Motion of a Charge in Magnetic Fields

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LEVELJEE Advanced

A particle of mass $m = 1.6 \times 1 0^{- 27}$ kg and charge $q = 1.6 \times 1 0^{- 19}$ C enters at $E$ in a region of uniform magnetic field of strength $1$ T along the direction shown in figure. The speed of the particle is $1 0^{7}$ m/s. (a) The magnetic field is directed along the inward normal to the plane of the paper. The particle leaves the region of the field at the point $F$ . Find the distance $E F$ and the angle $θ$ . (b) If the direction of the field is along the outward normal to the plane of the paper, find the time spent by the particle in the region of the magnetic field after entering it at $E$ .

JEE Advanced 2018

LEVELJEE Advanced

In the $x y$ -plane, the region $y > 0$ has a uniform magnetic field $B_{1} \hat{k}$ and the region $y < 0$ has another uniform magnetic field $B_{2} \hat{k}$ . A positively charged particle is projected from the origin along the positive $Y$ -axis with speed $v_{0} = π ms^{- 1}$ at $t = 0$ , as shown in figure. Neglect gravity in this problem. Let $t = T$ be the time when the particle crosses the $X$ -axis from below for the first time. If $B_{2} = 4 B_{1}$ , the average speed of the particle, in $ms^{- 1}$ , along the $X$ -axis in the time interval $T$ is ......... .

JEE Advanced 1993

LEVELJEE Advanced

An electron gun $G$ emits electrons of energy $2 keV$ travelling in the positive $x$ -direction. The electrons are required to hit the spot $S$ where $GS = 0.1 m$ , and the line $GS$ makes an angle of $6 0^{\circ}$ with the $x$ -axis as shown in figure. A uniform magnetic field $B$ parallel to $GS$ exists in the region outside the electron gun. Find the minimum value of $B$ needed to make the electrons hit $S$ .

JEE Advanced 2004

LEVELJEE Main

An electron moving with a speed $u$ along the positive $x$ -axis at $y = 0$ enters a region of uniform magnetic field $B = - B_{0} \hat{k}$ which exists to the right of $y$ -axis. The electron exits from the region after sometime with the speed $v$ at coordinate $y$ , then

(A)

v > u, y < 0

(B)

v = u, y > 0

(C)

v > u, y > 0

(D)

v = u, y < 0

JEE Advanced 2017

LEVELJEE Advanced

Comprehension Passage

A charged particle (electron or proton) is introduced at the origin (

x = 0, y = 0, z = 0

) with a given initial velocity

v

. A uniform electric field

E

and a uniform magnetic field

B

exist everywhere. The velocity

v

, electric field

E

and magnetic field

B

are given in columns 1, 2 and 3, respectively. The quantities

E_{0}, B_{0}

are positive in magnitude. $\begin{array}{lll} \hline \text{Column 1} & \text{Column 2} & \text{Column 3} \\ \hline \text{(I) Electron with } \mathbf{v} = 2\frac{E_0}{B_0}\hat{x} & \text{(i) } \mathbf{E} = E_0\hat{z} & \text{(P) } \mathbf{B} = -B_0\hat{x} \\ \text{(II) Electron with } \mathbf{v} = \frac{E_0}{B_0}\hat{y} & \text{(ii) } \mathbf{E} = -E_0\hat{y} & \text{(Q) } \mathbf{B} = B_0\hat{x} \\ \text{(III) Proton with } \mathbf{v} = 0 & \text{(iii) } \mathbf{E} = -E_0\hat{x} & \text{(R) } \mathbf{B} = B_0\hat{y} \\ \text{(IV) Proton with } \mathbf{v} = 2\frac{E_0}{B_0}\hat{x} & \text{(iv) } \mathbf{E} = E_0\hat{x} & \text{(S) } \mathbf{B} = B_0\hat{z} \\ \hline \end{array}$

Question 1:

In which case would the particle move in a straight line along the negative direction of Y-axis (i.e. move along $- \overset{y}{^}$ )?

In which case will the particle move in a straight line with constant velocity?

In which case will the particle describe a helical path with axis along the positive z-direction?

A uniform magnetic field $B$ exists in the region between $x = 0$ and $x = \frac{3 R}{2}$ (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge $+ Q$ and momentum $p$ directed along $X$ -axis enters region 2 from region 1 at point $P_{1} (y = - R)$ . Which of the following option(s) is/are correct?

* Multiple Correct Options

(A)

When the particle re-enters region 1 through the longest possible path in region 2, the magnitude of the change in its linear momentum between point

P_{1}

and the farthest point from

Y

-axis is

\frac{p}{2}

(B)

For

B = \frac{8}{13} \frac{p}{QR}

, the particle will enter region 3 through the point

P_{2}

X

-axis.

(C)

For

B > \frac{2}{3} \frac{p}{QR}

, the particle will re-enter region 1.

(D)

For a fixed

B

, particles of same charge

Q

and same velocity

v

, the distance between the point

P_{1}

and the point of re-entry into region 1 is inversely proportional to the mass of the particle.

LEVELJEE Main

Two particles $X$ and $Y$ having equal charges, after being accelerated through the same potential difference, enter a region of uniform magnetic field and describe circular paths of radii $R_{1}$ and $R_{2}$ respectively. The ratio of the mass of $X$ to that of $Y$ is

(A)

(\frac{R _{1}}{R _{2}})^{1/2}

(B)

\frac{R _{2}}{R _{1}}

(C)

(\frac{R _{1}}{R _{2}})^{2}

(D)

\frac{R _{1}}{R _{2}}

JEE Advanced 2008

LEVELJEE Advanced

A particle of mass $m$ and charge $q$ , moving with velocity $v$ enters Region II normal to the boundary as shown in the figure. Region II has a uniform magnetic field $B$ perpendicular to the plane of the paper. The length of the Region II is $l$ . Choose the correct choice (s).

* Multiple Correct Options

(A)

The particle enters Region III only if its velocity

v > \frac{q l B}{m}

(B)

The particle enters Region III only if its velocity

v < \frac{q l B}{m}

(C)

Path length of the particle in Region II is maximum when velocity

v = \frac{q l B}{m}

(D)

Time spent in Region II is same for any velocity

v

as long as the particle returns to Region I.

LEVELJEE Main

A proton moving with a constant velocity passes through a region of space without any change in its velocity. If $E$ and $B$ represent the electric and magnetic fields respectively. Then, this region of space may have

* Multiple Correct Options

(A)

E = 0, B = 0

(B)

E = 0, B \neq = 0

(C)

E \neq = 0, B = 0

(D)

E \neq = 0, B \neq = 0

LEVELJEE Advanced

A particle of charge $+ q$ and mass $m$ moving under the influence of a uniform electric field $E \hat{i}$ and uniform magnetic field $B \hat{k}$ follows a trajectory from $P$ to $Q$ as shown in figure. The velocities at $P$ and $Q$ are $v \hat{i}$ and $- 2 v \hat{j}$ . Which of the following statement(s) is/are correct ?

* Multiple Correct Options

(A)

E = \frac{3}{4} [\frac{m v ^{2}}{q a}]

(B)

Rate of work done by the electric field at

P

\frac{3}{4} [\frac{m v ^{3}}{a}]

(C)

Rate of work done by the electric field at

P

is zero

(D)

Rate of work done by both the fields at

Q

is zero

JEE Advanced 2015

LEVELJEE Advanced

Comprehension Passage

In a thin rectangular metallic strip a constant current

I

flows along the positive

x

-direction, as shown in the figure. The length, width and thickness of the strip are

l

w

and

d

, respectively. A uniform magnetic field

B

is applied on the strip along the positive

y

-direction. Due to this, the charge carriers experience a net deflection along the

z

-direction. This results in accumulation of charge carriers on the surface

P QR S

and appearance of equal and opposite charges on the face opposite to

P QR S

. A potential difference along the

z

-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons.

Question 1:

Consider two different metallic strips (1 and 2) of the same material. Their lengths are the same, widths are $w_{1}$ and $w_{2}$ and thicknesses are $d_{1}$ and $d_{2}$ , respectively. Two points $K$ and $M$ are symmetrically located on the opposite faces parallel to the $x$ - $y$ plane (see figure). $V_{1}$ and $V_{2}$ are the potential differences between $K$ and $M$ in strips 1 and 2, respectively. Then, for a given current $I$ flowing through them in a given magnetic field strength $B$ , the correct statements is/are

* Multiple Correct Options

(A)

w_{1} = w_{2}

and

d_{1} = 2 d_{2}

, then

V_{2} = 2 V_{1}

(B)

w_{1} = w_{2}

and

d_{1} = 2 d_{2}

, then

V_{2} = V_{1}

(C)

w_{1} = 2 w_{2}

and

d_{1} = d_{2}

, then

V_{2} = 2 V_{1}

(D)

w_{1} = 2 w_{2}

and

d_{1} = d_{2}

, then

V_{2} = V_{1}

Question 2:

Consider two different metallic strips (1 and 2) of same dimensions (length $l$ , width $w$ and thickness $d$ ) with carrier densities $n_{1}$ and $n_{2}$ , respectively. Strip 1 is placed in magnetic field $B_{1}$ and strip 2 is placed in magnetic field $B_{2}$ , both along positive $y$ -directions. Then $V_{1}$ and $V_{2}$ are the potential differences developed between $K$ and $M$ in strips 1 and 2, respectively. Assuming that the current $I$ is the same for both the strips, the correct options is/are

* Multiple Correct Options

(A)

B_{1} = B_{2}

and

n_{1} = 2 n_{2}

, then

V_{2} = 2 V_{1}

(B)

B_{1} = B_{2}

and

n_{1} = 2 n_{2}

, then

V_{2} = V_{1}

(C)

B_{1} = 2 B_{2}

and

n_{1} = n_{2}

, then

V_{2} = 0.5 V_{1}

(D)

B_{1} = 2 B_{2}

and

n_{1} = n_{2}

, then

V_{2} = V_{1}

JEE Advanced 2011

LEVELJEE Main

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

* Multiple Correct Options

(A)

They will never come out of the magnetic field region

(B)

They will come out travelling along parallel paths

(C)

They will come out at the same time

(D)

They will come out at different times

JEE Advanced 2013

LEVELJEE Advanced

A particle of mass $M$ and positive charge $Q$ , moving with a constant velocity $u_{1} = 4 \hat{i} ms^{- 1}$ , enters a region of uniform static magnetic field normal to the $x$ - $y$ plane. The region of the magnetic field extends from $x = 0$ to $x = L$ for all values of $y$ . After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity $u_{2} = 2 (3 \hat{i} + \hat{j}) ms^{- 1}$ . The correct statement(s) is (are)

* Multiple Correct Options

(A)

the direction of the magnetic field is

- z

direction.

(B)

the direction of the magnetic field is

+ z

direction

(C)

the magnitude of the magnetic field is

\frac{50 π M}{3 Q}

units.

(D)

the magnitude of the magnetic field is

\frac{100 π M}{3 Q}

units.

JEE Advanced 1981

LEVELJEE Main

A potential difference of $600 V$ is applied across the plates of a parallel plate condenser. The separation between the plates is $3 mm$ . An electron projected vertically, parallel to the plates, with a velocity of $2 \times 1 0^{6} m/s$ moves undeflected between the plates. Find the magnitude and direction of the magnetic field in the region between the condenser plates. (Neglect the edge effects). (Charge of the electron = $1.6 \times 1 0^{- 19} C$ )

JEE Advanced 1998

LEVELJEE Advanced

A particle of mass $m$ and charge $q$ is moving in a region where uniform, constant electric and magnetic fields $E$ and $B$ are present. $E$ and $B$ are parallel to each other. At time $t = 0$ , the velocity $v_{0}$ of the particle is perpendicular to $E$ (Assume that its speed is always $<< c$ , the speed of light in vacuum). Find the velocity $v$ of the particle at time $t$ . You must express your answer in terms of $t$ , $q$ , $m$ , the vector $v_{0}$ , $E$ and $B$ and their magnitudes $v_{0}$ , $E$ and $B$ .

LEVELJEE Main

$H^{+}$ , $He^{+}$ and $O^{2 +}$ all having the same kinetic energy pass through a region in which there is a uniform magnetic field perpendicular to their velocity. The masses of $H^{+}$ , $He^{+}$ and $O^{2 +}$ are $1 amu$ , $4 amu$ and $16 amu$ respectively. Then

* Multiple Correct Options

(A)

H^{+}

will be deflected most

(B)

O^{2 +}

will be deflected most

(C)

He^{+}

and

O^{2 +}

will be deflected equally

(D)

all will be deflected equally

JEE Advanced 2007

LEVELJEE Main

A magnetic field $B = B_{0} \hat{j}$ exists in the region $a < x < 2 a$ and $B = - B_{0} \hat{j}$ , in the region $2 a < x < 3 a$ , where $B_{0}$ is a positive constant. A positive point charge moving with a velocity $v = v_{0} \hat{i}$ , where $v_{0}$ is a positive constant, enters the magnetic field at $x = a$ . The trajectory of the charge in this region can be like

A uniform magnetic field with a slit system as shown in figure is to be used as momentum filter for high-energy charged particles. With a field $B$ Tesla, it is found that the filter transmits $α$ -particles each of energy $5.3 MeV$ . The magnetic field is increased to $2.3 B$ Tesla and deuterons are passed into the filter. The energy of each deuteron transmitted by the filter is ......MeV.

LEVELJEE Advanced

For a positively charged particle moving in a $x - y$ plane initially along the $x$ -axis, there is a sudden change in its path due to the presence of electric and/or magnetic fields beyond $P$ . The curved path is shown in the $x - y$ plane and is found to be non-circular.

(A)

E = 0; B = b \hat{j} + c \hat{k}

(B)

E = a \hat{i}; B = c \hat{k} + a \hat{i}

(C)

E = 0; B = c \hat{j} + b \hat{k}

(D)

E = a \hat{i}; B = c \hat{k} + b \hat{j}

JEE Advanced 2012

LEVELJEE Advanced

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $E = E_{0} \hat{j}$ and $B = B_{0} \hat{j}$ . At time $t = 0$ , this charge has velocity $v$ in the $x$ - $y$ plane, making an angle $θ$ with the $x$ -axis. Which of the following option(s) is(are) correct for time $t > 0$ ?

* Multiple Correct Options

(A)

θ = 0^{\circ}

, the charge moves in a circular path in the

x

z

plane.

(B)

θ = 0^{\circ}

, the charge undergoes helical motion with constant pitch along the

y

-axis.

(C)

θ = 1 0^{\circ}

, the charge undergoes helical motion with its pitch increasing with time, along the

y

-axis.

(D)

θ = 9 0^{\circ}

, the charge undergoes linear but accelerated motion along the

y

-axis.

JEE Advanced 1982

LEVELJEE Advanced

A particle of mass $1 \times 1 0^{- 26}$ kg and charge $+ 1.6 \times 1 0^{- 19}$ C travelling with a velocity $1.28 \times 1 0^{6}$ m/s in the $+ x$ direction enters a region in which a uniform electric field $E$ and a uniform magnetic field of induction $B$ are present such that $E_{x} = E_{y} = 0, E_{z} = - 102.4$ kV/m and $B_{x} = B_{z} = 0, B_{y} = 8 \times 1 0^{- 2}$ T. The particle enters this region at the origin at time $t = 0$ . Determine the location ( $x, y$ and $z$ coordinates) of the particle at $t = 5 \times 1 0^{- 6}$ s. If the electric field is switched off at this instant (with the magnetic field still present), what will be the position of the particle at $t = 7.45 \times 1 0^{- 6}$ s ?

JEE Advanced 1997

LEVELJEE Main

A proton, a deutron and an $α$ -particle having the same kinetic energy are moving in circular trajectories in a constant magnetic field. If $r_{p}, r_{d}$ and $r_{α}$ denote, respectively the radii of the trajectories of these particles, then

(A)

r_{α} = r_{p} < r_{d}

(B)

r_{α} > r_{d} > r_{p}

(C)

r_{α} = r_{d} > r_{p}

(D)

r_{p} = r_{d} = r_{α}

LEVELJEE Main

A particle of mass $m$ and charge $q$ moves with a constant velocity $v$ along the positive $x$ -direction. It enters a region containing a uniform magnetic field $B$ directed along the negative $z$ -direction, extending from $x = a$ to $x = b$ . The minimum value of $v$ required so that the particle can just enter the region $x > b$ is

(A)

\frac{q b B}{m}

(B)

\frac{q ( b - a ) B}{m}

(C)

\frac{q a B}{m}

(D)

\frac{q ( b + a ) B}{m}

JEE Advanced 1984

LEVELJEE Main

A neutron, a proton, an electron and an alpha particle enter a region of constant magnetic field with equal velocities. The magnetic field is along the inward normal to the plane of the paper. The tracks of the particles are labelled in figure. The electron follows track…… and the alpha particle follows track……

JEE Advanced 2004

LEVELJEE Main

A proton and an alpha particle, after being accelerated through same potential difference, enter uniform magnetic field, the direction of which is perpendicular to their velocities. Find the ratio of radii of the circular paths of the two particles.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In which case would the particle move in a straight line along the negative direction of Y-axis (i.e. move along −y^​)?

In which case will the particle move in a straight line with constant velocity?

In which case will the particle describe a helical path with axis along the positive z-direction?

Comprehension Passage

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

A proton and an alpha particle, after being accelerated through same potential difference, enter uniform magnetic field, the direction of which is perpendicular to their velocities. Find the ratio of radii of the circular paths of the two particles.

In which case would the particle move in a straight line along the negative direction of Y-axis (i.e. move along $- \overset{y}{^}$ )?