Question

Figure 28-31 gives snapshots for three situations in which a positively charged particle passes through a uniform magnetic field $\overrightarrow{B}$. The velocities$\overrightarrow{V}$of the particle differ in orientation in the three snapshots but not in magnitude. Rank the situations according to (a) the period, (b) the frequency, and (c) the pitch of the particle’s motion, greatest first.

Short answer

1. Ranking of situations according to the period is $3>2>1$
2. Ranking of situations according to frequency is$1>2>3$
3. Ranking of situations according to the pitch of the particle’s motion is$3>2>1$

Step-by-step solution

Step 1: Given

Mass, velocity, and charge of different particles are given in table.

Determining the concept

Equate centripetal force to magnetic force and find the relation for radius. Using radius, find the period, which depends upon the sine of the angle between magnetic field and velocity. From this, rank the situations according to the period. Then using the relation between period and frequency, rank the situations according to frequency. Substituting the period value in the formula for pitch, get the pitch value depending upon the cotangent of the angle between magnetic field and velocity. Using this, rank the situations according to the pitch of the particle’s motion.

Formulae are as follow:

$$\begin{gathered} r=\frac{mv}{\left|q\right|B} \\ T=\frac{2\mathrm{\pi r}}{V} \\ f=\frac{1}{T} \\ P=V_{parallel}T \end{gathered}$$

Where, r is radius, B is magnetic field, v is velocity, m is mass,q is charge on particle, T is time period, f is frequency, P is pitch.

(a) Determining the rank of the situations according to period

Rank the situations according to period:

In order to get circular motion, the centripetal force must be balanced by magnetic force.

$$\begin{gathered} \frac{m{v}^2}{r}=qVB\sin \theta \\ r=\frac{mv}{qBV\sin \theta } \\ r=\frac{m}{qB\sin \theta } \end{gathered}$$

But,

$T=\frac{2\mathrm{\pi r}}{V}$

Substituting $r$in period,

$T=\frac{2\mathrm{\pi }}{v}\frac{m}{qB\sin \theta }$

Velocity, magnetic field, and charge is constant. Hence, the period is inversely proportional to angle$\theta$between magnetic field and velocity vector. As angle decreases, period increases, and vice versa.

Hence, ranking of situations according to the period is $3>2>1$.

 (b) Determining the rank of the situations according to frequency

Rank situations according to frequency:

$f=\frac{1}{T}$

Hence, ranking of situations according to frequency is $1>2>3$.

(c) Determining the rank of the situations according to the pitch of the particle’s motion

Ranking according to pitch of particle’s motion:

The parallel component of velocity to magnetic field is given by,

$V_{parallel}=v\cos \theta$

Where,$\theta$is the angle between $\overrightarrow{V}$and $\overrightarrow{B}$.

Pitch $P=V_{parallel}T$

Hence,

$$\begin{gathered} P=v\cos \theta \frac{2\mathrm{\pi m}}{qB\sin \theta } \\ P=\frac{2\mathrm{\pi mv}}{qB}cot\theta \end{gathered}$$

Hence, as Ɵ decreases, the pitch value increases. Hence, situation 3 will has maximum value of pitch and situation 2 will have the smaller value of pitch. Situation 1 will have no pitch as it makes ${90}^o$angle.

Therefore, ranking of situations according to the pitch of the particle’s motion is

3>2>1 .

Therefore, equating centripetal force and magnetic force, rank the situation of particle’s motion in the magnetic field according to time period, frequency, and pitch of its motion.