Helical Motion

Q. When a charged particle is projected at some angle in a magnetic field, the path attained will be

1. Parabola
2. Ellipse
3. Helical
4. Circle

Q. The electric field lines for a system of two charges $Q_1$ and $Q_2$ fixed at two different points on the $\text{x-}$axis are as shown in the figure. Then $\frac{Q_1}{Q_2}$ might be

1. $\frac{9}{13}$
2. $\frac{13}{9}$
3. impossible to determine
4. $1$

Q. A charged particle enters in uniform magnetic at an angle $ta{n}^{-1}\left\{2\pi \right\}$. This angle is between velocity vector and magnegtic field vector. Its path becomes helix with radius of path ‘r” and pitch of “p”, then

1. $p=\sqrt{3}r$
2. $p=\frac{r}{\sqrt{3}}$
3. p = r
4. $p=\frac{r}{\sqrt{2}}$

Q. A charged particle enters in uniform magnetic at an angle $ta{n}^{-1}\left\{2\pi \right\}$. This angle is between velocity vector and magnegtic field vector. Its path becomes helix with radius of path ‘r” and pitch of “p”, then

1. $p=\sqrt{3}r$
2. $p=\frac{r}{\sqrt{3}}$
3. p = r
4. $p=\frac{r}{\sqrt{2}}$

Q. A particle of charge $q$ and mass $m$ starts moving from the origin under the action of an electric field $\overrightarrow{E}=E_0\hat{i}$ and $\overrightarrow{B}=B_0\hat{i}$ with velocity $\overrightarrow{v}=v_0\hat{j}$. The speed of the particle will become $2v_0$ after time

1. $t=\frac{2m{v}_0}{qE}$
2. $t=\frac{2Bq}{m{v}_0}$
3. $t=\frac{\sqrt{3}Bq}{m{v}_0}$
4. $t=\frac{\sqrt{3}m{v}_0}{qE}$

Q.

If the direction of the initial velocity of the charged particle is neither along nor perpendicular to that of the magnetic field, then the orbit will be

1. A straight line

2. An ellipse

3. A circle

4. A helix

Q. A particle of charge $q$ and mass $m$ starts moving from the origin under the action of an electric field $\overrightarrow{E}=E_0\hat{i}$ and $\overrightarrow{B}=B_0\hat{i}$ with velocity $\overrightarrow{v}=v_0\hat{j}$. The speed of the particle will become $2v_0$ after time

1. $t=\frac{2m{v}_0}{qE}$
2. $t=\frac{2Bq}{m{v}_0}$
3. $t=\frac{\sqrt{3}Bq}{m{v}_0}$
4. $t=\frac{\sqrt{3}m{v}_0}{qE}$

Q. A charged particle is fired at an angle $\theta$ to a uniform magnetic field directed along the $x–$axis. During its motion along a helical path, if the pitch of the helical path is equal to the maximum distance of the particle from the $x–$ axis, then

1. $\cos \theta =\frac{1}{\pi }$
2. $\sin \theta =\frac{1}{\pi }$
3. $\tan \theta =\frac{1}{\pi }$
4. $\tan \theta =\pi$

Q. A charged particle with some initial velocity is projected in a region where non-zero electric and/or magnetic fields are present. In column I, information about the existence of electric and/or magnetic field and direction of initial velocity of charged particle are given; while in Column II, the probable path of the charged particle is mentioned. Match the entries of Column I with the entries of Column II.
$\begin{array}{cccc}& \text{Colum I}& & \text{Colum II}\\ \text{i.}& \overrightarrow{E}=0,\overrightarrow{B}\ne 0\text{and initial velocity may}& \text{a.}& \text{Straight line}\\ & \text{be at any angle with}\overrightarrow{B}.& & \\ \text{ii.}& \overrightarrow{E}=0,\overrightarrow{B}=0\text{and initial velocity may}& \text{b.}& \text{Parabola}\\ & \text{be at any angle with}\overrightarrow{E}.& & \\ \text{iii}& \overrightarrow{E}\ne 0,\overrightarrow{B}\ne 0,\overrightarrow{E}||\overrightarrow{B}\text{and initial velocity is}& \text{c.}& \text{Circular}\\ & \perp \text{to both.}& & \\ & & \text{d.}& \text{Helical path}\end{array}$

1. $\text{(i)}\to \text{(a), (b), (c); ~(ii)}\to \text{(a), (b);~ (iii)}\to \text{(a), (d)}$
2. $\text{(i)}\to \text{(a), (c), (d);~ (ii)}\to \text{(a), (b);~ (iii)}\to \text{(d)}$
3. $\text{(i)}\to \text{(a), (c), (d); ~(ii)}\to \text{(a), (b); ~(iii)}\to \text{(c), (d)}$
4. $\text{(i)}\to \text{(c), (d); ~(ii)}\to \text{(d), (b);~ (iii)}\to \text{(d)}$

Q.
An electron gun is emitting electrons of various speed in a region of uniform magnetic field (parallel to OP) as shown in figure. Which electrons will hit point P?

1. An electron with speed $\frac{eBd}{6\pi mcos\theta }$
2. An electron with speed $\frac{eBd}{\pi mcos\theta }$
3. An electron with energy $\frac{e^2{B}^2{d}^2}{{\pi }^{2}mco{s}^2\theta }$
4. An electron with momentum $\frac{eBd}{4\pi cos\theta }$

Q. A charged particle with some initial velocity is projected in a region where non-zero electric and/or magnetic fields are present. In column I, information about the existence of electric and/or magnetic field and direction of initial velocity of charged particle are given; while in Column II, the probable path of the charged particle is mentioned. Match the entries of Column I with the entries of Column II.
$\begin{array}{cccc}& \text{Colum I}& & \text{Colum II}\\ \text{i.}& \overrightarrow{E}=0,\overrightarrow{B}\ne 0\text{and initial velocity may}& \text{a.}& \text{Straight line}\\ & \text{be at any angle with}\overrightarrow{B}.& & \\ \text{ii.}& \overrightarrow{E}=0,\overrightarrow{B}=0\text{and initial velocity may}& \text{b.}& \text{Parabola}\\ & \text{be at any angle with}\overrightarrow{E}.& & \\ \text{iii}& \overrightarrow{E}\ne 0,\overrightarrow{B}\ne 0,\overrightarrow{E}||\overrightarrow{B}\text{and initial velocity is}& \text{c.}& \text{Circular}\\ & \perp \text{to both.}& & \\ & & \text{d.}& \text{Helical path}\end{array}$

1. $\text{(i)}\to \text{(a), (b), (c); ~(ii)}\to \text{(a), (b);~ (iii)}\to \text{(a), (d)}$
2. $\text{(i)}\to \text{(a), (c), (d);~ (ii)}\to \text{(a), (b);~ (iii)}\to \text{(d)}$
3. $\text{(i)}\to \text{(a), (c), (d); ~(ii)}\to \text{(a), (b); ~(iii)}\to \text{(c), (d)}$
4. $\text{(i)}\to \text{(c), (d); ~(ii)}\to \text{(d), (b);~ (iii)}\to \text{(d)}$

Q. A charged particle is fired at an angle $\theta$ to a uniform magnetic field directed along the $x–$axis. During its motion along a helical path, if the pitch of the helical path is equal to the maximum distance of the particle from the $x–$ axis, then $\tan \theta$ is equal to