The explanation involves analyzing the motion of charged particles in a magnetic field using the concept of radius of curvature. When a charged particle moves perpendicularly to a magnetic field, the radius of curvature $ r $ of its path is given by:

$ r = \frac{mv}{qB} = \frac{p}{qB} $

Where:

$ m $ is the mass of the particle.

$ v $ is the velocity.

$ q $ is the charge.

$ B $ is the magnetic field strength.

$ p $ is the momentum of the particle.

From this formula, we can see that the radius of curvature $ r $ is directly proportional to the momentum $ p $ and inversely proportional to the charge $ q $.

In the given scenario:

The Oxygen ion $\mathrm{O}^{-2}$ and the Hydrogen ion $\mathrm{H}^{+}$ both enter the magnetic field with equal momentum.

Since $\mathrm{O}^{-2}$ has a charge of $-2e$ and $\mathrm{H}^{+}$ has a charge of $+e$, and given that $r \propto \frac{1}{q}$, the path of $\mathrm{O}^{-2}$ will indeed have a smaller radius of curvature compared to that of $\mathrm{H}^{+}$.

Therefore, the Assertion $ \mathbf{A} $ is true.

The Reason $ \mathbf{R} $ states that a proton and an electron with the same linear momentum will form paths of different radii upon entering a magnetic field, with the proton having a smaller radius of curvature. This reasoning is based on the same principle, since the mass and charge of the electron and proton are different. However, this explanation does not directly justify the assertion about Oxygen and Hydrogen ions, so Reason $ \mathbf{R} $ is false in the context of explaining Assertion $ \mathbf{A} $. Thus, the correct conclusion is that $\mathbf{A}$ is true, but $\mathbf{R}$ is false.