Imagine that you have a cylinder of radius $a$, with total charge $Q$ uniformly distributed through the surface, infinite height and coaxial with $z$ axis. Let be $s$, $\phi$, z (cylindrical) our system of coordinates.
Imagine also that we have a certain circumferential electric field $\vec{E(s)}=E(s)\vec{u_{\phi}}$, whose value depends only upon $s$ coordinate (the distance from axis $z$), and we´re asked for the torque exerted to the cylinder by the field. In the problem where I´ve found this situation, it´s due to the fact that the cylinder is inside a solenoid whose current is changing, and therefore, according to Faraday´s law, a circumferential electric field is created by a changing magnetic field, but I don´t find it relevant for the purpose of my question.
According to my sources, the torque is simply $\tau=\vec{r} \times Q\vec{E}= a\vec{u_{s}} \times Q\vec{E}$
This is equivalent to suppose that the cylinder is a point charge $Q$ located at a distance $a$ from axis $z$, what can seem "logical", but I´m unable to prove it.
My questions are:
How can this fact be proven from the point of view of rigid body mechanics?
Is there a more general theorem for rigid body that says that, given certain symmetries in the problem, such as uniformity of the force on the surface or the shape of the body, torque or angular momentum can be written this way?