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Torque on current carrying loop - class-XII

Description: torque on current carrying loop
Number of Questions: 73
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Tags: physics magnetic effects of current and magnetism
Attempted 0/73 Correct 0 Score 0

The torgue required to keep a magnet of length $20cm$ at $30^o$ to a uniform field is $2 \times^{-5}N-m$. The magnetic force on each pole is 

  1. $2\times ^{-4}N$

  2. $4 \times 10^{-4}N$

  3. $1\times 10^{-3}N$

  4. $2\times 10^{-6}N$


Correct Option: A
Explanation:
We have,

Length of magnetic field is $0.20m$, Magnetic field $2\times10^{-5}Am$

So, $\tau=mB\sin30^0=0.20\times2\times10^{-5}\times\dfrac{1}{2}=2\times10^{-4}N$

A magnetic dipole placed in two perpendicular magnetic fields $\vec{B}$ and $\vec{B} _\circ{}$ is in equilibrium making an angle $\theta $ with $\vec{B}$ then

  1. $B = B _\circ{}$

  2. $B \cos \theta = B _\circ{}\sin\theta $

  3. $B \sin \theta = B _\circ{}\cos\theta $

  4. $B = B _\circ{} \tan \theta$


Correct Option: C
Explanation:

For equilibrium of dipole net torque on the loop should be zero.
So, $\vec{Z} = \vec{M}\times \vec{B}$
$\vec{Z} _1 = \vec{Z} _o$
$\vec{M}\times \vec{B} = \vec{M}\times \vec{B} _o$
$MB \sin \theta = M B _o \sin (90^o-\theta)$
$B \sin\theta = B _o \cos \theta$   $(\because \sin(90^o-\theta) = \cos \theta)$

A bar magnet of moment $\overline{M}$ is in a magnetic field of induction $\overline{B}$. Then the couple is:

  1. $\overline{M}\times \overline{B}$

  2. $\overline{B}\times \overline{M}$

  3. $\overline{M} . \overline{B}$

  4. $\overline{B} . \overline{M}$


Correct Option: A
Explanation:

$Torque = IAB \sin\theta$
$m = IA$
$T = mB \sin\theta$
$T = m \times B$

A current carrying a coil suspended freely in a uniform magnetic field is in stable equilibrium, if the angle between its magnetic dipole moment vector and the magnetic field is 

  1. $180^0$

  2. zero

  3. $45^0$

  4. $90^0$


Correct Option: B
Explanation:

A current carrying coil behaves as a magnetic dipole. 

Now we know that most stable position for dipole placed in a magnetic field is when angle between its magnetic moment and the external magnetic field is zero. 
In this position potential energy of the system is minimum therefore, it is in stable equilibrium position.

A current carrying loop is placed in a uniform magnetic field. The torque acting on it does depend upon

  1. shape of the loop keeping perimeter fix

  2. area of the loop

  3. value of the current

  4. magnetic field


Correct Option: A,B,C,D
Explanation:

Torque $\vec{\tau} = \vec{\mu} \times \vec{B}$ where $\vec{\mu} = iA\hat{n}$
$i$ is the current in the loop and $A$ is the area of the loop. $\hat{n}$ is a unit vector perpendicular to the plane of the loop.
Area will depend on the shape. Hence, $\vec{\tau}$ will depend on  area, current and  magnetic field.

A coil carrying electric current is placed in a uniform magnetic field with its axis making a nonzero angle $\theta$ with the field, then

  1. torque is applied on the coil

  2. emf is induced

  3. both (a) and (b) are correct

  4. a net force acts on the coil


Correct Option: A
Explanation:

Due to rotating current in the coil, a magnetic moment $M$ is produced. 
The torque exerted on the coil=$\vec{M}\times \vec{B}=MB\sin\theta\neq 0$ for $\theta\neq 0$.

A current carrying coil is subjected to a uniform magnetic field. The coil will orient so that its plane becomes

  1. inclined at $45^{o}$ to the magnetic field

  2. inclined at an arbitrary angle to the field

  3. parallel to the magnetic field

  4. perpendicular to the magnetic field


Correct Option: D
Explanation:

A current carrying coil behaves as a magnetic dipole. Therefore, in a uniform magnetic field coil will get aligned such that the dipole moment of the coil is parallel to the magnetic field. And we know that dipole moment of a coil is perpendicular to its plane.
Therefore, coil will align itself such that its plane is perpendicular the direction of magnetic field.
option(D)

A magnetic field is produced and directed along y-axis. A magnet is placed along x-axis .The direction of the torque on the magnet is:

  1. in the x-y plane

  2. along z-axis

  3. along y-axis

  4. torque will be zero


Correct Option: B
Explanation:

$\tau =\vec{M}\times \vec{B}$
$\hat{i}\times \hat{j}=\hat{k}$
The direction of torque will be always $Z-axis (+ve)$

There is no couple acting when two bar magnets are placed co-axially separated by a distance because :

  1. there are no forces on the poles

  2. the forces are parallel and their lines of action do not cincide

  3. the forces are perpendicular to each other

  4. the forces act along the same line


Correct Option: D
Explanation:

$\vec{\tau} =\vec{M}\times \vec{B}=0$

$MB\sin \theta =0$
$\sin \theta =0$
$\Rightarrow \theta =0$
$\Rightarrow $ Magnetic field lines of both the magnets are in the same line which results in forces due to both magnets act along the same line

A circular loop of area 0.02 m$^2$ carrying a current of 10A, is held with its plane perpendicular to a magnetic field induction 0.2 T. The torque acting on the loop is

  1. 0.01 Nm

  2. 0.001 Nm

  3. zero

  4. 0.8 Nm


Correct Option: C

If a current carrying loop is placed in non uniform magnetic field, then the loop
a) experiences a force
b) experiences a torque
c) will develop induced current
d) oscillates

  1. a, c are correct

  2. a, b, c are correct

  3. b,c,d are correct

  4. a,b,d are correct


Correct Option: B
Explanation:

Current carrying loop in a non-uniform magnetic field experiences a torque, force and current is induced in the loop.

A conducting circular loop of radius $r$ carries a constant current $i$. It is placed in a uniform magnetic field $\bar{B} _{o}$ such that $\bar{B} _{o}$ is perpendicular to the plane of the loop. The magnetic force acting on the loop is

  1.  $ir\bar{B} _{o}$

  2. $2\pi ri \bar{B} _{o}$

  3. $0$

  4. $\pi  ri\bar{B} _{o}$


Correct Option: C
Explanation:

The direction of magnetic force acting on a small current carrying element is given by $\vec{dF}=i(\vec{dl}\times \vec{B})$

$\implies  \vec{F}=\oint i(\vec{dl}\times \vec{B})$ 
Since the vectors $\vec{dl}$ form a closed loop with angle between $\vec{dl}$ and $\vec{B}$ equal to $90^{\circ}$, the net force is zero.

A rectangular coil of wire carrying a current is kept in a uniform magnetic field. The torque acting on the coil will be maximum when

  1. the plane of the coil is perpendicular to the field

  2. the normal to the plane of the coil is parallel to the field

  3. the normal to the plane of the coil is perpendicular to the field

  4. the plane of the coil is making an angle of $45^o$ with the field


Correct Option: C
Explanation:

$\vec{\tau} = \vec{M}\times \vec{B}$
$= niAB \sin\theta$
So, $\sin \theta$ is maximum when $\theta = 90^o$
and $\theta$ is normal between normal of plane of coil and field.
So, the normal to the plane of the coil should be perpendicular to the field to produce maximum torque.

The torque acting on a magnetic dipole of moment $P _{m}$ when placed in a magnetic field is

  1. $P _{m}$B

  2. $\bar{P _{m}}\times \bar{B}$

  3. $\bar{P _{m}}.\bar{B}$

  4. $P _{m}$/B


Correct Option: B
Explanation:

Torque $(\tau) = \vec{M}\times \vec{B}$
                   $= \vec{P _m}\times \vec{B}$

A current carrying loop in a uniform magnetic field will experience

  1. force only

  2. torque only

  3. both torque and force

  4. neither torque nor force


Correct Option: B
Explanation:

A current carrying loop in a uniform magnetic field will experience torque only given by
$\vec{\tau}= \vec{M}\times \vec{B}$
Where M is magnetic moment of loop B is magnetic field.
Force is zero on a current carrying loop in a uniform magnetic field.
Note : Magnetic field must be uniform for net force to be zero.
For a non uniform field, net force may not be zero

A rectangular coil of wire carrying a current is kept in a uniform magnetic field. The torque acting on the coil will be zero when

  1. the plane of the coil is perpendicular to the field

  2. the normal to the plane of the coil is making an angle of 45$^o$ with the field

  3. the normal to the plane of the coil is perpendicular to the field

  4. the plane of the coil is parallel to the field


Correct Option: A
Explanation:

$\vec{\tau} = \vec{M}\times \vec{B}$
$= niAB \sin\theta$
So, for torque to be zero $\sin \theta$ should be zero and for that $\theta$ should be zero, where $\theta$ is angle between normal to the plane of coil and field.
So the plane of the coil and field should be perpendicular to each other

The coil of a galvanometer has $500$ turns and each turn has an average area of $3\times 10^{-4} m^2$. when a current of $0.5$ A passes through it. If a torque of $1.5 Nm$ is required for this coil carrying same current to set it parallel to a magnetic field, calculate the strength of the magnetic field.

  1. 10

  2. 20

  3. 22

  4. 30


Correct Option: B

If X amount of work is required to rotate a bar magnet in a magnetic field by $60^0$, from a position parallel to the field, What is the torque required to maintain it in new position.

  1. $\sqrt3 5X$

  2. $\sqrt3 X$

  3. $\sqrt3 2X$

  4. $\sqrt3 3X$


Correct Option: B
Explanation:

work done in rotating a magnet from $0$ to $60$ will be,

                        $X =$$M\times B\times (1- cos\theta)$
                            =$M\times B\times (0.5)$
                         $2X=M\times B$              taking $\theta =60$
      torqur required to maintain in that position will be ,
                               $\tau=M\times B\times Sin\theta$
                               $\tau=2X\times \dfrac{\sqrt(3)}{2}$     taking $\theta =60$
                                        =${\sqrt3} X$

A circular coil of $25$ turns and radius of $12$cm is placed in a uniform magnetic field of $0.5$ T normal to the plane of coil. If the current in the coil is $5$A, then total torque experienced by the coil is

  1. $1.5$N m

  2. $2.5$N m

  3. $3.5$ N m

  4. zero


Correct Option: D
Explanation:

Here, n = $25$ turns, r = $12$cm, B = $0.5$T
Since the coil is placed in uniform magnetic field normal to the place of the coil. Hence the angle between magnetic moment and magnetic field direction is zero $(i.e. \theta = 0)$
$therefore = mB sin \theta = mB sin 0$
$\therefore T = 0$

A closely wound solenoid of 2000 turns and area of cross-section $1.5 \times 10^{-4}m^2$ carries a current of 2.0 A. It is suspended through its centre and perpendicular to its length, allowing it to turn in a horizontal plane in a uniform magnetic field $5 \times 10^{-2}T$, making an angle of 30$^o$ with the axis of the solenoid. The torque on the solenoid will be

  1. $3 \times 10^{-3} N -m$

  2. $1.5 \times 10^{-3} N -m$

  3. $1.5 \times 10^{-2} N -m$

  4. $3 \times 10^{-2} N -m$


Correct Option: C
Explanation:

Given :   $N = 2000$ turns        $A = 1.5\times 10^{-4} m^2$              $I = 2.0$ A                $B = 5\times 10^{-2}$ T            $\theta = 30^o$

Torque    $\tau = NI (A\times B) =NIAB \sin \theta$                 
$\therefore$  $\tau = (2000) (2.0) (1.5\times 10^{-4}) (5\times 10^{-2}) (0.5) $                           $(\because \sin 30^o =0.5)$
$\implies$   $\tau = 1.5\times 10^{-2}$  $N-m$

At a place the horizontal component of earth's field is $0.5 \times 10^{-4}T$. A bar magnet suspended horizontally perpendicular to earth's field experiences a torque of $4.5\times 10^{-4}N-m$ at that place. The magnetic moment of the magnet is: 

  1. $2.25\times 10^{-8}J/T$

  2. $1/9 J/T$

  3. $2.25 J/T$

  4. $9 J/T$


Correct Option: D
Explanation:

$\tau = MB sin\theta$
$4.5 \times 10^{-4}= M\times 0.5 \times 10^{-4}. sin 90^0$
$M= \dfrac{4.5 \times 10^{-4}}{0.5\times 10^{-4}}= 9 J/T$

A coil in the shape of equilateral tringale of side $0.02\ m$  is suspended from the vertex such that it is hanging in a  vertical place between the pole-pieces of a permanent magnet  producing a horizontal magnetic field of $5 \times 10^{-2}\  T.$ When a current of $0.1\ A$ passed through it and the  magnetic field is parallel to its plane then couple acting on  the coil is :

  1. $8.65 \times 10^{-7}\ N-m$

  2. $6.65 \times 10^{-7}\ N-m$

  3. $3.35 \times 10^{-7}\ N-m$

  4. $3.91 \times 10^{-7}\ N-m$


Correct Option: A
Explanation:

The torque acting on a coil is given by,

$\tau $ = N B I A sin$\theta $
Here, A = $\frac{1}{2}  \times  base  \times $ height
$\begin{array}{l} =\frac { 1 }{ 2 } \times 0.02\times \sqrt { { { 0.02 }^{ 2 } }-{ { 0.01 }^{ 2 } } }  \ =1.732\times { 10^{ -4 } }{ m^{ 2 } } \end{array}$
$\theta $ = Angle between direction of the magnetic field and normal to the plane of the coil. Also, as the magnetic field is parallel to the plane of the coil, so $\theta $ = 90
thus, 
$\tau$  = $1\times 5\times { 10^{ -2 } }\times 0.1\times 1.732\times { 10^{ -4 } }\times 1$
    = 8.6$\times10^{-7}$ Nm



A bar magnet when placed at an angle of $30^o$ to the direction of magnetic field of induction of $ 5\times10^{-5} T$, experiences a moment of a couple $2.5\times10^{-6} N-m$. If the length of the magnet is $5\ cm$ its pole strength is:

  1. $2\times10^{-2}\ Am$

  2. $5\times10^{-2}\ Am$

  3. $2\ Am$

  4. $5\ Am$


Correct Option: C
Explanation:

Torque experienced by bar magnet is $\tau =MB\sin \theta $ where $M$magnetic moment, $B$ is magnetic field induction and $\theta $ is the angle between $M\,\,and\,\,B$

It is given that

  $ \tau =2.5\times {{10}^{-6}}N $

 $ B=5\times {{10}^{-5}}T $

 $ L=0.05m $

 $ 2.5\times {{10}^{-6}}=M\times 5\times {{10}^{-5}}\,\times \dfrac{1}{2} $

 $ M={{10}^{-1}} $

 $ M=m\times L $

 $ {{10}^{-1}}=m\times 0.05 $

 $ m=2 Am $

A bar magnet is held perpendicular to a uniform magnetic field. If the couple acting on the magnet is to be halved by rotating it, then the angle by which it is to be rotated is:

  1. $30^o$

  2. $45^o$

  3. $60^o$

  4. $90^o$


Correct Option: A
Explanation:

We know that,

$ \tau =MB\sin \theta  $

$ \tau \propto \sin \theta  $

So,

$ \dfrac{{{\tau } _{1}}}{{{\tau } _{2}}}=\dfrac{MB\sin {{\theta } _{1}}}{MB\sin {{\theta } _{2}}} $

$ \dfrac{{{\tau } _{1}}}{{{\tau } _{2}}}=\dfrac{\sin {{\theta } _{1}}}{\sin {{\theta } _{2}}} $

$ \dfrac{\tau }{\frac{\tau }{2}}=\dfrac{\sin {{90}^{0}}}{\sin {{\theta } _{2}}} $

$ \sin {{\theta } _{2}}=\dfrac{1}{2} $

$ {{\theta } _{2}}={{30}^{0}} $

Hence, the angle is ${{30}^{0}}$


If a current is passed through a loop which is placed in a magnetic field, then the acting torque will not depend on

  1. Shape of the loop

  2. Area of the loop

  3. The current value

  4. Magnetic field


Correct Option: A
Explanation:

Since,  $T _B=MB\sin\theta$ where $\theta$ is angle between body and the magnetic field$\implies $ shape of loop does not affect torque since $B$ depends on the area and current.

A charged particle is moving with uniform velocity $V\hat {j}$ through a uniform magnetic field $B(-\hat {i})$ and a unifom electric field $\vec {E}$. Then $\vec {E}$ is

  1. $-Bv\hat {k}$

  2. $Bv\hat {k}$

  3. $\dfrac {v}{B}\hat {k}$

  4. $\dfrac {-B}{v}\hat {k}$


Correct Option: B

At some location on earth the horizontal components of earth's magnetic field is $18 \times 10^-6 T.$ At this location, magnetic needle of length 0.12 m and pole strength 1.8 Am is suspended from its mid-point using a thread, it makes $45^0$ angle with horizontal in equilibrium to keep this needle horizontal, the vertical force that should be applied at one of its ends is:

  1. $3.6 \times 10^-5 N$

  2. $6.5 \times 10^-5 N$

  3. $1.3 \times 10^-5 N$

  4. $1.8 \times 10^-5 N$


Correct Option: B
Explanation:

At $45 ^o, B _H = B _V$

$F \dfrac{l}{2} = MB _V = m \times l \times B _V$
$F = \dfrac{2 m l B _V} { l} = 3.6 \times 18 \times 10^{-6} $
$ = 6.5 \times 10^{-5} N$

Two bar magnets with magnetic moments2$\mathrm { M }$ and $\mathrm { M }$ are fastened together at right angles to each other at their centres to forma crossed system, which can rotate freelyabout a vertical axis through the centre. The crossed system sets in earth's magnetic fieldmaking an angle $\theta$ with the magnetic merid-ian such that

  1. $\theta = \tan ^ { - 1 } \left( \dfrac { 1 } { \sqrt { 3 } } \right)$

  2. $\theta = \tan ^ { - 1 } ( \sqrt { 3 } )$

  3. $\theta = \tan ^ { - 1 } \left( \dfrac { 1 } { 2 } \right)$

  4. $\theta = \tan ^ { - 1 } \left( \dfrac { 3 } { 4 } \right)$


Correct Option: C

The magnetic moment of a magnetic wire of length L is M . It is bent at its middle point such that of makes an angle of ${60^0}$ . The magnetic moment of this bent wire will be :

  1. M

  2. M/2

  3. 2M

  4. $\frac{{\sqrt 3 }}{{2M}}$


Correct Option: C

If the net force acting on the loop is zero then : 

  1. no torque acts on loop

  2. loop performs translational motion

  3. torque can act on the loop if lines of force do not coincide

  4. both a) and b)


Correct Option: A
Explanation:

If the net force acting on the loop is zero then, 


1) The loop performs rotational motion.


2) No torque acts on the loop.

The correct option is A.

A bar magnet of magnetic moment 1.5 J/T is along the direction of the uniform magnetic field of 0.22T. The work done in turning the magnet opposite to the field direction and the torque required to keep in that position are 

  1. $0.33J$ and $0.33 N-m$

  2. $0.66J$ and $0.66 N-m$

  3. $0.33J$ and $0 N-m$

  4. $0.66J$ and $0 N-m$


Correct Option: D

On applying a uniform magnetic field on a current-carrying coil the coil rotates in such a way that its plane

  1. becomes perpendicular to magnetic field

  2. becomes parallel to magnetic field

  3. makes an angle of $45^o$ with the magnetic field

  4. makes any angle with the magnetic field


Correct Option: A
Explanation:

On applying a uniform magnetic field on a current-carrying coil, the lines of force are at right angle to the plane of coil. Hence, the coil rotates in such a way that its plane becomes perpendicular to magnetic field.

A very long magnet of pole strength 16 A-m is placed vertically with its one pole on the table. At what distance from the pole, there will be a neutral point on the table. $(B _H =4 \times 10^{-5} \ Wbm^{-2})$

  1. 0.4 m

  2. 0.2 m

  3. 0.5 m

  4. 0.8 m


Correct Option: A

The torque $(\vec t)$ experienced by a current - loop of magnetic moment $(\vec M)$ placed in magnetic field $\vec B$ is -

  1. $\vec t = \vec M \times \vec B$

  2. $\vec t = \vec B \times \vec M$

  3. $\vec t = \frac{\vec M}{\vec B}$

  4. $\vec t = \vec M.\vec B$


Correct Option: A

A coil of area 0.01 m$^2$ is lying in a perpendicular magnetic field of 0.1 Tesla. If a current of 10 A is passed in it then the maximum torque acting on the coil will be

  1. 0.01 N/m

  2. 0.001 N/m

  3. 1.1 N/m

  4. 0.8 N/m


Correct Option: A
Explanation:

Magnetic moment = $ I \vec A = 0.01 \times 10   A/m^2$ perpendicular to the field.
Maximum torque on the magnetic moment is when angle between magnetic moment and the field is $90^{\circ}= (I \vec A ) \times \vec B = I A B = 10 \times 0.01 \times 0.1 = 0.01 Nm $

A flat coil carrying a current has a magnetic moment $\vec{\mu}$. It is placed in a magnetic field $\vec B$. The torque on the coil is $\vec{\tau}$

  1. $\vec{\tau} = \vec{\mu} \times \vec B$

  2. $\vec{\tau} = \vec{B} \times \vec{\mu} $

  3. $|\vec{\tau}| = \vec{\mu} \cdot \times \vec B$

  4. $\vec{\tau}$ is perpendicular to both $\vec{\mu}$ and $\vec{B}$.


Correct Option: A,D
Explanation:

The magnetic moment is defined as a vector relating the aligning torque on the object from an externally applied magnetic field to the field vector itself. The relationship is given by:

$ \tau = \vec{\mu} \times \vec{B} $

where  $\tau$ is the torque acting on the dipole and $B$ is the external magnetic field, and $\mu$  is the magnetic moment. Direction of torque is given by the right hand rule.

A current-carrying loop suspended freely in a uniform magnetic field  will experience 

  1. torque only

  2. force only

  3. neither torque nor force

  4. both


Correct Option: A
Explanation:

A current carrying loop behaves as a magnetic dipole. and we know that a dipole placed in uniform magnetic field only experiences torque.

Asteady current 'I' flows in a small square loop of wire of side $L$ in a horizontal plane. The loop is now folded about its middle such that half of it lies in a vertical plane. Let $\overline{\mu} _1$ and $\overline{\mu} _2$ respectively denote the magetic moments of the current loop before and after folding. Then:

  1. $\overline{\mu} _2 = 0$

  2. $\overline{\mu} _1$ and $\overline{\mu} _2$ are in the same direction

  3. $\dfrac{|\overline{\mu} _1|}{|\overline{\mu} _2|} = \sqrt{2}$

  4. $\dfrac{|\overline{\mu} _1|}{|\overline{\mu} _2|} = \dfrac{1}{\sqrt{2}}$


Correct Option: C

A current carring wire is bent to from a circuital coil. If this coil is placed in any other magnetic filed the maximum torque on the coil, the number of turns will be 

  1. 1

  2. 2

  3. 4

  4. 8


Correct Option: C

The plane of a rectangular loop of wire with sides $0.05 m$ and $0.08 m$ is parallel to a uniform magnetic field of induction $1.5\times 10^{-2}T$ . A current of $10.0 A$ flows through the loop. If the side of length $0.08 m$ is normal and the side of length $0.05 m$ is parallel to the lines of field, then the torque acting on it is

  1. $6000N-m$ 

  2. $zero$ 

  3. $1.2\times 10^{-2}N-m$

  4. $6\times 10^{-4}N-m$ 


Correct Option: D
Explanation:

Torque on the loop $\vec{\tau} = \vec{\mu} \times \vec{B}$ where $\vec{\mu}=i\vec{A}$ is the magnetic moment of the loop and $\vec{B}$  is the magnetic field.
$ \therefore \tau = i\vec{A} \times \vec{B}$
Since $\vec{B}$ is in the plane of the loop, $\vec{A} \perp \vec{B}$
$ \therefore \vec{\tau} = 10 \times ( 0.05 \times 0.08) \times( 1.5 \times 10^{-2})= 6 \times 10^{-4}N-m$

A flat coil carrying a current has a magnetic moment $\vec{\mu}$. It is placed in a magnetic field $\vec B$ such that $\vec{\mu}$ is antiparallel to $\vec B$. The coil is

  1. not in equilibrium

  2. in stable equilibrium

  3. in unstable equilibrium

  4. in neutral equilibrium


Correct Option: C
Explanation:

At  present situation force of magnetic moment = $ \vec{\mu} \times \vec{B} =0 $
If it is disturbed from this position, the force that would develop would take it away from this current position and destabilize the current equilibrium. Hence it is at untstable equilibrium.
In stable equilibrium, the force developed on the system upon disturbing brings the system back to its original position. Here the opposite is the case.
In neutral equilibrium, upon being disturbed from the original position, the system attains a new equilibrium at the new position.   

A dipole of dipole moment p is kept at the centre of a ring of radius R and charge Q. The dipole moment has direction along the axis of the ring. The resultant force on the ring due to the dipole is

  1. zero

  2. $\frac{k P Q}{R^3}$

  3. $\frac{2k P Q}{R^3}$

  4. $\frac{k P Q}{R^3}$ only if the charge is uniformly distributed on the ring


Correct Option: A

A current loop in a magnetic field

  1. can be in equilibrium in one orientation.

  2. can be in equilibrium in two orientations, both the equilibrium states are unstable.

  3. can be in equilibrium in two orientations, one stable while the other is unstable.

  4. experiences a torque where the field is uniform or non-uniform in all orientations.


Correct Option: C
Explanation:

A current loop is a magnetic field is in equilibrium in two orientations one is stable and another unstable.
$\because \vec{\tau} = \vec M \times \vec B = M  B  sin  \theta$
If $\theta = 0^o \Rightarrow \tau = 0$    (stable)
If $\theta = \pi \Rightarrow \tau = 0$     (unstable)
Do not experience a torque in some orientations
Hence option (c) is correct.

A current carrying loop lies on a smooth horizontal place. Then

  1. it is possible to establish a uniform magnetic field in the region so that the loop starts rotating about its own axis.

  2. it is possible to establish a uniform magnetic field in the region so that the loop will tip over about any of the point.

  3. it is not possible that loop will tip over about any of the point whatever be the direction of established magnetic field (uniform).

  4. both A and B are correct.


Correct Option: B
Explanation:

As the loop is placed in horizontal plane, so area vector is along vertical direction. From $\vec {\tau}=I(\vec A\times \vec B)$, as $\vec A$ is in vertical direction, $\vec {\tau}$ would be the plane of loop only. So, option (a) is wrong because for rotation of loop about its own axis $\vec {\tau}$ must be along vertical direction. (b) is correct because we can produce torque in the plane of the loop and due to this the loop can tip over.

The plane of a rectangular loop of wire with sides $0.05\ m$ and $0.08\ m$ is parallel to a uniform magnetic field of induction $1.5\times 10^{-2}\ T$. A current of  $10.0\ ampere$ flows through the loop. If the side of length $0.08\ m$ is normal and the side of length 0.05m is parallel to the lines of induction, then the torque acting on the loop is

  1. $6000\ Nm$

  2. $zero$

  3. $1.2\times 10^{-2}\ Nm$

  4. $6\times 10^{-4}\ Nm$


Correct Option: D
Explanation:

Torque $\tau$ acting on a current carrying coil of area A placed in a magnetic field of induction B is given by,


$\tau=NIBA \sin\theta$

where $I=$ current in the coil, $\theta=$ angle which the normal the plane of the coil makes with the lines of induction $B$.

Here, $N=1, B=1.5\times 10^{-2}\ T$

$A=0.05\times 0.08=40\times 10^{-4}\ m^2$


$I=10.0 amp, \theta=90^o=\pi /2$

$\tau=(1.5\times 10^{-2})(10.0)\times (1)(40\times 10^{-4}\sin (\pi /2)$

$\tau =6\times 10^{-4}\ Nm$

When a current carrying coil is placed in a uniform magnetic field with its magnetic moment anti-parallel to the field

  1. torque on it is maximum

  2. torque on it is zero

  3. potential energy is maximum

  4. dipole is in unstable equilibrium


Correct Option: B,C,D
Explanation:

Given, $\vec{M} \nparallel \vec{B}$, It is anti parallel.
$\therefore \vec{M} \times \vec{B} =0$ since angle is $180^{\circ}$
Potential energy $U =-\vec{M}.\vec{B}= -MB \cos 180^{\circ}= MB \times 1= MB$
Potential energy is maximum is an equilibrium but is an unstable equilibrium.

A coil carrying electric current is placed in uniform magnetic field. Then : 

  1. torque id formed

  2. emf is induced

  3. both $(a)$ and $(b)$ are correct

  4. none of the above


Correct Option: A
Explanation:

As magnetic field is constant therefore there will be no change in flux therefore no induced emf. current carrying coil has magnetic dipole moment. Hence a torque p = m x B acts on it in magnetic field.

A current carrying coil is subjected to a uniform magnetic field. The coil will orient so that its plane becomes

  1. inclined at 45$^o$ to the magnetic field

  2. inclined at any arbitrary angle to the magnetic field

  3. parallel to the magnetic field

  4. perpendicular to magnetic field


Correct Option: D
Explanation:

$\tau = (\vec M \times \vec B)$, where $|\vec M| = i .A$

$= M B   sin  \theta$
where
$\theta$ is angle between Magnetic moment & $\vec B$.
For $\theta = 0   \vec{\tau} = 0$ & coil is in stable equilibrium.
Hence plane of coil must be perpendicular to magnetic field.

A current carrying loop is free to turn in a uniform magnetic field. The loop will then come into equilibrium when it's plane is inclined at

  1. $0^{0}$ to the direction of the field.

  2. $45^{0}$ to the direction of the field.

  3. $90^{0}$ to the direction of the field.

  4. $60^{0}$to the direction of the field.


Correct Option: C
Explanation:

loop will come to equilibrium when torque is zero
So, $\vec{\tau} = 0 = \vec{M}\times \vec{B}$
                            $= MB sin \theta$
                            $= NiA Bsin \theta$
So, angle between area vector and field is zero than only $\theta$ is 0 so that $sin \theta = 0$.
So, for area vector along field direction plane is perpendicular to the field.

A coil of area $100\ cm^{2}$ having $500$ turns carries a current of $1\ m A$. It is suspended in a uniform magnetic field of induction $10^{-3}Wb /m^{2}$ . It's plane makes an angle of $60^{o}$ with the lines of induction. The torque acting on the coil is :

  1. $250\times 10^{-8}Nm$

  2. $25\times 10^{-8}Nm$

  3. $2.5\times 10^{-8}Nm$

  4. $0.2\times 10^{-8}Nm$


Correct Option: A
Explanation:

$J=\vec { M } \times \vec { B } $
$=ni\ \vec { A}\times   \vec { B } $
$=1\times { 10 }^{ -3 }\times 100\times { 10 }^{ -4 } \times  { 10 }^{ -3 }\times 500\times \sin { { 30 }^{ o  } } $
$=25\times { 10 }^{ -7 }Nm.$

Since. Plane makes an angle of $60^o$ with the lines of induction, it will makes an angle of $30^o$ with it's normal.

A circular coil of $1$ turn and area $0.01 m$ carries a current of $10 A$. It is placed in a uniform magnetic field of induction $0.1T$ such that the plane of the circle is perpendicular to the direction of the field, the torque acting on the coil is

  1. $0.1N m$

  2. $0.001Nm$

  3. $0.01N m$

  4. $0$


Correct Option: D
Explanation:

Magnetic moment vector is parallel to that of field. Hence, Torque $=\vec { M } \times \vec { B } =Q\quad $

A current carrying coil tends to set itself____

  1. parallel to an external magnetic field.

  2. parallel to its own magnetic field

  3. perpendicular to the external magnetic field.

  4. perpendicular to the geographic meridian


Correct Option: C
Explanation:

When current carrying coil is placed in the magnetic field torque acts on the coil. The torque acting on the coil will be zero if the coil and the magnetic field are perpendicular. Thus it orients such that the plane of coil is perpendicular to the field direction.

A rectangular coil of wire of area $400 \ cm^{2}$ contains $500$ turns. It is placed in a magnetic field of induction $4\times 10^{-3}\ T$ and it makes an angle $60^o$ with the field. A current of $0.2 \ A$ is passed through it. The torque on the coil is :

  1. $8\sqrt{3}\times 10^{-3}\ N-m$

  2. $8\times 10^{-3}\ N-m$

  3. $8\sqrt{3}\times 10^{-4}\ N-m$

  4. $8\times 10^{-4}\ N-m$


Correct Option: B
Explanation:

$A=400\times { 10 }^{ -4 }{ m }^{ 2 }$
 $n = 500$
$B=4\times { 10 }^{ -3 }$
Area vector makes complementary angle with field therefor, angle between magnetic field and normal of coil is $\theta=90^o-60^{ o }={30  }^{ o  }$
$i = 0.2$

$\tau= $$\vec { M } \times \vec { B } $
 $ = niA$$ B\sin\theta$
  $= 500 \times 0.2 \times 400\times 10^{-4} \times 4 \times 10^{-3} \times \dfrac{1}{2} $
$ = 8 \times 10^{-3}Nm$

A rectangular coil of length $10 cm$ and breadth $20 cm$ is placed in uniform magnetic field of induction $20 Wbm^{-2}$ . A current of $2A$ is passed through the coil. If it consists of $100$ turns, the maximum torque experienced is :

  1. $40Nm$

  2. $80Nm$

  3. $4000Nm$

  4. $8000Nm$


Correct Option: B
Explanation:

$J= ni AB\sin\theta $
maximum at ,
$ \sin\theta=1$
$\therefore J=niAB=100\times 2\times 20\times 10\times 20\times { 10 }^{ -4 }=80 Nm$

When a coil of area $\overline{\mathrm{A}}$ carrying current $\mathrm{i}$ is suspended in a magnetic field of induction $\overline{\mathrm{B}}$ , then the torque on the coil is

  1. $(\overline{\mathrm{A}} \times \overline{\mathrm{B}})$

  2. $\hat{i}(\overline{\mathrm{A}}-\overline{\mathrm{B}})$

  3. $\hat{i}(\overline{\mathrm{A}} \times \overline{\mathrm{B}})$

  4. $\overline{A}(\hat{i} \times \overline{\mathrm{B}})$


Correct Option: C
Explanation:

Torque on magnetic dipole, $\hat i (A\times B)$
where $i$ is current 
$A$ is area vector normal to the area surface
$B$ is magnetic field

A rectangular coil of wire carrying a current is suspended in a uniform magnetic field. The plane of the coil is making an angle of $30^\circ{}$ with the direction of the field and the torque experienced by it is $\tau _{1}$ and when the plane of the coil is making an angle of $60^\circ{}$ with the direction of the field the torque experienced by it is $\tau _{2}$, then the ratio $\tau _{1}$ : $\tau _{2}$ is :

  1. $1:\sqrt{3}$

  2. $\sqrt{3}:1$

  3. $1 : 3$

  4. $3 : 1$


Correct Option: B
Explanation:

${ \tau } _{ 1 }=\vec { M } \times \vec { B }  = MB  \sin{ 60 }^{ o  }$
${ \tau } _{ 2 }=\vec { M } \times  \vec { B }  = MB  \sin{ 30 }^{ o }$

If coil is making an angle of $\theta$ with field, then it makes an angle of $(90 -\theta)$ with area vector, i.e. magnetic moment vector.

    $\therefore \quad \dfrac { { \tau } _{ 1 } }{ { \tau } _{ 2 } } =\dfrac { \sqrt { 3 }  }{ 1 } $

A vertical rectangular coil of sides $5 cm \times 2 cm$ has $10$ turns and carries a current of $2A$. The torque(couple) on the coil when it is placed in a uniform horizontal magnetic field of $0.1T$ with its plane perpendicular to the field is 

  1. $4\times 10^{-3}N-m$

  2. $0$

  3. $2\times 10^{-3}N-m$

  4. $10^{-3}N-m$


Correct Option: B
Explanation:

If plane is perpendicular to the field, normal vector will be parallel to field 
$\sin \theta=0$
$J = MB \sin\theta = 0$

A coil in the shape of an equilateral triangle of side $l$ is suspended between the pole pieces of a permanent magnet such that $B$ is in the plane of the coil. If due to current $i$ in the triangle, a torque $\tau $ acts on it. The side $l$ of the triangle is

  1. $\dfrac { 2 }{ \sqrt { 3 } } \left( \dfrac { \tau }{ Bi } \right) $

  2. $2{ \left( \dfrac { \tau }{ \sqrt { 3 } Bi } \right) }^{ { 1 }/{ 2 } }$

  3. $\dfrac { 2 }{ \sqrt { 3 } } { \left( \dfrac { \tau }{ Bi } \right) }^{ { 1 }/{ 2 } }$

  4. $\dfrac { 1 }{ \sqrt { 3 } } \dfrac { \tau }{ Bi } $


Correct Option: B
Explanation:

The current flowing clockwise in an equilateral triangle has a magnetic field in the direction of $\hat { k }$.
$\tau =BiNA\sin { \theta  } $
$\tau =BiNA\sin { { 90 }^{ o } } $
$\tau =Bi\times \dfrac { \sqrt { 3 }  }{ 4 } { I }^{ 2 }\times 1$
[$\because $ Area of equilateral triangle $=\dfrac { \sqrt { 3 }  }{ 4 } { I }^{ 2 }$ and $N=1$]
(assuming)
$\Rightarrow { I }^{ 2 }=\dfrac { 4\tau  }{ \sqrt { 3 } Bi } \Rightarrow I=2{ \left[ \dfrac { \tau  }{ Bi\sqrt { 3 }  }  \right]  }^{ { 1 }/{ 2 } }$

A charge $q$ is spread uniformly over an insulated loop of radius $r$. If it is rotated with an angular velocity $\omega$ with respect to normal axis then the magnetic moment of the loop is

  1. $\dfrac {1}{2}q \omega r^{2}$

  2. $\dfrac {4}{3}q \omega r^{2}$

  3. $\dfrac {3}{2}q \omega r^{2}$

  4. $q \omega r^{2}$


Correct Option: A
Explanation:

Let us take an element at an angle $\theta$ subtending an angle $d \theta$

The charge $dq$ the element has can be written as
$dq = \dfrac{q}{2 \pi} d \theta$
We know that $i = \dfrac{dq}{dt}= \dfrac{q}{2 \pi} \times \dfrac{d \theta}{dt}$
The time $dt$ can be written as
$dt= \dfrac{d\theta}{w} $
Hence $i = \dfrac{q w}{2 \pi}$
magnetic moment = $i A$
Hence Manetic moment $M= \dfrac{qw}{2 \pi} \times \pi r^2=\dfrac{qwr^2}{2}$

An electric dipole is placed at an angle of ${30}^{o}$ with an electric field of intensity $2 \times { 10 }^{ 5 }N\quad { C }^{ -1 }$. It experiences a torque equal to $4 \ N$ $m$. The charge on the dipole of the dipole length is $2 \ cm$ is

  1. $8 \ mC$

  2. $4 \ mC$

  3. $2 \ mC$

  4. $6 \ mC$


Correct Option: D
Explanation:

Here $E=2\times { 10 }^{ 5 }N{ C }^{ -1 },l=2cm,\tau =4Nm\quad $
Torque $\vec { \tau  } =\vec { p } \times \vec { E } =pE\sin { \theta  } $
$\therefore 4=p\times 2\times { 10 }^{ 5 }\times \sin { { 30 }^{ o } } $
or $p=4\times { 10 }^{ -5 }Cm$
$\therefore$ Charge $q=\cfrac { p }{ l } =\cfrac { 4\times { 10 }^{ -5 }Cm }{ 0.02m } =2\times { 10 }^{ -3 }C=2mC$

The potential energy of a bar magnet of magnetic moment $M$ placed in a magnetic field of induction $B$. The position of stable equilibrium of the magnet is at the angular position given by $\theta$ equal to:
[where $\theta$ is the angle between B and M]
  1. ${0}^{o}$

  2. ${90}^{o}$

  3. ${45}^{o}$

  4. ${180}^{o}$


Correct Option: A

A circular coil of 25 turns and radius 12 cm is placed in a uniform magnetic field of 0.5 T normal to the plane of the coil. If the current in the coil is 6 A then total torque acting on the coil is

  1. zero

  2. 3.4 N m

  3. 3.8 N m

  4. 4.4 N m


Correct Option: A
Explanation:

The torque acting on the coil $\tau$=|$\bar m$ X $\bar B$ | =$ mB\sin\Theta$
Here the circular coil is placed normal to the direction of magnetic field then the angle between the. direction of magnetic moment ($\bar m$) and magnetic field ($\bar B$) is zero, then
$\tau$ = $mBsin\Theta$ = $mBsin 0$ = 0                $\tau = 0$

The magnitude of torque experienced by a square coil of side 12 cm which consists of 25 turns and carries a current 10 A suspended vertically and the normal to the plane of coil makes an angle of 30 with the direction of a uniform horizontal magnetic field of magnitude 0.9 T is:

  1. 1.6 Nm

  2. 1.2 Nm

  3. 1.4 Nm

  4. 1.8 Nm


Correct Option: A
Explanation:

Here,
$N=25,\\ I=10A, \\B= 0.9T,\\\theta=30^0$

$A=a^2=12 \times 10^{-2} \times 12 \times 10^{-2} =144 \times 10^{-4}m^2$

       $\tau$= $NIA\sin\theta$

$\therefore \tau = 25 \times 10\times 144 \times 10^{-4} \times 0.9 \times \sin30^0 = 1.6Nm$  

A circular coil of 70 turns and radius 5 cm carrying a current of 8 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1.5 T. The field lines make an angle of 30 with the normal of the coil then the magnitude of the counter torque that must be applied to prevent the coil from turning is :

  1. 33 N m

  2. 3 N m

  3. 3.3 x $10^{2}$N m

  4. 3.3 x $10^{-4}$N m


Correct Option: C
Explanation:
Number of turns on the circular coil, $n = 70$

Radius of the coil, $r = 5.0 cm = 0.05\ m$

Area of the coil , $A= \pi r^2=3.14\times(0.05)^2= $

Current flowing in the coil, $I = 8.0\ A$

Magnetic field strength, $B = 1.5\ T$

Angle between the field lines and normal with the coil surface,

$θ = 30^o$

The coil experiences a torque in the magnetic field. Hence, it turns. The counter torque applied to prevent the coil from turning is given by the relation,

$\tau = n IBA sinθ$ … (i)

$\tau= 70 × 8 × 1.5 × 0.00785× sin30^o$

$\tau = 3.3\ N m$


A dipole of magnetic moment $\vec{m}=30\hat{j}$A $m^2$ is placed along the y-axis in a uniform magnetic field $\bar{B}=(2\hat{i}+5\hat{j})$T. The torque acting on it is?

  1. $-40\hat{k}$N m

  2. $-50\hat{k}$N m

  3. $-60\hat{k}$N m

  4. $-70\hat{k}$N m


Correct Option: C
Explanation:

Given that,

Magnetic moment of a dipole, $\vec m=30\hat j\ Am^2$
Magnetic field, $\vec B= (2\hat i+ 5\hat j)\ T$

Torque acting on the dipole , $\vec{\tau}= \vec m \times \vec B$

$\implies  \vec{\tau}= 30\hat j \times( 2\hat i +5\hat j)= -60 \hat k \ Nm $

A circular coil of $100$ turns, radius $10$cm carries a current of $5$A. It is suspended vertically in a uniform horizontal magnetic field of $0.5$T and the field lines make an angle of $60^0$ with the plane of the coil. The magnitude of the torque that must be applied on it to prevent it from turning is going to be

  1. $2.93$ N m

  2. $3.43$ N m

  3. $3.93$ N m

  4. $4.93$ N m


Correct Option: C
Explanation:

Given that,


Number of turns, $N=100$
Radius of coil, $r=10\ cm=0.1 \ m$
Current in the coil, $i=5\ A$
Magnetic moment of the coil, $M= Ni(\pi r^2)$
Magnetic field, $B=0.5\ T$
$\theta=90^o-60^o=30^o$

Torque that must be applied to the coil, $\tau = Ni(\pi r^2) \times B \times sin\ \theta=100\times 5\times (3.14\times 0.1^2)\times0.5\times sin 30^o=3.93\ Nm$

So, correct option is $(C)$

The torque and magnetic potential energy of a magnetic dipole in most stable position in a uniform magnetic field $(\bar{B})$ having magnetic moment $(\bar{m})$ will be

  1. $-mB, zero$

  2. $mB, zero$

  3. $zero, mB$

  4. $zero, -mB$


Correct Option: D
Explanation:

Torque, $\bar{T} = \bar{m} \times \bar{B} = mB sin \theta$
and magnetic potential energy $U _m = \bar{m}. \bar{B} = mB cos \theta$ at $\theta = 0^0$ the dipole will be in most stable position
$T = mB sin\theta = mB sin 0^0 = 0$
and $U _m = -mB cos\theta = -mB cos 0^0 = -mB$

The final torque on a coil having magnetic moment 25 A $m^2$ in a 5 T uniform external magnetic field, if the coil rotates through an angle of 60 under the influence of the magnetic field is:

  1. 216.5 Nm

  2. 108.25 Nm

  3. 102.5 Nm

  4. 58.1 Nm


Correct Option: B
Explanation:
Here, 
$m=25 A m^2 ; \\ \theta  = 60^0 ;\\ B = 5T$


$|\vec {\tau} = |\vec m \times \vec B |=mB\sin\Theta$

$\therefore  = 25 \times 5\times  \sin60^0$ or

 $\tau= 125 \times \dfrac{\sqrt3}{2} = 108.25 Nm$

The magnetic moment of a short bar magnet placed with its magnetic axis at $30^0$ to an external field of $900$G and experiences a torque of $0.02$N m is going to be

  1. $0.35 A m^2$

  2. $0.44 A m^2$

  3. $2.45 A m^2$

  4. $1.5 A m^2$


Correct Option: B
Explanation:

Here, $B = 900 Gauss = 900 \times 10^{-4}T. = 9 \times 10^{-2}T$
$T = 0.02 N m and \theta = 30^0$
$\therefore T = mB sin \theta$
$\Rightarrow 0.02 = m \times 9 \times 10^{-2} \times sin 30^0$
$m = \dfrac{0.02 \times 2 }{9 \times 10^{-2}}= 0.44 A m^2$.

If a solenoid is having magnetic moment of $0.65 JT^{-1}$ is free to turn about the vertical direction and has a uniform horizontal magnetic field of $0.25$T applied. What is the magnitude of the torque on the solenoid when its axis makes an angle of $30^0$ with the direction of applied field is?

  1. $0.075$N m

  2. $0.080$ N m

  3. $0.081$ N m

  4. $0.091$ N m


Correct Option: C
Explanation:
Given that, $M=0.65 JT^{-1}$ , $B=0.25\ T$ and $\theta= 30^o$

$\therefore$ Torque on the solenoid $, \tau =MB\sin \theta =0.65\times 0.25\times \sin 30^o$
$=0.65\times 0.25\times \dfrac{1}{2}=0.08125=0.081\ Nm$

So, option $(C)$ is correct. 

A bar magnet has a magnetic moment of $200 A m^2$. The magnet is suspended in a magnetic field of $0.30 N A^{-1} m^{-1}$. The torque required to rotate the magnet from its equilibrium position through an angle of $30^0$, will be then

  1. $30$Nm

  2. $30 \sqrt3$Nm

  3. $60$Nm

  4. $60 \sqrt3$Nm


Correct Option: A
Explanation:

Torque experienced by a magnet suspended in a uniform magnetic field B is given by
$T = MB sin \theta$
Here, $M = 200 A m^2, B = 0.30 N A^{-1}m^{-1}  and \  \theta = 30^0$
$\therefore T = 200 \times 0.30 \times sin 30^0$
$T = 30 N m$

A magnetic needle kept non parallel to the magnetic field in a non uniform magnetic field experience.

  1. A force but not a torque

  2. A torque but not a force

  3. Both a force and a torque

  4. Neither a force nor a torque


Correct Option: C
Explanation:

Magnetic needle can be considered as a magnetic dipole since it is placed in a non uniform magnetic field it will experience af force as magnetic force on both magnetic force will be different. Also, needle is placed non parallel to the field, so it will experience a torque also.

Therefore, C is correct option. 

A uniform horizontal magnetic field of $7.5\times 10^{-2}$T is set up at angle of $30^o$ with the axis of an solenoid and the magnetic moment associated with it is $1.28$J $T^{-1}$. Then the torque on it is?

  1. $4.8\times 10^{-2}$N m

  2. $1.6\times 10^{-2}$N m

  3. $1.2\times 10^{-2}$N m

  4. $4.8\times 10^{-4}$N m


Correct Option: A
Explanation:

Torque, $\tau =MB\sin\theta$
Here, $M=1.28J$ $T^{-1}$, $B=7.5\times 10^{-2}T, \theta =30^o$
$\therefore \tau =1.28\times 7.5\times 10^{-2}\sin 30^o$
$=1.28\times 7.5\times 10^{-2}\times \dfrac{1}{2}$
$=0.64\times 7.5\times 10^{-2}=4.8\times 10^{-2}$N m.

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