Question 14: Magnetism and Matter

Question

A current carrying solenoid is placed vertically and a particle of mass $m$ with charge $Q$ is released from rest. The particle moves along the axis of solenoid. If $g$ is acceleration due to gravity then the acceleration (a) of the charged particle will satisfy :

Accepted Answer

A current-carrying solenoid produces a magnetic field ( $\overrightarrow{\mathrm{B}}$ ). For an ideal solenoid, the magnetic field inside is uniform and directed parallel to its axis.
The magnetic force $\left(\vec{F}_might)$ on a moving charge $Q$ with velocity $\vec{v}$ is given by the Lorentz force formula :
$$ \overrightarrow{\mathrm{F}}_{\mathrm{m}}=\mathrm{Q}(\overrightarrow{\mathrm{v}} 	imes \overrightarrow{\mathrm{B}}) $$
In this case, the charged particle is released from rest along the axis. The magnetic field $\vec{B}$ is also along the axis. Therefore, the velocity $\vec{v}$ (once the particle starts moving) and the magnetic field $\vec{B}$ are parallel to each other.
The magnitude of the magnetic force is the cross product of $\mathrm{Q}(\overrightarrow{\mathrm{v}} 	imes \overrightarrow{\mathrm{B}})$,
$$ F_m=Q v B \sin 	heta $$
Since the angle $	heta$ between the velocity and the magnetic field is $0^{\circ}$ :
$$ \mathrm{F}_{\mathrm{m}}=\mathrm{Q} v \mathrm{~B} \sin 0^{\circ} $$
The magnetic field exerts no force on the particle moving along the axis.
The solenoid is placed vertically. When the particle of mass $m$ is released, the only non-zero force acting on it is the force of gravity $\left(\overrightarrow{\mathrm{F}}_{\mathrm{g}}ight)$ :
$$ \overrightarrow{\mathrm{F}}_{\mathrm{g}}=\mathrm{mg} $$
This force acts vertically downwards along the axis of the solenoid.
According to Newton's Second Law ( $\mathrm{F}_{	ext {net }}=\mathrm{ma}$ ) :
$$ \mathrm{mg}=\mathrm{ma} $$
$$ \Rightarrow \mathrm{a}=\mathrm{g} $$
Since the magnetic force is zero the particle falls freely under gravity. Thus, its acceleration is exactly equal to g.
Hence, the correct option is (D).

Answer

$a>g$

Answer

$a=0$

Answer

$0 < a < g$

Answer

$a=g$

Magnetism and Matter

A current carrying solenoid is placed vertically and a particle of mass m with charge Q is released from rest. The particle moves along the axis of solenoid. If g is acceleration due to gravity then the acceleration (a) of the charged particle will satisfy :

Choose the correct answer: