Helical Motion

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

Q. A long straight wire along the z-axis carries a current I in the negative z direction. The magnetic vector field $\overrightarrow{B}$ at a point having coordinates (x, y) in the z = 0 plane is

1. $\frac{{\mu }_{0}I(x\hat{i}+y\hat{j})}{2\pi (x^2+y^2)}$
   
2. $\frac{{\mu }_{0}I(y\hat{i}-x\hat{j})}{2\pi (x^2+y^2)}$
   
3. $\frac{{\mu }_{0}I(x\hat{j}-y\hat{i})}{2\pi (x^2+y^2)}$
   
4. $\frac{{\mu }_{0}I(x\hat{i}-y\hat{j})}{2\pi (x^2+y^2)}$

Q. Under the influence of a uniform magnetic field, a charged particle moves with constant speed $v$ in a circle of radius $R$. The time period of rotation of the particle

1. depends on $R$ and not on $v$.
2. is independent of both $v$ and $R$.
3. depends on both $v$ and $R$.
4. depends on $v$ and not on $R$.

Q. Choose the correct option :

1. True dip is not mathematically related to apparent dip.
2. True dip is always equal to the apparent dip.
3. True dip is always greater than the apparent dip.
4. True dip is less than apparent dip.

Q.

A charged particle projected in a magnetic field $B=(3\hat{i}+4\hat{j})\times {10}^{-2}T$ and the acceleration of the particle is found to be $a=(x\hat{i}+2\hat{j})m{s}^{-2}$. Then find the value of $x$.

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

Q.

The value of the angle of dip is zero at the magnetic equator because on it:

1. V and H are equal

2. V and H are zero

3. V is zero

4. V is infinity

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

Q.

A particle of specific charge $\alpha$ starts moving from the origin under the action of an electric field $\overrightarrow{E}=E_0\hat{i}$ and magnetic field $\overrightarrow{B}=B_0\hat{k}$. Its velocity at $(x_0,y_0,0)$ is $(4\hat{i}+3\hat{j})$. The value of $x_0$ is

1. $\frac{25}{2\alpha E_0}$

2. $\frac{16\alpha B_0}{E_0}$

3. $\frac{5\alpha }{2B_0}$

4. $\frac{13\alpha E_0}{2B_0}$

Q. At a place, the earth's horizontal component of magnetic field is $0.38\times {10}^{-4}\text{T}$. If the angle of dip at that place is ${60}^{\circ }$, then the vertical component of earth's magnetic field at that place will be approximately -

1. $0.12\times {10}^{-4}\text{T}$
2. $0.24\times {10}^{-4}\text{T}$
3. $0.40\times {10}^{-4}\text{T}$
4. $0.65\times {10}^{-4}\text{T}$

Q.

Three point charges 3nC, 6nC and 9nC are placed at the corners of an equilateral triangle of side 0.1 m. The potential energy of the system is:

1. 9910 J

2. 8910 J

3. 99100 J

4. None of these

Q. The true value of angle of dip at a place is ${60}^{\circ }$. The apparent dip, in a plane inclined at an angle of ${30}^{\circ }$ with the magnetic meridian is:

1. ${\tan }^{-1}\left(\frac{1}{2}\right)$
2. ${\tan }^{-1}\left(2\right)$
3. ${\tan }^{-1}\left(\frac{2}{3}\right)$
4. ${\tan }^{-1}\left(\frac{1}{3}\right)$

Q.

A cyclotron’s oscillating frequency is$10MHz$. The radius of dee is$60cm$. What is the kinetic energy of proton produced by the accelerator, if the magnetic field is$0.65Tesla$?

Q. Uniform electric and magnetic fields with strength E and induction B, respectively, are along $y$ -axis as shown in the figure. A particle with specific charge $q/m$ leaves the origin 0 in the direction of $x$ -axis with an initial non relativistic velocity $v_0.$ The coordinate $y_n$ of the particle when it crosses the $y$ -axis for the $n^{\text{th}}$ time is

1. $\frac{{\pi }^{2}m{n}^{2}E}{3q{B}^2}$
2. $\frac{\sqrt{3}p{i}^{2}m{n}^{2}E}{q{B}^2}$
3. $\frac{{\pi }^{2}m{n}^{2}E}{q{B}^2}$
4. $\frac{2{\pi }^{2}m{n}^{2}E}{q{B}^2}$

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 neutral particle at rest in a magnetic field decays into two charged particles of different masses but have equal amount of charges. The energy released goes into their kinetic energies. Then, what can be the path(s) of the two particles.

Neglect any interaction between the two charges.

1. 
2. 
3. 
4. 

Q. The horizontal component of earth’s magnetic field at a place is $0.2\text{G}$ whereas its total magnetic field is $0.4\text{G}$. The angle of dip at the place is

1. ${60}^{\circ }$
2. ${45}^{\circ }$
3. ${90}^{\circ }$
4. ${30}^{\circ }$

Q. ntIf an electron of velocity (2i +3j) is subjected to magnetic field of 4k , then itsn ntA, path will changen ntB, speed does not changen ntC, path must be circularn ntD, momentum is constantn ntExplain in detailn

Q.

A charged particle of specific charge 0.2 C/kg has velocity of (2i + 3j) at some instant in uniform magnetic field (5i + 2j). What is the acceleration of the particle at this instant?

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. If a bar magnet of magnetic moment $M$ is freely suspended in a uniform magnetic field of strength $B$, the work done in rotating the magnet through an angle $\theta$ is

1. $MB(1-\sin \theta )$
2. $MB\sin \theta$
3. $MB\cos \theta$
4. $MB(1-\cos \theta )$

Q. At an angle ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$ to the magnetic meridian, the apparent dip is ${60}^{\circ }$. The true dip at the place is :

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${37}^{\circ }$
4. ${60}^{\circ }$

Q. Point charge q moves from point P to point S along the path PQRS (figure shown) in a uniform electric field E pointing coparallel to the positive direction of the axis. The coordinates of the points and are and respectively. The work done by the field in the above process is given by the expression

1. q E a
2. - q E a
3. $qEa\sqrt{2}$
4. $qE\sqrt{\left[\right(2a{)}^2+b^2]}$

Q. If positively charged pendulum is oscillating in a uniform electric field as shown in figure. Its time period as compared to that when it was uncharged:

1. will increase
2. will decrease
3. will not change
4. will first increase then decrease

Q. A proton of mass $1.67\times {10}^{-27}kg$ and charge $1.6\times {10}^{-19}C$ is projected with a speed of $2\times {10}^{-6}m/s$ at an angle of ${60}^{\circ }$ with the X-axis. If a uniform magnetic field of $0.104T$ is applied along Y-axis, the path of the proton is

1. A circle of radius $=4.6\times {10}^{-14}m$ and time period $9\times {10}^{-13}s$ approximately
2. A circle of radius $=3\times {10}^{-14}m$ and time period $5\times {10}^{-13}s$ approximately
3. A helix of radius $=9.6\times {10}^{-14}m$ and time period $6\times {10}^{-7}s$ approximately
4. A helix of radius $=2\times {10}^{-14}m$ and time period $4\times {10}^{-13}s$ approximately

Q. A dip circle is adjusted so that its needle moves freely in the magnetic meridian. In this position, the angle of dip is ${40}^{\circ }$. Now, the dip circle is rotated so that the plane in which the needle moves, makes an angle of ${30}^{\circ }$ with the magnetic meridian. In this position the needle will dip by an angle

1. ${40}^{\circ }$
2. ${30}^{\circ }$
3. More than ${40}^{\circ }$
4. Less than ${40}^{\circ }$

Q. A capacitor of capacitance $C$, charged to potential difference $V$ is connected to another capacitor of capacitance $3C$, charged to potential difference $3V$ as shown in the figure. The final charges on capacitor of capacitance $C$ and $3C$ are respectively

1. $\frac{5CV}{2},-\frac{15CV}{2}$
2. $\frac{15CV}{2},\frac{5CV}{2}$
3. $-\frac{5CV}{2},\frac{10CV}{2}$
4. $\frac{5CV}{2},\frac{15CV}{2}$

Q. The path of a moving charged particle with constant velocity entering a uniform magnetic field making an angle $\left(i\right)0^{\circ }\left(ii\right){45}^{\circ }\text{and}\left(iii\right){90}^{\circ }$ with the field will trace the path of-

1. Circle ; helix ; parabola
2. Straight line ; Circle ; ellipse
3. Parabola ; straight line ; circle
4. Straight line ; helix ; circle

Q. A point charge q moves from ‘P’ to ‘S’ along path PQRS in a uniform electric field E directied parallel to positive X-axis. The co-ordinates of the points P.Q.R and S are (a, b, 0), (2a, 0, 0), (0, –b, 0) and (0, 0, 0) respectively. The work done by field in the above process is given by

1. $qEa$
2. $-qEa$
3. $qEa\sqrt{2}$
4. $qE\sqrt{(2a{)}^2+b^2}$

Q. At a place in a town, the horizontal component of earth’s magnetic field is $2.5\text{G}$ and the angle of dip at that place is ${60}^{\circ }$. The magnetic field of earth at this location is?

1. $1.25\text{G}$
2. $3\text{G}$
3. $2.25\text{G}$
4. $5\text{G}$