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Calculation of the Self Force using the Extended-Object Approach

We present here the extended-object approach for the explanation and calculation of the self-force phenomenon. In this approach, one considers a charged extended object of a finite size $ε$ that accelerates in a nontrivial manner, and calculates the total force exerted on it by the electromagnetic field (whose source is the charged object itself). We show that at the limit $ε\to 0$ this overall electromagnetic field yields a universal result, independent on the object's shape, which agrees with the standard expression for the self force acting on a point-like charge. This approach has already been considered by many authors, but previous analyses ended up with expressions for the total electromagnetic force that include $O(1/ε)$ terms which do not have the form required by mass-renormalization. (In the special case of a spherical charge distribution, this $\propto 1/ε$ term was found to be 4/3 times larger than the desired quantity.) We show here that this problem was originated from a too naive definition of the notion of ''total electromagnetic force'' used in previous analyses. Based on energy-momentum conservation combined with proper relativistic kinematics, we derive here the correct notion of total electromagnetic force. This completely cures the problematic $O(1/ε)$ term, for any object's shape, and yields the correct self force at the limit $ε\to 0$. In particular, for a spherical charge distribution, the above ''4/3 problem'' is resolved.