The Electric Field at the Center of an Accelerating, Spherical Shell of Charge
1. Problem Statement.
Given:A non-rotating, uniform spherical shell, of charge q and radius R, with constant acceleration along the x-axis.
Task: Compute Ex at the sphere’s center at the instant the shell’s center coincides with the origin, which is defined to be the moment when the sphere is at rest.
2. The Point Charges Axiom.
Ex for the spherical shell is identical to the field of a semicircle of sufficiently small point charges that lie on the shell’s intersection with the xy-plane (with y>0).
3. Computing Ex.
Method: Compute each incremental point charge’s retarded coordinates, velocity and acceleration. Then use the point charge field solutions to compute that charge increment’s Ex at the shell’s center. Finally, sum the results to obtain the net Ex at the shell’s center.
Variables: Let dq=p/N, where N is sufficiently large. And let qi=(i+1/2)dq, where 0<i<N. The arc subtended by (i-1/2)dq and (i+1/2)dq defines a narrow, circular band centered on the x-axis. The area of this band is
The charge in the band is
4. Retarded Quantities.
When the spherical shell’s center is at the origin, dqi is at
Fig. 4_1 depicts the retarded coordinates (with tr, the retarded time, greatly exaggerated).
Present and Retarded Positions at t=0
Referring to Fig. 4_1, we find that
where tr is found by the modeling software.
5. The Point Charge Solution for dEx.
Let r be the displacement vector from dqi’s retarded position to the origin (or to the shell’s center at time t=0). And let v be the retarded velocity and a be the retarded acceleration (which is the present acceleration in this constant acceleration case). Define utility vector u to be
The electric field for dqi is then
6. Ex as a Function of the Acceleration.
Fig. 6_1 plots Ex(a), using values of q=-1 coulomb and R=1 meter. The acceleration varies over the range 0<a<1E6 m/sec2. Note that for negative q, Ex is positive. (if q were positive, Ex would be negative.)
Ex(a), q=-1 coul, R=1 m
If a positive charge were at the spherical shell’s center, then this positive charge’s own acceleration-induced Ex would point in the negative x-direction. Thus the surrounding negative "cloud’s" Ex cancels out part of the central, positive charge’s Ex. Similarly, the central charge’s a-induced Ex, at points in the negative spherical shell, would partly cancel out the shell’s own, a-induced Ex. The net result is that the "atom’s" inertia (or the system mass in F=ma) would be less than the summed inertias if the positive and negative charges were infinitely separated.
7. Ex as a Function of R.
Fig. 7_1 plots Ex(R) over the range 1 meter < R < 10 meters. Values of a=1E6 m/sec2 and q=-1 coul are used.
Ex(R), q=-1 coul, a=1e6 m/sec2
Note that Ex at the shell’s center decreases as R increases.
8. Clouds of Virtual Charge.
Recent work by a Purdue team suggests that virtual charge pairs may briefly pop into and out of existence around electrons. As previously discussed, such virtual charge pairs may help resolve the discrepancy between an electron’s measured mass and the electromagnetic mass theoretically expected from the electron’s small radius. Sect. 7 of this article indicates that, if the positive and negative virtual charges are distributed in spherical clouds, with the positive cloud’s radius less than the negative cloud’s, then the positive charge’s lessening effect on the electron’s mass will be more pronounced than the negative charge’s magnifying effect. And the Purdue group did find reason to believe that, in the case of an electron, the positive half of the virtual pair resides closer to the electron than the negative half does during the virtual pair’s brief existence. Noteworthy in Fig. 7_1 is the exponential growth of the spherical cloud’s Ex at the shell’s center, as R decreases. Thus in the case of an electron, whose virtual charge pair would presumably have magnitudes of only 1.6E-19 coul, the very small radius of the inner, positive cloud could result in an Ex of magnitude comparable (but of opposite sign) to the electron’s own, a-induced Ex. In brief, the lessening effect on the electron’s electromagnetic mass could be significant.