-> E' field of accelerated charges
E <------- total electric field
E₀ <-- electric field of cap
______
/ \
• <-- | × |
q v \_______/
test charge "cap" charged shell
The Coulomb field of the "cap" and the remaining charged shell make the electric field E at q larger than what H
Gauss's law predicts.
The charge q also "sees" some charges in the shell being accelerated radially outward from the center × of the shell. These accelerated charges on the surface of the shell form a ring which is symmetric around the line from q to the center ×.
The field of the accelerating ring of charges
In our January 3, 2025 we compared two configurations. In one, the entire shell is expanding. In the other, only the "cap" moves radially, and the test charge sees the accelerating ring of charges arounf it.
To restore Gauss's law, the field of the ring must produce just the right transient electric field E' which is radial at q.
How can a such a field B arise from a system which is axially symmetric around the line between q and ×?
× × B transient
---> E' transient
○ ○ B transient
The field should satisfy Maxwell's equations. If we calculate B from the transient E', does E' satisfy the corresponding equation for the transient B?
Bent lines of force
If there is a transient tangential loop of the magnetic field B around the line × to q, then there will be a transient radial electric field E at q. Can the ring of accelerating charges produce such a loop around q?
Let us first use a naive interpretation of the field. Because of symmetry, there obviously cannot be a tangential component of B at q. By the same argument, there cannot be a tangential component of B anywhere. Thus, for the radial component of E:
dEr / dt = 0.
But this is a contradiction. The "cap" argument shows that there would be a transient change in Er.
Maybe we must not consider the actual current value of B (current in the laboratory time coordinate), but the value which q "derives" from what q "sees".
The ring of accelerating charges probably produces a tangential loop of the magnetic field B around q:
B a <-- •
○○
• q × center
××
B a <-- •
accelerating
ring
The tangential component of B is zero at q, but nonzero close to q.
The familiar Edward M. Purcell diagram shows the electric field lines of a charge which was suddenly accelerated to the left. The bends in the lines of force, and the "tangential" lines in the diagram, represent the sudden acceleration. The beautiful radial lines represent the old motion (the charge was static) and the new motion with a constant velocity.
The retardation formula and the diagram of Edward M. Purcell are only approximate
Assuming that Maxwell's equations hold, the process described in Purcell's diagram is very complex. There has to be backscattering of the wave which was created by the accelerated charge in the diagram. The lines of force cannot consist of straight line segments.
Approximate retardation formula. If a charge Q moves at a constant velocity v in the laboratory frame, the a test charge q sees the electric field as if Q would be at its current position, where "current" is in the laboratory time coordinate.
We have been using the formula above in our analysis of retardation. Now we realize that it is only approximate. The retardation formula is seen in the straight radial lines of force in the Edward M. Purcell diagram. But the lines cannot be perfectly straight. The formula is only approximate.
Assume that Gauss's law holds: the test charge q must not see any difference between the whole shell or a part expanding
Case A
______
<-- E / \
• q <-- | × |
v \_______/
only the "cap" and ring move, shell static
Case B
^ v
|
__________
/ \
<-- E / \
• q | × |
\ /
\___________/
|
v v
entire shell expands
The speed of light is finite. The test charge q cannot know if the entire uniform shell of charge is expanding, or if just the "cap" and the ring of charges around it are moving.
Let us assume that Gauss's law holds. The electric field measured at q must be the exact same E in cases A and B. Otherwise, q would receive information faster than light.
If we assume Gauss's law, then calculating the electric field E in Case B is trivial.
Case A, actually, is not unique. The acceleration in the ring of charges may happen in many ways. There may be phases of slow and fast acceleration. In all these cases, the electric field E should have the exact same value as in Case B. Why would that be?
It is because q sees in its light cone exactly the same history in A and B. The electric field E has to be the same!
We showed that Gauss's law holds for the electric field E, after all. Our error on January 3, 2025 was a superficial treatment of the charges in the accelerating ring.
Let us suddenly freeze both A and B. The potential of q is then different in A and B. Thus, there is retardation in the potential of q.
Conclusions
We made an elementary error on January 3, 2025. Gauss's law holds for the electric field.
Retardation in gravity is a more complex phenomenon. We will next look at retardation in the rate of clocks.
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