An electron moving horizontally from left to right in a region where a magnetic field exists vertically downwards describes: (when seen from above)

Options

A. horizontal clock-wise circular path
B. horizontal anti clock-wise circular path
C. vertical clock-wise circular path
D. vertical anti clock-wise circular path

Answer

To solve the problem of an electron moving horizontally from left to right in a vertically downward magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Directions**:
- The electron is moving from left to right, which we can denote as the positive x-direction (x̂).
- The magnetic field is directed vertically downward, which we can denote as the negative z-direction (-ẑ).

2. **Determine the Velocity Vector**:
- The velocity vector of the electron can be represented as:
\[ \mathbf{v} = v \hat{x} \]
where \( v \) is the speed of the electron.

3. **Determine the Magnetic Field Vector**:
- The magnetic field vector can be represented as:
\[ \mathbf{B} = -B \hat{z} \]
where \( B \) is the magnitude of the magnetic field.

4. **Apply the Lorentz Force Law**:
- The force acting on a charged particle in a magnetic field is given by:
\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \]
where \( q \) is the charge of the electron (which is negative).

5. **Calculate the Cross Product**:
- We need to calculate the cross product \( \mathbf{v} \times \mathbf{B} \):
\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) = q (v \hat{x} \times (-B \hat{z})) \]
- Using the right-hand rule for the cross product:
\[ \hat{x} \times \hat{z} = \hat{y} \]
- Therefore:
\[ \mathbf{F} = -qvB \hat{y} \]

6. **Direction of the Force**:
- Since the charge of the electron is negative, the actual force acting on the electron will be in the opposite direction of the calculated force:
\[ \mathbf{F}_{\text{real}} = -(-qvB \hat{y}) = qvB \hat{y} \]
- This means the force will act in the positive y-direction (upward).

7. **Determine the Motion Path**:
- The upward force will cause the electron to move in a circular path in the x-y plane due to the centripetal nature of the magnetic force.
- Since the force is acting in the positive y-direction, the motion will be clockwise when viewed from above.

8. **Conclusion**:
- The electron will describe a horizontal clockwise circular path.

### Final Answer:
The electron moving horizontally from left to right in a region where a magnetic field exists vertically downwards describes a **horizontal clockwise circular path**.

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