Problem of dielectric cylinder in uniform electric field is well known. For example, Jackson textbook or Griffiths textbook or online solution here.
Solution always given for case of long cylinder. So that $z$ dependence is removed from solution of Laplace equation $\nabla^2 \phi = 0$ in cylindrical coordinates $(r, \theta, z)$. Solution in form $\phi = \phi (r, \theta)$ is given here.
Now consider the case of nonlong cylinder. What are the additional boundary conditions on potential $\phi = \phi (r, \theta, z)$, compared to those of long cylinder case? Or, how to treat end effects?