If electron velocity is `2i+4j` and it is subjected to magnetic field of `4k`, then its

Options

A. speed will change
B. path will change
C. velocity is Constant
D. momentum is Constant

Answer

To solve the problem, we need to analyze the motion of an electron with a given velocity in a magnetic field. The velocity of the electron is given as \( \vec{v} = 2\hat{i} + 4\hat{j} \) and the magnetic field is given as \( \vec{B} = 4\hat{k} \).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Velocity of the electron: \( \vec{v} = 2\hat{i} + 4\hat{j} \)
- Magnetic field: \( \vec{B} = 4\hat{k} \)

2. **Use the Lorentz Force Equation:**
The force \( \vec{F} \) 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 electron.

3. **Calculate the Cross Product \( \vec{v} \times \vec{B} \):**
We need to compute \( \vec{v} \times \vec{B} \):
\[ \vec{v} \times \vec{B} = (2\hat{i} + 4\hat{j}) \times (4\hat{k}) \]

Using the properties of the cross product:
- \( \hat{i} \times \hat{k} = -\hat{j} \)
- \( \hat{j} \times \hat{k} = \hat{i} \)

We can compute:
\[ \vec{v} \times \vec{B} = 2\hat{i} \times 4\hat{k} + 4\hat{j} \times 4\hat{k} \]
\[ = 8(-\hat{j}) + 16\hat{i} \]
\[ = -8\hat{j} + 16\hat{i} \]

4. **Substituting into the Force Equation:**
The force on the electron is then:
\[ \vec{F} = q(-8\hat{j} + 16\hat{i}) \]

5. **Direction of the Force:**
The force has components in both the \( \hat{i} \) (x-direction) and \( \hat{j} \) (y-direction). The negative \( \hat{j} \) component indicates that the force is directed in the negative y-direction, while the positive \( \hat{i} \) component indicates that the force is directed in the positive x-direction.

6. **Effect on the Electron's Motion:**
Since the electron is subjected to a magnetic force, its velocity will change direction due to the force acting on it. However, the magnitude of the velocity (speed) will remain constant because the magnetic force does not do work on the charged particle.

### Conclusion:
- The speed of the electron remains constant.
- The path of the electron will change due to the magnetic force acting on it.
- The velocity of the electron will change direction.

### Final Answer:
The properties that will change are:
- The path of the electron will change.