An electron is moving with an initial velocity `vecv=v_(0)hati` and is in a magnetic field `vecB=B_(0)hatj`. Then it's de-Broglie wavelength

Options

A. remains constant
B. increases with time
C. decreases with time
D. increases and decreases periodically

Explanation

To solve the problem of finding the de Broglie wavelength of an electron moving in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Quantities**:
- The initial velocity of the electron is given as \( \vec{v} = v_0 \hat{i} \).
- The magnetic field is given as \( \vec{B} = B_0 \hat{j} \).
- The charge of the electron is \( q = -e \) (where \( e \) is the elementary charge).

2. **Calculate the Force on the Electron**:
- The force acting on a charged particle in a magnetic field is given by the Lorentz force equation:
\[ \vec{F} = q \vec{v} \times \vec{B} \]
- Substituting the values:
\[ \vec{F} = -e (v_0 \hat{i}) \times (B_0 \hat{j}) \]
- Using the right-hand rule for the cross product:
\[ \hat{i} \times \hat{j} = \hat{k} \]
- Therefore, we have:
\[ \vec{F} = -e v_0 B_0 \hat{k} \]

3. **Determine the Motion of the Electron**:
- The force \( \vec{F} \) acts perpendicular to the velocity \( \vec{v} \), causing the electron to move in a circular path. The magnitude of the velocity remains constant, but the direction changes.

4. **Calculate the Momentum of the Electron**:
- The momentum \( p \) of the electron is given by:
\[ p = mv \]
- Since the magnitude of the velocity does not change, we can write:
\[ p = m v_0 \]

5. **Apply de Broglie's Wavelength Formula**:
- The de Broglie wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{p} \]
- Substituting the expression for momentum:
\[ \lambda = \frac{h}{mv_0} \]

6. **Conclusion**:
- The de Broglie wavelength of the electron moving in the magnetic field is:
\[ \lambda = \frac{h}{mv_0} \]
- This wavelength remains constant as long as the speed \( v_0 \) does not change.