`H^(+),He^(+)` and `O^(2+)` all having the same kinetic energy pass through a region in which there is a uniform magnetic field perpendicular to their velocity. The masses of `H^(+), He^(+)` and `O^(2+)` are 1 amu, 4 amu, and 16 amu respectively. Then :

Options

A. `H^(+)` will be deflected most
B. `O^(2+)` will be deflected most
C. `He^(+)` and `O^(2+)` will be deflected equally
D. All will be deflected equally

Explanation

To solve the problem, we need to determine the radius of the circular path taken by each ion in a magnetic field, as this will help us understand which ion will deflect the most. The radius of the path \( r \) of a charged particle moving in a magnetic field is given by the formula:

\[ r = \frac{mv}{qB} \]

Where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

Since all three ions (H\(^+\), He\(^+\), and O\(^{2+}\)) have the same kinetic energy \( K \), we can express the velocity \( v \) in terms of kinetic energy:

\[ K = \frac{1}{2} mv^2 \implies v = \sqrt{\frac{2K}{m}} \]

Substituting this expression for \( v \) into the radius formula gives:

\[ r = \frac{m \sqrt{\frac{2K}{m}}}{qB} = \frac{\sqrt{2Km}}{qB} \]

Now we can analyze each ion:

1. **For H\(^+\)**:
- Mass \( m_H = 1 \) amu
- Charge \( q_H = 1 \) e
- Radius \( r_H = \frac{\sqrt{2K \cdot 1}}{1B} = \frac{\sqrt{2K}}{B} \)

2. **For He\(^+\)**:
- Mass \( m_{He} = 4 \) amu
- Charge \( q_{He} = 1 \) e
- Radius \( r_{He} = \frac{\sqrt{2K \cdot 4}}{1B} = \frac{2\sqrt{2K}}{B} \)

3. **For O\(^{2+}\)**:
- Mass \( m_O = 16 \) amu
- Charge \( q_O = 2 \) e
- Radius \( r_O = \frac{\sqrt{2K \cdot 16}}{2B} = \frac{4\sqrt{2K}}{2B} = \frac{2\sqrt{2K}}{B} \)

Now we can summarize the radii:

- \( r_H = \frac{\sqrt{2K}}{B} \)
- \( r_{He} = \frac{2\sqrt{2K}}{B} \)
- \( r_O = \frac{2\sqrt{2K}}{B} \)

### Comparison of Radii:
- Since \( r_H < r_{He} = r_O \), the hydrogen ion (H\(^+\)) will have the smallest radius and will deflect the most.
- Helium (He\(^+\)) and Oxygen (O\(^{2+}\)) have the same radius and will deflect equally.

### Conclusion:
- The ion that deflects the most is H\(^+\).
- Helium and Oxygen will deflect equally.

### Final Answer:
- H\(^+\) will deflect the most, and He\(^+\) and O\(^{2+}\) will deflect equally.

---