An electron is projected with uniform velocity along the axis of a current carrying long solenoid. Which of the following is true?

Options

A. The electron will be accelerated along the axis.
B. The electron path will be circular about the axis.
C. The electron will experience a force at `45@` to the axis and hence execute a helical path.
D. The electron will contnue to move with uniform velocity along the axis of the solenoid.

Explanation

To solve the problem, we need to analyze the situation of an electron moving through a current-carrying solenoid. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the setup
We have a long solenoid carrying a current, which creates a magnetic field inside it. The electron is projected with a uniform velocity along the axis of the solenoid.

**Hint:** Remember that the magnetic field inside a long solenoid is uniform and directed along the axis of the solenoid.

### Step 2: Determine the direction of the magnetic field
The magnetic field (B) inside a long solenoid is directed along the axis of the solenoid. For a solenoid with current flowing in a certain direction, the magnetic field lines run parallel to the axis of the solenoid.

**Hint:** Use the right-hand rule to determine the direction of the magnetic field based on the direction of the current in the solenoid.

### Step 3: Apply the Lorentz force law
The force (F) experienced by a charged particle moving in a magnetic field is given by the Lorentz force law:
\[ F = Q(\mathbf{V} \times \mathbf{B}) \]
where:
- \( Q \) is the charge of the particle,
- \( \mathbf{V} \) is the velocity vector of the particle,
- \( \mathbf{B} \) is the magnetic field vector.

**Hint:** Remember that the charge of the electron is negative, and that the cross product will depend on the angle between the velocity and the magnetic field.

### Step 4: Analyze the angle between velocity and magnetic field
In this case, the electron is moving along the axis of the solenoid, which means the angle (θ) between the velocity vector (V) and the magnetic field vector (B) is 0 degrees.

**Hint:** Recall that the sine of 0 degrees is 0, which will affect the calculation of the force.

### Step 5: Calculate the force
Using the formula:
\[ F = QVB \sin(\theta) \]
Since \( \theta = 0 \):
\[ F = QVB \sin(0) = 0 \]
Thus, the force acting on the electron is zero.

**Hint:** Consider what happens to an object when no net force acts upon it.

### Step 6: Conclusion about the motion of the electron
Since the force acting on the electron is zero, according to Newton's first law of motion, the electron will continue to move with uniform velocity along the axis of the solenoid.

**Hint:** Think about the implications of Newton's first law in the context of motion with no net external force.

### Final Answer
The correct option is: **The electron will continue to move with uniform velocity along the axis of the solenoid.**