Radiation reaction force and Newton's third law

[goran d]

There seems to be a problem with radiation reaction force.

Let's look at charged sphere, attached on a rod to an object with larger mass, and rotating around its centre together with it.
The rotating charged sphere emits radiation with power P.
The sphere's speed is V.

Assuming energy is conserved, the rotation is slowed down by radiation reaction force.
The radiation reaction force's magnitude is F
P=FV
F=P/V
Thus we find the force.

Now let's find how much maximum "force" would the emitted radiation create.
The energy of radiation:
E=mc²
The impulse of radiation:
p=mc

p=E/c

we differentiate both sides with respect to time

F₁=P/c
Where F₁ is the force that the radiation would create if all of it were to move away in the same direction as the motion of the sphere.Clearly F₁<F, so the radiation is not sufficient to supply the radiation reaction force. And even more, not all radiation is going to "shoot" in the same direction, so the difference is even greater. So the radiation reaction force must therefore be unbalanced, contrary to Newton's third law!

F₁=F(v/c)

 

[xAxis]

"The radiation reaction force's magnitude is F
P=FV
F=P/V
Thus we find the force."
Where does this come from? What kind of force you refer to here?