# Accelerated charge inside sphere (again!)

Imagine a stationary charge ##q##, with mass ##m##, at the center of a stationary hollow spherical dielectric shell with radius ##R##, mass ##M## and total charge ##-Q##.

I apply a force ##\mathbf{F}## to charge ##q## so that it accelerates:

$$\mathbf{F} = m \mathbf{a}$$

The accelerating charge ##q## produces a retarded (forwards in time) radiation electric field at the sphere. When integrated over the sphere this field leads to a total force ##\mathbf{f}## on the sphere given by:

$$\mathbf{f} = \frac{2}{3} \frac{qQ}{4\pi\epsilon_0c^2R}\mathbf{a}.$$

So I apply an external force ##\mathbf{F}## to the system (charge + sphere) but a total force ##\mathbf{F}+\mathbf{f}## operates on the system.

Isn't there an inconsistency here?

As the acceleration of charge ##q## is constant there is no radiation reaction force reacting back on it from its electromagnetic field - so that's not the answer.

Instead maybe there is a reaction force back from the charged shell, ##-\mathbf{f}##, to the charge ##q## so that the equation of motion for the charge is given by:

$$\mathbf{F} - \mathbf{f} = m\mathbf{a}\ \ \ \ \ \ \ \ \ \ \ (1)$$

This reaction force might be mediated by an advanced electromagnetic interaction going backwards in time from the shell to the charge so that it acts at the moment the charge is accelerated.

Now the total force acting on the system is the same as the force supplied:

$$\mathbf{F} - \mathbf{f} + \mathbf{f} = \mathbf{F}.$$

If one rearranges Equation (1) one gets:

$$\mathbf{F} = (m + \frac{2}{3} \frac{qQ}{4\pi\epsilon_0c^2R}) \mathbf{a}$$

Thus the effective mass of the charge ##q## has increased.

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Dale
Mentor
2020 Award
So I apply an external force ##\mathbf{F}## to the system (charge + sphere) but a total force ##\mathbf{F}+\mathbf{f}## operates on the system.
Don't forget to include the fields as part of your system (i.e. charge + sphere + fields). Then you see clearly that f is an internal force and therefore causes no acceleration of the system.

If you insist on keeping the fields separate from the objects then there are two forces (transfers of momentum). First, momentum is transfered from the internal charge to the field, then the same amount of momentum is transfered from the field to the sphere.

Btw, this approach of yelling "inconsistency" is quite tiresome and must stop.

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