An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then :

The correct answer is: the electron will continue to move with uniform velocity along the axis of the solenoid.

To understand why, we need to recall the fundamental law governing the force on a moving charge in a magnetic field — the Lorentz Magnetic Force.

The magnetic force on a charged particle is given by:

\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B})

where $q$ is the charge of the particle, $\overrightarrow{v}$ is its velocity, and $\overrightarrow{B}$ is the magnetic field.

Step 1: Understand the magnetic field inside a solenoid.

A current-carrying solenoid produces a uniform magnetic field along its axis. If the solenoid's axis lies along the X-direction, then the magnetic field inside is:

\overrightarrow{B}=B\widehat{x}

This means the field lines run parallel to the axis of the solenoid.

Step 2: Understand the direction of the electron's velocity.

The electron is projected along the axis of the solenoid. So the velocity of the electron is also directed along the same axis (X-direction):

\overrightarrow{v}=v\widehat{x}

Step 3: Calculate the cross product $\overrightarrow{v}\times \overrightarrow{B}$.

Since both $\overrightarrow{v}$ and $\overrightarrow{B}$ are directed along the same axis (X-axis), the angle $\theta$ between them is 0°.

\overrightarrow{v}\times \overrightarrow{B}=\left|v\right|\left|B\right|sin\left(\mathrm{0°}\right)=0

The cross product of two parallel (or anti-parallel) vectors is always zero.

Step 4: Calculate the magnetic force on the electron.

Substituting into the Lorentz force formula:

\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B})=q\times 0=0

The net magnetic force on the electron is zero.

Step 5: Apply Newton's First Law and draw the conclusion.

Since there is no magnetic force acting on the electron, and there is no electric force either (a solenoid only produces a magnetic field inside — no electric field along the axis), the electron experiences zero net force.

By Newton's First Law of Motion: an object continues in its state of uniform motion unless acted upon by an external net force.

Therefore, the electron will continue to move with uniform velocity along the axis of the solenoid — its speed and direction remain completely unchanged.

Why are the other options wrong?

- Option 1 (Accelerated along axis): There is no net force along the axis, so no acceleration is possible.
- Option 3 (Circular path about axis): Circular motion requires a centripetal force perpendicular to velocity. Since $\overrightarrow{F}=0$, no such force exists.
- Option 4 (Helical path at 45°): A helical path occurs only when the velocity has a component perpendicular to $\overrightarrow{B}$. Here, the velocity is entirely parallel to $\overrightarrow{B}$, so no helical motion occurs.

Key Takeaway: The magnetic force $\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B})$ is zero whenever the velocity is parallel to the magnetic field. This is the core physical principle behind this question.