A beam of electrons moving with a momentum p enters a uniform magnetic field of flux density B perpendicular to its motion. Which of the following statement(s) is (are) true?

Options

A. Energy gained is `p^(2)/(2m)`
B. Centripetal force on the electron is `Bem/p`
C. Radius of the electron's path is `p/(Be)`
D. Work done on the electrons by the magnetic field is zero

Answer

To solve the problem, we need to analyze the motion of a beam of electrons entering a uniform magnetic field. Here are the steps to determine which statements are true:

### Step 1: Understand the Motion of Electrons in a Magnetic Field
When charged particles, such as electrons, enter a magnetic field, they experience a magnetic force given by the Lorentz force law:
\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \]
where \( q \) is the charge of the particle, \( \mathbf{v} \) is its velocity, and \( \mathbf{B} \) is the magnetic field.

### Step 2: Determine the Direction of the Force
Since the electrons are negatively charged, the direction of the force will be opposite to that predicted by the right-hand rule. If the magnetic field \( \mathbf{B} \) is perpendicular to the velocity \( \mathbf{v} \), the force will cause the electrons to move in a circular path.

### Step 3: Calculate the Radius of the Circular Path
The magnetic force acts as a centripetal force, which can be expressed as:
\[ F = \frac{mv^2}{r} \]
Equating the magnetic force to the centripetal force gives:
\[ qvB = \frac{mv^2}{r} \]
From this, we can derive the radius \( r \) of the circular path:
\[ r = \frac{mv}{qB} \]
Since momentum \( p \) is defined as \( p = mv \), we can rewrite the radius as:
\[ r = \frac{p}{qB} \]

### Step 4: Work Done by the Magnetic Field
The work done by a magnetic field on a charged particle is given by:
\[ W = \mathbf{F} \cdot \mathbf{s} \]
Since the magnetic force is always perpendicular to the displacement of the charged particle, the work done is:
\[ W = 0 \]
This implies that the kinetic energy of the electrons does not change as they move through the magnetic field.

### Step 5: Analyze the Given Statements
1. **Centripetal force on the electron is \( \frac{BEM}{P} \)**: This is incorrect. The centripetal force is given by \( qvB \) or \( EVB \), not in the form presented.
2. **Work done by the magnetic field is zero**: This is correct, as established.
3. **Energy gained is \( \frac{P^2}{2M} \)**: This is incorrect because the work done is zero, indicating no change in kinetic energy.
4. **Radius of the circular path is \( \frac{P}{qB} \)**: This is correct, as derived.

### Conclusion
The true statements are:
- The work done by the magnetic field is zero.
- The radius of the circular path is \( \frac{P}{qB} \).

### Final Answer
The correct statements are:
- Work done by the magnetic field is zero.
- The radius of the circular path is \( \frac{P}{qB} \).