An electron enters a magnetic field along perpendicular direction. Following quantity will remain constant -

Options

A. momentum
B. kinetic energy
C. velocity
D. all of the above

Answer

To solve the problem of determining which quantity remains constant when an electron enters a magnetic field perpendicularly, we can follow these steps:

### Step 1: Understand the Scenario
An electron enters a magnetic field at a perpendicular angle. This means that the velocity vector of the electron is perpendicular to the magnetic field vector.

**Hint:** Remember that the direction of the magnetic field and the velocity of the charged particle are crucial in determining the forces acting on it.

### Step 2: Apply the Lorentz Force Law
The force acting on a charged particle moving in a magnetic field is given by the Lorentz force equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
where \( q \) is the charge of the particle, \( \vec{v} \) is the velocity vector, and \( \vec{B} \) is the magnetic field vector.

**Hint:** The cross product indicates that the force is perpendicular to both the velocity and the magnetic field.

### Step 3: Determine the Work Done
Since the magnetic force is always perpendicular to the velocity of the electron, the work done by the magnetic force on the electron is zero. This can be expressed as:
\[ W = \vec{F} \cdot \vec{d} = F d \cos(90^\circ) = 0 \]

**Hint:** Recall that work done is related to the angle between the force and displacement vectors.

### Step 4: Relate Work Done to Kinetic Energy
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy:
\[ W = \Delta KE \]
Since the work done is zero, we have:
\[ \Delta KE = 0 \]
This implies that the kinetic energy of the electron remains constant.

**Hint:** Kinetic energy is given by the formula \( KE = \frac{1}{2} mv^2 \). If it remains constant, then \( v \) must also remain constant in magnitude.

### Step 5: Analyze Velocity and Momentum
While the kinetic energy remains constant, the velocity of the electron is changing in direction as it moves in a circular path due to the magnetic force acting as a centripetal force. Therefore, the velocity vector itself is not constant.

Additionally, since momentum is the product of mass and velocity, and the direction of the velocity is changing, the momentum of the electron will also not remain constant.

**Hint:** Consider the definitions of velocity and momentum and how they relate to direction.

### Conclusion
From the analysis, we conclude that the only quantity that remains constant when an electron enters a magnetic field perpendicularly is the kinetic energy.

**Final Answer:** Kinetic Energy remains constant.