If a electron velocity is (2hatixx3hatj)` and it subjected to a magnetic field `4hatk` then

Options

A. speed of electon will charge
B. path of electron will charge
C. Both (a) and (b)
D. None of above

Answer

To solve the problem, we need to analyze the motion of an electron moving in a magnetic field. We will use the formula for the magnetic force acting on a charged particle, which is given by:

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

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

### Step 1: Identify the given quantities

The velocity of the electron is given as:

\[ \mathbf{v} = 2\hat{i} + 3\hat{j} \]

The magnetic field is given as:

\[ \mathbf{B} = 4\hat{k} \]

### Step 2: Calculate the cross product \( \mathbf{v} \times \mathbf{B} \)

To find the magnetic force, we need to calculate the cross product \( \mathbf{v} \times \mathbf{B} \).

Using the determinant form for the cross product:

\[ \mathbf{v} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 0 \\ 0 & 0 & 4 \end{vmatrix} \]

Calculating this determinant:

\[ \mathbf{v} \times \mathbf{B} = \hat{i} \begin{vmatrix} 3 & 0 \\ 0 & 4 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 0 \\ 0 & 4 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & 3 \\ 0 & 0 \end{vmatrix} \]

Calculating the 2x2 determinants:

\[ = \hat{i} (3 \cdot 4 - 0 \cdot 0) - \hat{j} (2 \cdot 4 - 0 \cdot 0) + \hat{k} (2 \cdot 0 - 3 \cdot 0) \]
\[ = 12\hat{i} - 8\hat{j} + 0\hat{k} \]

Thus,

\[ \mathbf{v} \times \mathbf{B} = 12\hat{i} - 8\hat{j} \]

### Step 3: Calculate the force \( \mathbf{F} \)

The charge of an electron is \( q = -1.6 \times 10^{-19} \, \text{C} \). Therefore, the magnetic force is:

\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) = -1.6 \times 10^{-19} (12\hat{i} - 8\hat{j}) \]

Calculating this gives:

\[ \mathbf{F} = -1.6 \times 10^{-19} \cdot 12 \hat{i} + 1.6 \times 10^{-19} \cdot 8 \hat{j} \]
\[ = -1.92 \times 10^{-18} \hat{i} + 1.28 \times 10^{-18} \hat{j} \]

### Step 4: Analyze the motion of the electron

Since the force \( \mathbf{F} \) is perpendicular to the velocity \( \mathbf{v} \), the speed of the electron will remain constant, but its direction will change. This means that the electron will move in a circular path due to the magnetic force acting as a centripetal force.

### Conclusion

The electron's speed will remain constant, but its path will change due to the magnetic force acting on it, which is perpendicular to its velocity.