JEE Advanced PYQ Physics – Moving Charges and Magnetism

Q 1 :

A magnetic field $\vec{B} = B_{0} \hat{j},$ exists in the region $a < x < 2 a$ and $\vec{B} = - B_{0} \hat{j},$ in the region $2 a < x < 3 a,$ where $B_{0}$ is a positive constant. A positive point charge moving with a velocity $\vec{v} = v_{0} \hat{i},$ 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 [2007]

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(1)

Use the vector form of $\vec{B}$ and $\vec{v}$ in the formula $\vec{F} = q (\vec{v} \times \vec{B})$ to get the instantaneous direction of force at $x = a$ and $x = 2 a$ .

In the region $a < x < 2 a$ force must be in the direction $\hat{i} \times \hat{j}$ i.e., $+ z$ -direction, so vertically upward. And in the region $2 a < x < 3 a$ , in $- z$ -direction vertically downward.

Q 2 :

An electron travelling with a speed $u$ along the positive $x$ -axis enters into a region of magnetic field where $B = - B_{0} \hat{k}$ $(x > 0) .$ It comes out of the region with speed $v$ then [2004]

$v = u$ at $y > 0$
$v = u$ at $y < 0$
$v > u$ at $y > 0$
$v > u$ at $y < 0$

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(2)

Magnetic force on the charged particle does not change its speed.

$∴$ $u = v$

The force acting on the electron will be perpendicular to the direction of velocity till the electron remains in the magnetic field. Therefore, the electron will follow the path as given below.

Hence, the electron comes out with speed $v = u$ and at $y < 0$ .

Q 3 :

For a positively charged particle moving in an $x - y$ 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. Which one of the following combinations is possible? [2003]

$\vec{E} = 0; \vec{B} = b \hat{i} + c \hat{k}$
$\vec{E} = a \hat{i}; \vec{B} = c \hat{k} + a \hat{i}$
$\vec{E} = 0; \vec{B} = c \hat{j} + b \hat{k}$
$\vec{E} = a \hat{i}; \vec{B} = c \hat{k} + b \hat{j}$

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(2)

When $\vec{E} = 0$ , then the path of the charged particle beyond P will be a helix; hence options (1) and (3) are incorrect. The velocity at P is in the $X$ -direction (given).

Let $\vec{v} = k \hat{i}$

After P, the positively charged particle gets deflected in the $x - y$ plane towards $- y$ direction and the path is non-circular.

$Now, \vec{F} = q (\vec{v} \times \vec{B})$

$\Rightarrow \vec{F} = q [k \hat{i} \times (c \hat{k} + a \hat{i})] for option (2)$

$= q [k c \hat{i} \times \hat{k} + k a \hat{i} \times \hat{i}]$ $= k c q (- \hat{j})$

Since in option (2), electric field is also present $\vec{E} = a \hat{i},$ therefore it will also exert a force in the $+ X$ direction. The net result of the two forces will be a non-circular path.

Only option (2) fits for the above logic.

Q 4 :

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 [2002]

$\frac{q b B}{m}$
$\frac{q (b - a) B}{m}$
$\frac{q a B}{m}$
$\frac{q (b + a) B}{2 m}$

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(2)

When a charged particle is moving in a uniform magnetic field normal to its motion,

$q v B = \frac{m v^{2}}{R}$

Width of the magnetic field region $(b - a) \leq R,$ where $' R'$ is its radius of curvature inside the magnetic field.

$∴$ $R = \frac{m v}{q B} \geq (b - a)$ $\Rightarrow v_{\min} = \frac{(b - a) q B}{m}$

Q 5 :

Two particles A and B of masses $m_{A}$ and $m_{B}$ respectively and having the same charge are moving in a plane. A uniform magnetic field exists perpendicular to this plane. The speeds of the particles are $v_{A}$ and $v_{B}$ respectively and the trajectories are as shown in the figure. Then [2001]

$m_{A} v_{A} < m_{B} v_{B}$
$m_{A} v_{A} > m_{B} v_{B}$
$m_{A} < m_{B}$ and $v_{A} < v_{B}$
$m_{A} = m_{B}$ and $v_{A} = v_{B}$

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(2)

When a charged particle is moving normal to the magnetic field, then a force acts on it which behaves as a centripetal force and moves the particle in circular motion.

$∴$ $F = q v B = \frac{m v^{2}}{r}$ $\Rightarrow r = \frac{m v^{2}}{q B}$

If $q$ and $B$ are same for both, then, $r \propto m v$

$Since r_{A} > r_{B}$

$\Rightarrow m_{A} v_{A} > m_{B} v_{B}$

Q 6 :

An ionized gas contains both positive and negative ions. If it is subjected simultaneously to an electric field along the $+ x$ -direction and a magnetic field along the $+ z$ -direction, then [2000]

positive ions deflect towards $+ y$ -direction and negative ions towards $- y$ direction
all ions deflect towards $+ y$ -direction
all ions deflect towards $- y$ -direction
positive ions deflect towards $- y$ -direction and negative ions towards $+ y$ -direction.

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(3)

Given electric field along $+ x$ direction, $\vec{E} = E . \hat{i}$ and magnetic field along $+ z$ direction, $\vec{B} = B . \hat{k}$

Velocity of the ionised particle will be along the direction determined by $q . \vec{E}$

or velocity $\vec{v} = A q E \hat{i},$ where A is a positive constant.

Here, A, E, and B are positive constants.

Charge on ions $(q)$ may be positive or negative.

Magnetic force $\vec{F} = q (\vec{v} \times \vec{B})$

$∴$ $\vec{F} = q [(A q E \hat{i}) \times (B \hat{k})] = q^{2} A E B (\hat{i} \times \hat{k})$

$= q^{2} A E B (- \hat{j}) = {(\pm q)}^{2} A F B,$ along the negative $y$ -direction.

As magnetic force is along the negative $y$ -axis, hence all ions, whether positive or negative, will deflect towards the negative $y$ -axis.

Q 7 :

A particle of charge q and mass m moves in a circular orbit of radius r with an angular speed $ω$ . The ratio of the magnitude of its magnetic moment to that of its angular momentum depends on [2000]

$ω$ and $q$
$ω, q$ and $m$
$q$ and $m$
$ω$ and $m$

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(3)

The angular momentum of the particle

$L = m v r = m r^{2} ω$ where $ω = 2 πn$ $[∵ v = r ω]$

$∴$ $Frequency n = \frac{ω}{2 π};$ $Further,$ $i = \frac{q}{t} = q \times n = \frac{ω q}{2 π}$

Magnetic moment, $M = i A = \frac{ω q}{2 π} \times π r^{2}$ $\Rightarrow M = \frac{ω q r^{2}}{2}$

$∴$ $\frac{M}{L} = \frac{ω q r^{2}}{2 m r^{2} ω} = \frac{q}{2 m}$

Q 8 :

A positive, singly ionized atom of mass number $A_{M}$ is accelerated from rest by the voltage 192 V. Thereafter, it enters a rectangular region of width $w$ with magnetic field ${\vec{B}}_{0} = 0.1 \hat{k} Tesla,$ as shown in the figure. The ion finally hits a detector at the distance $x$ below its starting trajectory. [Given: Mass of neutron/proton $= (\frac{5}{3}) \times 10^{- 27} kg,$ charge of the electron $= 1.6 \times 10^{- 19}$ C] [2024]

Which of the following option(s) is(are) correct?

The value of $x$ for $H^{+}$ ion is 4 cm.
The value of $x$ for an ion with $A_{M} = 144$ is 48 cm.
For detecting ions with $1 \leq A_{M} \leq 196$ , the minimum height $(x_{1} - x_{0})$ of the detector is 55 cm.
The minimum width $w$ of the region of the magnetic field for detecting ions with $A_{M} = 196$ is 56 cm.

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Select one or more options

(1, 2)

From figure $x = 2 R$

$\Rightarrow x = 2 \frac{P}{q B} \Rightarrow x = \frac{2 \sqrt{2 m q V}}{q B} \Rightarrow x = \frac{2}{B} \sqrt{\frac{2 m V}{q}}$

For $H^{+}$ $\to m = \frac{5}{3} \times 10^{- 27} kg$ and $V = 192$ given

$∴$ $x = \frac{2}{0.1} \sqrt{\frac{2 \times \frac{5}{3} \times 10^{- 27} \times 192}{1.6 \times 10^{- 19}}} = 4 cm$

so option (1) is correct.

For $A_{m} = 144$ ,

$x = \frac{2}{0.1} \sqrt{\frac{2 \times 144 \times \frac{5}{3} \times 10^{- 27} \times 192}{1.6 \times 10^{- 19}}} = 48 cm$

so option (2) is correct.

For $A_{m} = 1$ ,

$x = 4 cm$ and for $A_{m} = 196$

$x = 56 cm$

so $x_{0} = 4 cm \times x_{1} = 56 cm$

$∴$ $x_{1} - x_{0} = 52 cm$

so option (3) is incorrect.

For $A_{M} = 196$ ,

$Minimum width W_{\min} = \frac{P}{q B} = \frac{\sqrt{2 m q V}}{q B} = \frac{1}{B} \sqrt{\frac{2 m V}{q}}$

$∴$ $W_{\min} = R = \frac{1}{0.1} \sqrt{\frac{2 \times 196 \times \frac{5}{3} \times 10^{- 27} \times 192}{1.6 \times 10^{- 19}}} = 28 cm$

so option (4) is incorrect.

Q 9 :

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 the $x$ -axis enters region 2 from region 1 at point $P_{1}$ $(y = - R)$ . Which of the following option(s) are correct? [2017]

For $B > \frac{2}{3} \frac{p}{Q R},$ the particle will re-enter region 1.
For $B = \frac{8}{13} \frac{p}{Q R},$ the particle will enter region 3 through the point $P_{2}$ on the $x$ -axis.
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 the $y$ -axis is $\frac{p}{\sqrt{2}} .$
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.

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Select one or more options

(1, 2)

(1) For the charge $+ Q$ to return region 1,

$\frac{m v^{2}}{(3 R / 2)} = Q v B$ $\Rightarrow \frac{2 p}{3 R} = Q B$ $[Here, radius r = \frac{3}{2} R]$

$∴$ $B = \frac{2 p}{3 Q R}$

Therefore, for $B \geq \frac{2 p}{3 Q R},$ the particle will re-enter region 1.

(2) When $B = \frac{8 p}{13 Q R},$

$\frac{m v^{2}}{r} = Q v (\frac{8 p}{13 Q R})$ $∴$ $r = \frac{13 R}{8}$

Thus $' C'$ is the centre of the circular path of radius $\frac{13 R}{8}$

Also, $C P_{2} = \sqrt{C O^{2} + O P_{2}^{2}} = \sqrt{{(\frac{5 R}{8})}^{2} + {(\frac{3 R}{2})}^{2}}$

$∴$ $C P_{2} = \frac{13 R}{8}$

Thus the particle will enter region 3 through the point $P_{1}$ on the $X$ -axis.

(3) Change in momentum $= \sqrt{2} p$

(4) Further, $\frac{m v^{2}}{r} = q v B$ $∴$ $r = \frac{m v}{q B}$ $∴$ $r \propto m$

i.e., distance is directly proportional to mass.

Q 10 :

A conductor (shown in the figure) carrying constant current $I$ is kept in the $x - y$ plane in a uniform magnetic field $\vec{B}$ . If $F$ is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is (are) [2015]

If $\vec{B}$ is along $\hat{z}$ , $F \propto (L + R)$
If $\vec{B}$ is along $\hat{x}$ , $F = 0$
If $\vec{B}$ is along $\hat{y}$ , $F \propto (L + R)$
If $\vec{B}$ is along $\hat{z}$ , $F = 0$

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Select one or more options

(1, 2, 3)

Magnetic force acting on a current-carrying wire, placed in a uniform magnetic field, $\vec{F} = I (\vec{l} \times \vec{B})$

Here, $\vec{l} = displacement of the wire = 2 (L + R) \hat{x}$

$∴$ $\vec{F} = 2 I (L + R) (\hat{x} \times \vec{B})$

If $\vec{B} = B \hat{x},$ then

$\vec{F} = 2 I (L + R) (\hat{x} \times \hat{x}) B = 0$

If $\vec{B} = B \hat{y},$ then

$\vec{F} = 2 I (L + R) (\hat{x} \times \hat{y}) B = 2 I B (L + R) \hat{z}$

or $F \propto (L + R) .$

If $\vec{B} = B \hat{z},$ then

$\vec{F} = 2 I (L + R) (\hat{x} \times \hat{z}) B = - 2 I B (L + R) \hat{y}$

or, $F \propto (L + R) .$

Q 11 :

A particle of mass M and positive charge Q, moving with a constant velocity ${\vec{u}}_{1} = 4 \hat{i} {m s}^{- 1},$ 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 ${\vec{u}}_{2} = 2 (\sqrt{3} \hat{i} + \hat{j}) {ms}^{- 1} .$ The correct statement(s) is (are) [2013]

The direction of the magnetic field is $- z$ direction.
The direction of the magnetic field is $+ z$ direction.
The magnitude of the magnetic field is $\frac{50 π M}{3 Q}$ units.
The magnitude of the magnetic field is $\frac{100 π M}{3 Q}$ units.

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(1, 3)

According to Fleming's left-hand rule, magnetic field should be in the $- z$ direction.

From figure, $\tan θ = \frac{v_{y}}{v_{x}} = \frac{2}{2 \sqrt{3}}$

$∴$ $θ = \frac{π}{6}$

Angle rotated by the particle $= \frac{arc}{radius} = \frac{speed \times time}{radius}$

$\frac{π}{6} = \frac{4 \times 10 \times 10^{- 3}}{M \times v / Q B}$ $[∵ radius = \frac{M v}{Q B}]$

$∴$ $B = \frac{50 π M}{3 Q}$

Q 12 :

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E} = E_{0} \hat{j}$ and $\vec{B} = B_{0} \hat{j} .$ At time $t = 0$ , this charge has velocity $\vec{v}$ in the $x - y$ plane, making an angle $θ$ with the $x$ -axis. Which of the following option(s) is (are) correct for time $t > 0$ ? [2012]

If $θ = 0 °$ , the charge moves in a circular path in the $x - z$ plane.
If $θ = 0 °$ , the charge undergoes helical motion with constant pitch along the $y$ -axis.
If $θ = 10 °$ , the charge undergoes helical motion with its pitch increasing with time, along the $y$ -axis.
If $θ = 90 °$ , the charge undergoes linear but accelerated motion along the $y$ -axis.

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Select one or more options

(3, 4)

If $θ = 0 °$ , the charged particle is projected along the $x$ -axis. Due to the magnetic field $\vec{B}$ , the charged particle will tend to move in a circular path in the $y - z$ plane, but due to the force of the electric field $\vec{E}$ , the particle will move in a helical path with increasing pitch. Hence options (1) and (2) are wrong.

If $θ = 10 °$ , we can resolve velocity into two rectangular components: one along the $x$ -axis $(v \cos 10 °)$ and one along the $y$ -axis $(v \sin 10 °)$ . Due to $v \cos 10 °$ , the particle will move in a circular path and due to $v \sin 10 °$ plus the force due to the electric field, the particle will undergo helical motion with its pitch increasing.

If $θ = 90 °$ , the charge is moving along the magnetic field. Therefore the force due to the magnetic field is zero. But the force due to the electric field will accelerate the particle along the $y$ -axis.

Q 13 :

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? [2011]

They will never come out of the magnetic field region.
They will come out travelling along parallel paths.
They will come out at the same time.
They will come out at different times.

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Select one or more options

(2, 4)

The entry and exit of electron and proton in the magnetic field make the same angle with $A B$ as shown. Therefore both will come out travelling in parallel paths.

Also, $r = \frac{m v}{q B}$ or, $r \propto m$

$∵$ $m_{e} < m_{p}$ $∴$ $r_{e} < r_{p}$

and, $T = \frac{2 π m}{q B}$ or, $T \propto m$

$∴$ $T_{e} < T_{p},$ $t_{e} = \frac{T_{e}}{2}$ and $t_{p} = \frac{T_{p}}{2}$

or, $t_{e} < t_{p}$

Q 14 :

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). [2008]

The particle enters Region III only if its velocity $v > \frac{q ℓ B}{m}$
The particle enters Region III only if its velocity $v < \frac{q ℓ B}{m}$
Path length of the particle in Region II is maximum when velocity $v = \frac{q ℓ B}{m}$
Time spent in Region II is same for any velocity $v$ as long as the particle returns to Region I.

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Select one or more options

(1, 3, 4)

As $V ⊥ B$ and $B$ is uniform, the path of the charged particle in region-II is circular and the radius of the circular path is $r = \frac{m v}{q B}$

For the particle to enter region III, $r > ℓ$ (path shown by dashed line)

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

For maximum path length in region II, $r = ℓ$

$∴$ $ℓ = \frac{m v}{q B}$ $\Rightarrow v = \frac{q ℓ B}{m}$

The time taken by the particle to move in region II before coming back to region I is $t = \frac{T}{2} = \frac{π m}{q B},$

which is independent of $v$ , i.e., the time spent is same in region II is the same for any velocity.

Topic Question Set

Q 2 :

An electron travelling with a speed $u$ along the positive $x$ -axis enters into a region of magnetic field where $B = - B_{0} \hat{k}$ $(x > 0) .$ It comes out of the region with speed $v$ then [2004]

Q 6 :

An ionized gas contains both positive and negative ions. If it is subjected simultaneously to an electric field along the $+ x$ -direction and a magnetic field along the $+ z$ -direction, then [2000]

Q 7 :

A particle of charge q and mass m moves in a circular orbit of radius r with an angular speed $ω$ . The ratio of the magnitude of its magnetic moment to that of its angular momentum depends on [2000]

Q 10 :

A conductor (shown in the figure) carrying constant current $I$ is kept in the $x - y$ plane in a uniform magnetic field $\vec{B}$ . If $F$ is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is (are) [2015]

Q 13 :

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? [2011]

Q 14 :

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). [2008]

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Topic Question Set

Q 2 : An electron travelling with a speed u along the positive x-axis enters into a region of magnetic field where B=-B0k^ (x>0). It comes out of the region with speed v then [2004]

Q 5 : Two particles A and B of masses mA and mB respectively and having the same charge are moving in a plane. A uniform magnetic field exists perpendicular to this plane. The speeds of the particles are vA and vB respectively and the trajectories are as shown in the figure. Then [2001]

Q 6 : An ionized gas contains both positive and negative ions. If it is subjected simultaneously to an electric field along the +x-direction and a magnetic field along the +z-direction, then [2000]

Q 7 : A particle of charge q and mass m moves in a circular orbit of radius r with an angular speed ω. The ratio of the magnitude of its magnetic moment to that of its angular momentum depends on [2000]

Q 10 : A conductor (shown in the figure) carrying constant current I is kept in the x-y plane in a uniform magnetic field B→. If F is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is (are) [2015]

Q 13 : 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? [2011]

Q 14 : 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). [2008]

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Q 2 :

An electron travelling with a speed $u$ along the positive $x$ -axis enters into a region of magnetic field where $B = - B_{0} \hat{k}$ $(x > 0) .$ It comes out of the region with speed $v$ then [2004]

Q 6 :

An ionized gas contains both positive and negative ions. If it is subjected simultaneously to an electric field along the $+ x$ -direction and a magnetic field along the $+ z$ -direction, then [2000]

Q 7 :

A particle of charge q and mass m moves in a circular orbit of radius r with an angular speed $ω$ . The ratio of the magnitude of its magnetic moment to that of its angular momentum depends on [2000]

Q 10 :

A conductor (shown in the figure) carrying constant current $I$ is kept in the $x - y$ plane in a uniform magnetic field $\vec{B}$ . If $F$ is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is (are) [2015]

Q 13 :

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? [2011]

Q 14 :

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). [2008]