A proton and a deuteron with the same initial kinetic energy enter a magnetic field in a a direction perpendicular to the direction of the field . The ratio of the radii of the circular trajectories described by them is

Options

A. `1:4`
B. `1:sqrt(2)`
C. `1:1`
D. `1:2`

Answer

To solve the problem of finding the ratio of the radii of the circular trajectories described by a proton and a deuteron entering a magnetic field with the same initial kinetic energy, we can follow these steps:

### Step 1: Understand the motion of charged particles in a magnetic field
When a charged particle moves in a magnetic field perpendicular to its velocity, it experiences a magnetic force that causes it to move in a circular path. The magnetic force \( F \) acting on the particle is given by:
\[ F = qvB \]
where:
- \( q \) is the charge of the particle,
- \( v \) is the velocity of the particle,
- \( B \) is the magnetic field strength.

### Step 2: Relate the magnetic force to centripetal force
The magnetic force also acts as the centripetal force required to keep the particle in circular motion. Thus, we can equate the magnetic force to the centripetal force:
\[ qvB = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle and \( r \) is the radius of the circular path.

### Step 3: Solve for the radius \( r \)
Rearranging the equation gives us the expression for the radius \( r \):
\[ r = \frac{mv}{qB} \]

### Step 4: Express velocity in terms of kinetic energy
The kinetic energy \( K \) of the particle is given by:
\[ K = \frac{1}{2} mv^2 \]
From this, we can express the velocity \( v \) in terms of kinetic energy:
\[ v = \sqrt{\frac{2K}{m}} \]

### Step 5: Substitute velocity into the radius equation
Substituting \( v \) into the radius equation:
\[ r = \frac{m \sqrt{\frac{2K}{m}}}{qB} = \frac{\sqrt{2Km}}{qB} \]

### Step 6: Find the ratio of the radii for proton and deuteron
Let \( r_1 \) be the radius for the proton and \( r_2 \) be the radius for the deuteron. We have:
\[ r_1 = \frac{\sqrt{2K m_1}}{qB} \quad \text{and} \quad r_2 = \frac{\sqrt{2K m_2}}{qB} \]
The ratio of the radii is:
\[ \frac{r_1}{r_2} = \frac{\sqrt{m_1}}{\sqrt{m_2}} \]

### Step 7: Substitute the masses of the proton and deuteron
The mass of the proton \( m_1 \) is approximately \( 1 \, \text{amu} \) and the mass of the deuteron \( m_2 \) is approximately \( 2 \, \text{amu} \):
\[ \frac{r_1}{r_2} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Conclusion
Thus, the ratio of the radii of the circular trajectories described by the proton and the deuteron is:
\[ \frac{r_1}{r_2} = \frac{1}{\sqrt{2}} \]