An electron having a charged e moves with a velcoity v in postive x direction. A magnetic field acts on it in positive y direction. The force on the electron acts in (where outward direction is taken as positive z-axis)

Options

A. Negative direction of y-axis
B. Positive direction on y-axis
C. Positive direftion of Z-axis
D. Negative direction of z-axis

Answer

To determine the direction of the force acting on an electron moving in a magnetic field, we can use the Lorentz force law, which states that the force \( \mathbf{F} \) on a charged particle is given by:

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

where:
- \( q \) is the charge of the particle,
- \( \mathbf{v} \) is the velocity vector,
- \( \mathbf{B} \) is the magnetic field vector.

### Step 1: Identify the charge, velocity, and magnetic field vectors
- The charge of the electron is \( q = -e \) (negative charge).
- The velocity \( \mathbf{v} \) of the electron is in the positive x-direction, which can be represented as:
\[ \mathbf{v} = v \hat{i} \]
- The magnetic field \( \mathbf{B} \) is in the positive y-direction, represented as:
\[ \mathbf{B} = B \hat{j} \]

### Step 2: Calculate the cross product \( \mathbf{v} \times \mathbf{B} \)
To find the direction of the force, we need to compute the cross product \( \mathbf{v} \times \mathbf{B} \):
\[ \mathbf{F} = -e (\mathbf{v} \times \mathbf{B}) = -e (v \hat{i} \times B \hat{j}) \]

Using the right-hand rule for the cross product:
- Point your fingers in the direction of \( \hat{i} \) (positive x-direction),
- Curl them towards \( \hat{j} \) (positive y-direction),
- Your thumb will point in the direction of \( \hat{k} \) (positive z-direction).

Thus,
\[ \hat{i} \times \hat{j} = \hat{k} \]

### Step 3: Substitute the cross product into the force equation
Now substituting back into the force equation:
\[ \mathbf{F} = -e (vB \hat{k}) \]
Since the charge of the electron is negative, the force becomes:
\[ \mathbf{F} = -e v B \hat{k} \]

### Step 4: Determine the direction of the force
The negative sign indicates that the force is in the opposite direction of \( \hat{k} \). Therefore, the force acts in the negative z-direction (which is the negative z-axis).

### Conclusion
The direction of the force on the electron is in the negative z-direction.