The Electric Field at the Center of an Accelerating, Spherical Shell of Charge

G.R.Dixon, 4/11/2006

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 E

2. The Point Charges Axiom.

E_{x} 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 E_{x}.

* 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 E

* Variables: *Let dq=p/N, where N is sufficiently large. And let q

. (3_1)

The charge in the band is

. (3_2)

4. Retarded Quantities.

When the spherical shell’s center is at the origin, dq_{i} is at

, (4_1a)

, (4_1b)

. (4_1c)

Fig. 4_1 depicts the retarded coordinates (with t_{r}, the retarded time, greatly exaggerated).

Figure 4_1

Present and Retarded Positions at t=0

Referring to Fig. 4_1, we find that

, (4_2a)

, (4_2b)

where t_{r} is found by the modeling
software.

5. The Point Charge Solution for dE_{x}.

Let __r__ be the displacement vector from dq_{i}’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

. (5_1)

The electric field for dq_{i} is then

. (5_2)

Thus

. (5_3)

6. E_{x} as a Function of the Acceleration.

Fig. 6_1 plots E_{x}(a), using values of q=-1 coulomb and R=1 meter. The acceleration varies over the range 0__<__a<1E6 m/sec^{2}. Note that for negative q, E_{x} is positive. (if q were positive, E_{x} would be negative.)

Figure 6_1

E_{x}(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 E_{x} would point in the __negative__ x-direction. Thus the surrounding negative "cloud’s" E_{x} cancels out part of the central, positive charge’s E_{x}. Similarly, the central charge’s a-induced E_{x}, at points in the negative spherical shell, would partly cancel out the shell’s own, a-induced E_{x}. 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. E_{x} as a Function of R.

Fig. 7_1 plots E_{x}(R) over the range 1 meter __<__ R < 10 meters. Values of a=1E6 m/sec^{2} and q=-1 coul are used.

Figure 7_1

E_{x}(R), q=-1 coul, a=1e6 m/sec^{2}

Note that E_{x} 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 E_{x} 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 E_{x} of magnitude comparable (but of opposite sign) to the electron’s own, a-induced E_{x}. In brief, the lessening effect on the electron’s electromagnetic mass could be significant.