When an electronl beam passes through and electric field they gain kinetic energy. If the same electron beam passes through a magnetic field then their

Options

A. energy and momentum both ramain unchanged
B. potential energy increases
C. momentum increases
D. kinetic energy increases

Answer

To solve the question, we need to analyze the behavior of an electron beam when it passes through a magnetic field. Here's a step-by-step breakdown of the solution:

### Step 1: Understanding the Electron Beam in a Magnetic Field
When an electron beam (which consists of charged particles) passes through a magnetic field, the magnetic field exerts a force on the moving electrons. This force is given by the Lorentz force law, which states that the force \( \mathbf{F} \) acting on a charged particle is given by:

\[ \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) \]

where:
- \( q \) is the charge of the electron,
- \( \mathbf{v} \) is the velocity of the electron,
- \( \mathbf{B} \) is the magnetic field.

### Step 2: Analyzing the Force Direction
The direction of the force is perpendicular to both the velocity of the electron and the magnetic field. This means that the force will not do work on the electrons because the work done is given by:

\[ W = \mathbf{F} \cdot \mathbf{d} \]

where \( \mathbf{d} \) is the displacement. Since the force is perpendicular to the displacement, the work done is zero.

### Step 3: Implications on Kinetic Energy
Since the work done on the electrons is zero, there is no change in the kinetic energy of the electrons as they pass through the magnetic field. The kinetic energy remains constant.

### Step 4: Implications on Momentum
While the kinetic energy remains unchanged, the direction of the velocity of the electrons changes due to the magnetic force acting on them. This change in direction implies that the momentum of the electrons (which depends on both mass and velocity) will also change, even though the speed (magnitude of velocity) remains constant.

### Conclusion
Based on the analysis:
- The kinetic energy of the electrons remains unchanged.
- The momentum of the electrons changes due to the change in direction of their velocity.

### Final Answer
The correct option is: **Energy and momentum both remain unchanged** (for energy, it is unchanged; for momentum, it changes in direction).

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