  The self-force problem---which asks how self-interaction affects a body's
motion---has been poorly studied for spacetime dimensions $d \neq 4$. We remedy
this for all $d \geq 3$ by nonperturbatively constructing momenta such that
forces and torques acting on extended, self-interacting electromagnetic charges
have the same functional forms as their test body counterparts. The
electromagnetic field which appears in the resulting laws of motion is not
however the physical one, but a certain effective surrogate which we derive.
For even $d\geq 4$, explicit momenta are identified such that this surrogate
field satisfies the source-free Maxwell equations; laws of motion in these
cases can be obtained similarly to those in the well-known four-dimensional
Detweiler-Whiting prescription. For odd $d$, no analog of the Detweiler-Whiting
prescription exists. Nevertheless, we derive its replacement. These general
results are used to obtain explicit point-particle self-forces and self-torques
in Minkowski spacetimes with various dimensions. Among various characteristics
of the resulting equations, perhaps the most arresting is that an
initially-stationary charge which is briefly kicked in 2+1 dimensions
asymptotically returns to rest.
