Five equal-mass particles (A–E) enter a region of uniform magnetic field directed into the page. They follow the trajectories illustrated in the figure.
Which particle (if any) is neutral?
Which particle (if any) is negatively charged?
Rank the particles on the basis of their speed.
The concepts used in this problem are magnetic field, magnetic force and centripetal force. The ranking of the particles can be determined using the concept of magnetic field and magnetic force.
Magnetic field
The region around a magnetic material or moving charge where the magnetic force acts is called the magnetic field. An electric charge is affected under the influence of magnetic field.
Centripetal Force:
The linear acceleration is the rate of change of linear velocity but the rate of change of tangential velocity is the centripetal acceleration. The centripetal acceleration makes an object to move in a curved path. The force acts in this motion is centripetal force
The expression for centripetal force is:
${F_{\rm{c}}} = \frac{{m{v^2}}}{r}$
Here, $m$ is the mass, ${a_{\rm{c}}}$ is the centripetal acceleration, $v$ is the velocity and $r$ is the radial distance.
Magnetic force:
The magnetic force is perpendicular to the magnetic field and the velocity of the charge. The magnitude of magnetic force is given by:
$\begin{array}{c}\\{{\vec F}_{{\rm{mag}}}} = q\left( {\vec v \times \vec B} \right)\\\\{F_{{\rm{mag}}}} = qvB\sin \theta \\\end{array}$
Here, $q$ is the charge, $v$ is the velocity, $B$ is the magnetic field and $\theta$ is the angle between velocity and the magnetic field.
(Part A)
All the particles are deflecting except particle D, it means that except D all other particles are charged particles. So, particle D is neutral.
(Part B)
The direction of magnetic field is into the plane. The particles B, C and E are moving in counterclockwise direction which is the direction according to Right Hand Thumb Rule for positive charge.
So, for negative charge, the direction should be clockwise. Particle A moves in clockwise direction.
(Part D)
For a single particle, the magnetic force is:
${F_{{\rm{mag}}}} = qvB\sin \theta$
Substitute ${90^{\rm{o}}}$ for $\theta$ .
$\begin{array}{c}\\{F_{{\rm{mag}}}} = qvB\sin {90^{\rm{o}}}\\\\ = qvB\\\end{array}$
The magnetic force should be balanced by centripetal force.
${F_{\rm{c}}} = \frac{{m{v^2}}}{r}$
$\begin{array}{c}\\{F_{{\rm{mag}}}} = {F_{\rm{c}}}\\\\qvB = \frac{{m{v^2}}}{r}\\\\v = \frac{{qBr}}{m}\\\end{array}$
The mass of all the particle are equal, but no information is given about the magnitude of charge, so the ranking cannot be determined.
Ans: Part AThe neutral particle is particle D.
Part BThe negatively charge particle is particle A.
Part DThe correct ranking of the speed of the particles cannot be determined.
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