An electron is travelling along the x-direction. It encounters a magnetic field in the y-direction. Its subsequent motion wil be

Options

A. Straight line along the x-direction
B. A circule in the xz-plane
C. A circule in the yz-plane
D. A circle in the xy-plane

Answer

To solve the problem, we need to analyze the motion of an electron moving in a magnetic field. Here's a step-by-step breakdown of the solution:

### Step 1: Identify the initial conditions
- The electron is moving in the x-direction. We can denote its velocity vector as **v** = v * î, where v is the speed of the electron and î is the unit vector in the x-direction.
- The magnetic field is in the y-direction, which we can denote as **B** = B * ĵ, where B is the strength of the magnetic field and ĵ is the unit vector in the y-direction.

**Hint:** Identify the direction of the velocity and the magnetic field vectors to understand how they interact.

### Step 2: Use the Lorentz force equation
- The force acting on a charged particle moving in a magnetic field is given by the Lorentz force equation: **F** = q(**v** × **B**), where q is the charge of the particle.
- Since the electron has a negative charge (q = -e, where e is the elementary charge), we will consider the direction of the force after calculating the cross product.

**Hint:** Remember that the Lorentz force depends on the cross product of the velocity and magnetic field vectors.

### Step 3: Calculate the cross product **v** × **B**
- Using the right-hand rule, we can determine the direction of the cross product:
- **v** = v * î (in the x-direction)
- **B** = B * ĵ (in the y-direction)
- The cross product **v** × **B** can be calculated as follows:
- **v** × **B** = (v * î) × (B * ĵ) = vB (î × ĵ) = vB * k̂
- The direction of **v** × **B** is in the positive z-direction (k̂).

**Hint:** Use the right-hand rule to visualize the direction of the cross product.

### Step 4: Determine the direction of the force
- Since the electron is negatively charged, the force will be in the opposite direction of **v** × **B**:
- Therefore, **F** = -e(vB * k̂) = -evB * k̂, which means the force is in the negative z-direction (-k̂).

**Hint:** Remember that the charge's sign affects the direction of the force.

### Step 5: Analyze the motion of the electron
- The force acting on the electron is perpendicular to its velocity, which means the electron will undergo circular motion.
- The radius of the circular path can be determined by the centripetal force provided by the magnetic force, but for this question, we only need to describe the motion.
- The electron will move in a circular path in the xz-plane due to the constant magnetic force acting in the negative z-direction.

**Hint:** Circular motion occurs when a force is perpendicular to the velocity of the particle.

### Conclusion
The subsequent motion of the electron will be a circular path in the xz-plane.

**Final Answer:** The electron will move in a circular path in the xz-plane.

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