True
False
The purely electrical force on the northern hemisphere is in the direction given by
Equation (7c).
Equation (7f).
The electric repulsion of the hemispheres vanishes once the sphere is spinning.
True
False
If the sphere spins as shown with constant angular velocity, surface current is created on the sphere. However, the currents in the southern hemisphere exert no additional forces on the currents in the northern hemisphere because there is no relative motion of charge.
True
False
Wires placed side-by-side with currents that are parallel and flowing in the same direction
attract each other.
repel each other.
Due to the spinning motion, the northern hemisphere feels an additional force that is directed in the direction given by
Equation (7c).
Equation (7f).
If the sphere spins at constant angular velocity, the electrical and magnetic fields produced are static and do not change with time.
True
False
The Poynting vector (Equation (1)) automatically vanishes if the electric and magnetic fields are static.
True
False
The second term in Equation (5) vanishes in this case because the electric and magnetic fields of the spinning sphere are static.
True
False
The first term in Equation (5) involving the Maxwell Stress Tensor T vanishes in this case because the electric and magnetic fields of the spinning sphere are static.
True
False
From this point forward we IGNORE the purely electric fields and forces and concentrate only on the force arising from the magnetic fields.

To calculate the force on the northern hemisphere, the surface integral in Equation (5) must include both the northern and southern hemispheres because that is where the charges and currents are.
True
False
The surface integral in Equation (5) must conform exactly to the surface of the northern hemisphere because the northern hemisphere is the object on which we seek the force.
True
False
An acceptable surface for the surface integral could be the entire x-y plane and a hemisphere above it imagined at large (infinite) radius.
True
False
The basis for choosing the surface that includes the northern hemisphere is simply that which makes the computation easier.
True
False
The computation of the portion of the surface integral over a hemisphere at large (infinite) radius is simple because the fields are vanishingly small there.
True
False
An appropriate differential area for the portion of the surface integral in the x-y plane is given by
Equation (8a)
Equation (8b)
Equation (8c)
Equation (8d)
When the dot product of T and da is applied, the only surviving term in the surface integral is
Equation (9a)
Equation (9b)
Equation (9c)
An element of the Maxwell Stress Tensor (magnetic fields only) is given correctly by
Equation (10a)
Equation (10b)
Equation (10c)
Inside the sphere (r < R), the contribution to the surface integral is given by
Equation (11)
Equation (12)
Equation (13)
Equation (14)
Equation (15)
Equation (16)
Outside the sphere (r > R), the contribution to the surface integral is given by
Equation (11)
Equation (12)
Equation (13)
Equation (14)
Equation (15)
Equation (16)
The total magnetic (only) force on the northern hemisphere is given by
Equation (17)
Equation (18)
Equation (19)
Equation (20)
None of the above
EQUATIONS

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