A narrow beam of protons and deutrons, each having the same momentum, enters a region of uniform magnetic field directed perpendicular to their direction of momentum. The ratio of the radii of the circular paths described by them is

Options

A. `1:2`
B. `1:1`
C. `2:1`
D. `1:3`

Explanation

To solve the problem of finding the ratio of the radii of the circular paths described by protons and deuterons in a uniform magnetic field, we can follow these steps:

### Step 1: Understand the motion of charged particles in a magnetic field
When charged particles like protons and deuterons move in a magnetic field that is perpendicular to their velocity, they will follow a circular path due to the Lorentz force acting on them.

### Step 2: Write the formula for the radius of the circular path
The radius \( R \) of the circular path of a charged particle 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.

### Step 3: Relate momentum to mass and velocity
The momentum \( p \) of a particle is given by:
\[ p = mv \]
Since both protons and deuterons have the same momentum, we can express the radius in terms of momentum:
\[ R = \frac{p}{qB} \]

### Step 4: Set up the ratio of the radii for protons and deuterons
Let \( R_p \) be the radius for protons and \( R_d \) be the radius for deuterons. Then:
\[ \frac{R_p}{R_d} = \frac{p}{q_p B} \div \frac{p}{q_d B} = \frac{q_d}{q_p} \]
Since \( p \) and \( B \) are the same for both particles, they cancel out.

### Step 5: Determine the charges of protons and deuterons
The charge of a proton \( q_p \) is \( +e \) (where \( e \) is the elementary charge), and the charge of a deuteron \( q_d \) is also \( +e \) (since a deuteron consists of one proton and one neutron, and it carries the same charge as a proton).

### Step 6: Calculate the ratio
Since both charges are equal:
\[ \frac{q_d}{q_p} = \frac{e}{e} = 1 \]
Thus, the ratio of the radii is:
\[ \frac{R_p}{R_d} = 1 \]

### Conclusion
The ratio of the radii of the circular paths described by protons and deuterons is:
\[ \frac{R_p}{R_d} = 1 \]