Which of the following particles will trace out the smallest circle...

The radius of the circular path of a charged particle moving perpendicular to a uniform magnetic field is given by $r = \dfrac{mv}{qB}$.

Since the velocity $v$ and the magnetic field $B$ are identical for all the given particles, the radius $r$ is directly proportional to the mass-to-charge ratio $\dfrac{m}{q}$.

For the given particles, the magnitude of the charge $q$ is the same (equal to $e$). Thus, the radius is directly proportional to the mass of the particle ($r \propto m$).

Among $\text{He}^+$, $\text{Li}^+$, electron, and proton, the electron has the smallest mass ($m_e \approx \dfrac{m_p}{1836}$).

Therefore, the electron will trace out the smallest circle.