An electron moving through a magnetic does not experience any force. Under what conditions is this possible ?

Options

A. `theta=0^@` or `theta=90^@`
B. `theta=90^@` or `theta=180^@`
C. `theta=0^@` or `theta=180^@`
D. `theta=180^@`

Answer

To determine the conditions under which an electron moving through a magnetic field does not experience any force, we can analyze the situation using the formula for the magnetic force acting on a charged particle.

### Step-by-Step Solution:

1. **Understand the Magnetic Force Formula**:
The magnetic force \( F \) on a charged particle is given by the equation:
\[ F = q \cdot (\mathbf{v} \times \mathbf{B}) \]
where:
- \( F \) is the magnetic force,
- \( q \) is the charge of the particle,
- \( \mathbf{v} \) is the velocity vector of the particle,
- \( \mathbf{B} \) is the magnetic field vector.

2. **Identify the Cross Product**:
The term \( \mathbf{v} \times \mathbf{B} \) represents the cross product of the velocity and magnetic field vectors. The magnitude of this cross product can be expressed as:
\[ |\mathbf{v} \times \mathbf{B}| = vB \sin(\theta) \]
where \( \theta \) is the angle between the velocity vector \( \mathbf{v} \) and the magnetic field vector \( \mathbf{B} \).

3. **Set the Force to Zero**:
For the electron to experience no force, we need:
\[ F = q \cdot |\mathbf{v} \times \mathbf{B}| = 0 \]
Since the charge \( q \) (for an electron, \( q = -e \), where \( e = 1.6 \times 10^{-19} \) C) is not zero, we need the term \( |\mathbf{v} \times \mathbf{B}| \) to be zero.

4. **Analyze the Conditions for Zero Force**:
The expression \( |\mathbf{v} \times \mathbf{B}| = vB \sin(\theta) \) will be zero if:
- \( \sin(\theta) = 0 \)

This occurs when:
- \( \theta = 0^\circ \) (the velocity is parallel to the magnetic field)
- \( \theta = 180^\circ \) (the velocity is antiparallel to the magnetic field)

5. **Conclusion**:
Therefore, the electron will not experience any magnetic force when it is moving either parallel or antiparallel to the magnetic field. In summary, the conditions under which an electron does not experience any force while moving through a magnetic field are:
- The angle \( \theta \) between the velocity of the electron and the magnetic field is either \( 0^\circ \) or \( 180^\circ \).