Magnetic field inside a long solenoid is uniform and directed along its axis.
Magnetic force on a charged particle is given by Lorentz force:
\[ \vec{F} = q(\vec{v} \times \vec{B}) \]
Since the particle moves along the axis of the solenoid and magnetic field is also along the axis, the angle between velocity and magnetic field is 0°.
\[ F = qvB\sin 0^\circ = 0 \]
Therefore magnetic field exerts no force on the particle.
The only force acting is gravitational force \( mg \).
Hence acceleration of the particle is:
\[ a = g \]
Therefore correct answer is (D).
The concept involved in this problem is one of the most fundamental and conceptually powerful ideas in electromagnetism: the magnetic force on a moving charged particle. Many students initially assume that the presence of a magnetic field automatically implies a force on a charged particle. However, this is not true. The magnetic force depends critically on the direction of motion of the particle relative to the magnetic field.
The complete expression for the force on a charged particle in electromagnetic fields is given by the Lorentz force law:
\[ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \]
In this problem, there is no electric field mentioned. Therefore, the only electromagnetic force that can act is magnetic force:
\[ \vec{F}_B = q(\vec{v} \times \vec{B}) \]
The magnitude of this magnetic force is:
\[ F = qvB\sin\theta \]
where θ is the angle between velocity vector and magnetic field vector.
This formula reveals three extremely important conceptual conclusions:
1. Magnetic force depends on velocity. If velocity is zero, magnetic force is zero.
2. Magnetic force depends on sine of angle between v and B.
3. Magnetic force is maximum when motion is perpendicular to magnetic field.
In the present case, the solenoid is placed vertically. The magnetic field inside a long solenoid is uniform and parallel to its axis. If the particle moves along the axis, then velocity and magnetic field are parallel.
Therefore:
\[ \theta = 0^\circ \]
\[ \sin 0^\circ = 0 \]
Hence magnetic force becomes zero at all times.
This leads to one of the most important theoretical conclusions in magnetism:
Magnetic field does not exert force on a charged particle moving parallel to the field lines.
This is not just a numerical result but a deep vector property of cross product. The cross product of two parallel vectors is always zero.
Another extremely important concept related to this is that magnetic force is always perpendicular to velocity. Because of this, magnetic force never does work on a charged particle.
Work done by magnetic force:
\[ W = \vec{F} \cdot \vec{v} \]
Since \( \vec{F}_B \perp \vec{v} \),
\[ W = 0 \]
This implies magnetic field cannot change the speed (magnitude of velocity) of a charged particle. It can only change its direction.
This is a very high-yield concept for JEE Advanced as well.
Now let us analyze magnetic field of a solenoid. Using Ampere’s Law:
\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \]
For a long solenoid, the magnetic field inside is:
\[ B = \mu_0 n I \]
where n is number of turns per unit length.
The field is uniform and directed along axis. Outside the solenoid, field is nearly zero.
If a charged particle had initial velocity perpendicular to B, it would undergo circular motion with radius:
\[ r = \frac{mv}{qB} \]
Time period of circular motion:
\[ T = \frac{2\pi m}{qB} \]
If velocity had both parallel and perpendicular components, the motion would be helical.
But in this problem, velocity remains along axis because initial velocity is zero and acceleration due to gravity acts along axis.
Therefore magnetic field has no influence on motion.
This question tests conceptual clarity rather than computation.
Common mistakes students make:
• Assuming magnetic field reduces acceleration.
• Thinking magnetic force opposes gravity.
• Forgetting sinθ term.
• Not applying cross product reasoning.
Advanced conceptual extension:
If an electric field were also present inside solenoid, acceleration could change.
If solenoid current were time varying, induced electric field could appear.
But in steady current case, only static magnetic field exists.
In JEE Main, such conceptual magnetism questions are common because they differentiate between memorization and understanding.
Students must always visualize vector directions before concluding force behavior.
Summary of key principles:
• Magnetic force depends on cross product.
• Parallel motion → zero magnetic force.
• Magnetic field does no work.
• Solenoid field is axial and uniform.
• Only gravity determines acceleration here.
This concept forms foundation for advanced topics like cyclotron, mass spectrometer, velocity selector, charged particle beam motion and plasma physics.
Mastering this vector understanding ensures strong conceptual base for electromagnetic theory in competitive exams.
This educational content is prepared for practice and learning purposes for competitive exams such as JEE Main, JEE Advanced and NEET. Students are advised to verify calculations independently during preparation.