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

Answer

To solve the problem of determining the direction of the magnetic field required to make an electron 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 \( \vec{v} \) is directed along the x-axis, represented as \( \vec{v} = v \hat{i} \).

2. **Applying the Lorentz Force**:
- The force acting on a charged particle in a magnetic field is given by the Lorentz force equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
- For an electron, the charge \( q \) is negative, so we have:
\[ \vec{F} = -e (\vec{v} \times \vec{B}) \]
- Here, \( e \) is the magnitude of the charge of the electron.

3. **Direction of the Force**:
- The force must be directed towards the center of the circular path to maintain circular motion. Since the electron is moving anticlockwise in the x-y plane, at any point on the path, the force must be directed towards the center of the circle, which lies along the negative y-axis when the electron is moving along the positive x-axis.

4. **Using the Right-Hand Rule**:
- To find the direction of the magnetic field \( \vec{B} \), we can use the right-hand rule. For a positive charge, the thumb points in the direction of the velocity \( \vec{v} \) (positive x-direction), and the fingers point in the direction of the magnetic field \( \vec{B} \). The palm then points in the direction of the force \( \vec{F} \).
- However, since we are dealing with an electron (negative charge), the force will be in the opposite direction of what the right-hand rule indicates.

5. **Determining the Direction of the Magnetic Field**:
- Since the force must point in the negative y-direction (towards the center of the circular path), we can set up the equation:
\[ \vec{F} = -e (\vec{v} \times \vec{B}) \text{ must point in the negative } \hat{j} \text{ direction.} \]
- Given \( \vec{v} = v \hat{i} \), we need to find \( \vec{B} \) such that \( \hat{i} \times \vec{B} \) results in a vector in the negative y-direction.

6. **Finding the Magnetic Field Direction**:
- Let \( \vec{B} = B \hat{k} \) (where \( \hat{k} \) is the positive z-direction).
- Then, using the cross product:
\[ \hat{i} \times \hat{k} = \hat{j} \]
- Therefore, \( \vec{F} = -e (v \hat{i} \times B \hat{k}) = -e (vB \hat{j}) \).
- Since the force is in the negative y-direction, we conclude that the magnetic field \( \vec{B} \) must be in the positive z-direction.

### Final Answer:
The direction of the magnetic field required to make the electron move in an anticlockwise circular path in the x-y plane is along the positive z-axis.