An electron is moving along positive x-axis. To get it moving on an anticlockwise circular path in x-y plane, a magnetic field is applied

Options

A. along positive y-axis
B. along positive z-axis
C. along negative y-axis
D. along negative z-axis

Explanation

To solve the problem of determining the direction of the magnetic field required to make an electron moving along the positive x-axis move in an anticlockwise circular path in the x-y plane, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion of the Electron:**
- The electron is initially moving along the positive x-axis. This means its velocity vector \( \mathbf{v} \) can be represented as:
\[ \mathbf{v} = v \hat{i} \]
- Here, \( v \) is the speed of the electron, and \( \hat{i} \) is the unit vector in the x-direction.

2. **Direction of the Required Force:**
- To make the electron move in an anticlockwise circular path in the x-y plane, the magnetic force acting on the electron must be directed towards the center of the circular path.
- For an anticlockwise motion, the force must act in the positive y-direction. Therefore, we can denote the magnetic force \( \mathbf{F} \) as:
\[ \mathbf{F} = F \hat{j} \]
- Here, \( F \) is the magnitude of the force, and \( \hat{j} \) is the unit vector in the y-direction.

3. **Using the Lorentz Force Law:**
- The magnetic force on a charged particle is given by the Lorentz force equation:
\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \]
- For an electron, the charge \( q \) is negative, specifically \( q = -e \), where \( e \) is the elementary charge.

4. **Setting Up the Cross Product:**
- Substituting the values into the Lorentz force equation, we have:
\[ F \hat{j} = -e (\mathbf{v} \times \mathbf{B}) \]
- Since \( \mathbf{v} = v \hat{i} \), we can rewrite the equation as:
\[ F \hat{j} = -e (v \hat{i} \times \mathbf{B}) \]

5. **Determining the Direction of the Magnetic Field:**
- To find the direction of \( \mathbf{B} \), we need to ensure that the cross product \( \hat{i} \times \mathbf{B} \) results in a vector in the positive y-direction \( \hat{j} \).
- The cross product \( \hat{i} \times \hat{k} = \hat{j} \), where \( \hat{k} \) is the unit vector in the z-direction.

6. **Conclusion:**
- Therefore, for the magnetic force to be in the positive y-direction, the magnetic field \( \mathbf{B} \) must be directed along the positive z-axis:
\[ \mathbf{B} = B \hat{k} \]
- Hence, the direction of the magnetic field required to achieve the desired motion is along the positive z-axis.

### Final Answer:
The magnetic field should be applied along the positive z-axis.

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