Sphere Theorem: Electrostatic Failure

[MetaFrizzics]

An electrically charged conducting sphere 'pulses' radially i.e. its radius changes periodically with fixed amplitude. The charges on it's surface - acting as many dipole antennae - emit EM radiation.

 

What is the net pattern of radiation from the sphere?

Once again, we have a trick question, based upon mixing elements of different Electromagnetic theories, i.e., classical and modern.

 

In classical electrostatics we would apply the Sphere Theorem and simply declare there is no radiation outside the sphere, since the potential field does not change. A sphere of any size acts as if the charge were concentrated at the centre, and if the centre does not move, the field does not change. Likewise, the field inside the sphere does not change, since it remains zero at all times everywhere.

 

 

The only place the field actually changes is in the region between the smallest and largest diameter of the pulsing sphere. This region however does not exhibit 'EM radiation' per se, but rather instantaneous toggling of the field at any and every point in space as the surface barrier of the sphere passes it by. It simply switches instantaneously from zero to the predicted value for a point-mass at that location in an on-off quantized fashion.

 

That's right, even though we can assume the motion of the sphere surface is sinusoidal (and the acceleration involves reasonable speeds), the field effect at any and every point in space is a quantized two-state system. That's classical electrostatics!

 

Of course the result is physically absurd, due to quantization of charge. That is, other Electrostatic theory axioms cause the complete failure of the Sphere Theorem once again.

 

In fact, we can predict that there *is* a change in the field outside and inside the sphere due to the imbalance of forces caused by discrete localization of charges. This would also be true for gravitational radiation in classical Newtonian gravity theory if all the considerations are applied properly.

[MetaFrizzics]

But what about the Magnetic Field?

 

Here we see the magnetic field trying to form around the path of each charge as it moves outward. However, locally, the field-lines cancel. If the surface were a finite plane with an edge, there would be a 'residual' magnetic flow around the perimeter because of uncancelled field-lines. But because the sphere surface has no 'border' the magnetic field lines cancel everywhere, and there is no net field remaining.

 

However, when we analyze three-dimensionally, there is a twist:

Imagine we are riding upon one of the charges. From the point of view here, the adjacent charges are simply moving directly away on all sides in the 'plane' of the sphere's surface. This means the central (observer) charge sees horizontal magnetic fields formed around the receding charges. Now there *is* a residual force on the upper and lower boundary of the sphere's effective thickness.

 

 

Luckily, we can again imagine this is true for *every particle, and the complimentary residual currents again cancel out! Lucky that!