Magnetic Dipole and Dipole Moment and Torque on a Current-Carrying Planar Loop in Uniform Magnetic Field

Q 1 :

The magnetic moment of a bar magnet is 0.5 $A m^{2}$ . It is suspended in a uniform magnetic field of $8 \times 10^{- 2}$ T. The work done in rotating it from its most stable to most unstable position is [2024]

$16 \times 10^{- 2} J$
zero
$8 \times 10^{- 2} J$
$4 \times 10^{- 2} J$

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(3)

$At stable equilibrium, U = - M B \cos 0 ° = - M B$

$At unstable equilibrium, U = - M B \cos 180 ° = + M B$

$∆ U = 2 M B = 2 \times 0.5 \times 8 \times 10^{- 2}$

$Work done, W = Δ U = 2 M B$

$W = 8 \times 10^{- 2} J$

Q 2 :

A current of 200μA deflects the coil of a moving coil galvanometer through 60°. The current to cause deflection through $π$ /10 radian is [2024]

30μA
120μA
60μA
180μA

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(3)

$i \propto θ (angle of deflection)$

$∴ \frac{i_{2}}{i_{1}} = \frac{θ_{2}}{θ_{1}} \Rightarrow \frac{i_{2}}{200 μ A} = \frac{π / 10}{π / 3} = \frac{3}{10}$

$\Rightarrow i_{2} = 60 μ A$

Q 3 :

A coil having 100 turns, area of $5 \times 10^{- 3}$ $m^{2}$ , carrying current of 1 mA is placed in a uniform magnetic field of 0.20 T in such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through 90° is ____ μJ. [2024]

(100) $W = Δ U = U_{f} - U_{i}$

$W = | M B \cos 90 ° - M B \cos 0 ° | = M B$

$W = (NIA) B = (100 \times 5 \times 10^{- 3} \times 1 \times 10^{- 3}) \times 0.2 J$

$= 1 \times 10^{- 4} J = 100 μ J$

Q 4 :

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is $1.5 \times 10^{- 5}$ Tm. The magnetic moment of the dipole is _____. (Given: $μ_{0} / 4 π = 10^{- 7}$ $T m A^{- 1}$ ) [2024]

(6) The magnetic potential due to a magnetic dipole,

$V = \frac{μ_{0}}{4 π} \frac{M}{r^{2}}$

$\Rightarrow 1.5 \times 10^{- 5} = 10^{- 7} \times \frac{M}{{(20 \times 10^{- 2})}^{2}}$

$\Rightarrow M = \frac{1.5 \times 10^{- 5} \times 20 \times 20 \times 10^{- 4}}{10^{- 7}}$

$M = 1.5 \times 4 = 6$

Q 5 :

A moving coil galvanometer has 100 turns and each turn has an area of 2.0 cm². The magnetic field produced by the magnet is 0.01 T and the deflection in the coil is 0.05 radian when a current of 10 mA is passed through it. The torsional constant of the suspension wire is $x \times 10^{- 5}$ N-m/rad. The value of $x$ is _____. [2024]

(4) $Given, B = 0.01 T, i = 10 m A, N = 100$

$A = 2 \times 10^{- 4} m^{2} : K = ?$

$τ = K θ$ $\dots (i)$

$τ = B i N A \sin ϕ$ $\dots (i i)$

$From (i) and (i i), K θ = B i N A \sin 90 °$

$K = \frac{B i N A}{θ}$

$K = \frac{(0.01) \times (10 \times 10^{- 3} A) \times (100) \times (2 \times 10^{- 4} m^{2})}{(0.05)}$

$K = \frac{2 \times 10^{- 4}}{5} = \frac{20}{5} \times 10^{- 5} = 4 \times 10^{- 5} N - m / rad$

$x = 4$

Q 6 :

A uniform magnetic field of $2 \times 10^{- 3} T$ acts along positive Y-direction. A rectangular loop of sides 20 cm and 10 cm with current of 5A is in Y-Z plane. The current is in anticlockwise sense with reference to negative X axis. Magnitude and direction of the torque is [2024]

$2 \times 10^{- 4} N - m$ along positive Z-direction
$2 \times 10^{- 4} N - m$ along negative Z-direction
$2 \times 10^{- 4} N - m$ along positive X-direction
$2 \times 10^{- 4} N - m$ along positive Y-direction

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(2)

$\vec{M} = I \vec{A}$

$= 5 \times (0.2) \times (0.1) (- \hat{i}) = 0.1 (- \hat{i})$

$\vec{τ} = \vec{M} \times \vec{B} = 0.1 (- \hat{i}) \times (2 \times 10^{- 3}) (\hat{j})$

$= 2 \times 10^{- 4} (- \hat{k}) N-m$

Q 7 :

A circular coil having 200 turns, $2.5 \times 10^{- 4} m^{2}$ area and carrying 100 $μ A$ current is placed in a uniform magnetic field of 1 T. Initially, the magnetic dipole moment $(\vec{M})$ was directed along $\vec{B}$ . Amount of work, required to rotate the coil through 90° from its initial orientation such that $\vec{M}$ becomes perpendicular to $\vec{B}$ , is ____ $μ J$ . [2024]

(5)

Work done by external magnetic field $Δ U = U_{f} - U_{i}$

Initially, $U_{i} = - M B \cos θ = - M B \cos 0 ° = - M B$

Finally, $U_{f} = - M B \cos 90 ° = 0$

The magnetic moment of coil, $M = N i A$

$M = (100 \times 10^{- 6}) \cdot (200) \cdot (2.5 \times 10^{- 4})$

Work, $W = Δ U = M B = (5 \times 10^{- 6}) (1) = 5 μ J$

Q 8 :

A solenoid having area A and length ' $l$ ' is filled with a material having relative permeability 2. The magnetic energy stored in the solenoid is:: [2025]

$\frac{B^{2} A l}{μ_{0}}$
$\frac{B^{2} A l}{2 μ_{0}}$
$B^{2} A l$
$\frac{B^{2} A l}{4 μ_{0}}$

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(4)

$\frac{u}{V} = \frac{B^{2}}{2 μ_{r} μ_{0}} \Rightarrow u = \frac{B^{2}}{4 μ_{0}} V = \frac{B^{2}}{4 μ_{0}} A l$

Q 9 :

A magnetic dipole experiences a torque of $80 \sqrt{3} Nm$ when placed in uniform magnetic field in such a way that dipole moment makes angle of 60° with magnetic field. The potential energy of the dipole is: [2025]

80 J
$- 40 \sqrt{3} J$
–60 J
–80 J

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(4)

$| \vec{τ} |$ $=$ $| \vec{M} \times \vec{B} |$ $= M B \sin 60 °$

$80 \sqrt{3} = \frac{\sqrt{3}}{2} M B \Rightarrow M B = 160$

$U = - \vec{M} \cdot \vec{B} = - M B \cos 60 °$

$U = - 160 \times \frac{1}{2} = - 80 J$

Q 10 :

For a moving coil galvanometer, the deflection in the coil is 0.05 rad when a current of 10 mA is passed through it. If the torsional constant of suspension wire is $4.0 \times 10^{- 5} {Nm rad}^{- 1}$ , the magnetic field is 0.01 T and the number of turns in the coil is 200, the area of each turn (in ${cm}^{2}$ ) is [2023]

2.0
1.0
1.5
0.5

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(2)

$τ = K θ$

$N i A B = K θ$

$A = \frac{K θ}{N i B} = \frac{4 \times 10^{- 5} \times 0.05}{200 \times 10 \times 10^{- 3} \times 0.01}$

$On solving, A = 10^{- 4} m^{2} = 1 {cm}^{2}$

Q 11 :

A square loop of area 25 ${cm}^{2}$ has a resistance of $10 Ω$ . The loop is placed in uniform magnetic field of magnitude 40.0 T. The plane of loop is perpendicular to the magnetic field. The work done in pulling the loop out of the magnetic field slowly and uniformly in 1.0 sec, will be [2023]

$2.5 \times 10^{- 3}$ J
$1.0 \times 10^{- 3}$ J
$1.0 \times 10^{- 4}$ J
$5 \times 10^{- 3}$ J

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(2)

$l = 50 cm$

$t = 1 s$

$∴$ $V = \frac{0.05}{1} = 0.05 m/s$

$i = \frac{40 \times 0.05 \times 0.05}{10} = 0.01 A$

$∴$ $F = B_{I ∆ C} ℓ = 40 \times 0.01 \times 0.05$

$F = 0.02 N$

$∴$ $W = F \times ℓ = 0.02 \times 0.05$

$∴$ $W = 1 \times 10^{- 3} J$

Q 12 :

The magnetic moments associated with two closely wound circular coils A and B of radius $r_{A} = 10$ cm and $r_{B} = 20$ cm respectively are equal if (where $N_{A}, I_{A}$ and $N_{B}, I_{B}$ are number of turns and current of A and B respectively). [2023]

$2 N_{A} I_{A} = N_{B} I_{B}$
$N_{A} = 2 N_{B}$
$N_{A} I_{A} = 4 N_{B} I_{B}$
$4 N_{A} I_{A} = N_{B} I_{B}$

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(3)

$M = N I A$

$M_{A} = M_{B}$

$∴$ $N_{A} I_{A} A_{A} = N_{B} I_{B} A_{B}$

$∴$ $N_{A} I_{A} π {(0.1)}^{2} = N_{B} I_{B} π {(0.2)}^{2}$

$∴$ $N_{A} I_{A} = 4 N_{B} I_{B}$

Q 13 :

Given below are two statements:

Statement I: If the number of turns in the coil of a moving coil galvanometer is doubled, then the current sensitivity becomes double.

Statement II: Increasing current sensitivity of a moving coil galvanometer by only increasing the number of turns in the coil will also increase its voltage sensitivity in the same ratio.

In the light of the above statements, choose the correct answer from the options given below: [2023]

Statement I is false but Statement II is true
Both Statement I and Statement II are true
Both Statement I and Statement II are false
Statement I is true but Statement II is false

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(4)

$For a moving coil galvanometer$

$B i N A = k θ$

$θ = (\frac{B N A}{k}) i; current sensitivity = \frac{B N A}{k}$

$So, if N is doubled then current sensitivity is doubled.$

$Voltage sensitivity$

$B \frac{V}{R} N A = k θ$

$V = \frac{B N A}{R k} θ as N is doubled R is also doubled.$

$So, no change in voltage sensitivity.$

$Hence, option (4) is right.$

Q 14 :

The current sensitivity of moving coil galvanometer is increased by 25%. This increase is achieved only by changing the number of turns of coils and area of cross section of the wire while keeping the resistance of galvanometer coil constant. The percentage change in the voltage sensitivity will be: [2023]

+25%
−50%
Zero
−25%

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(1)

$I_{s} = \frac{N B A}{C} & V_{s} = \frac{N B A}{C G}$

$\Rightarrow V_{s} = \frac{I_{s}}{G}$

$If G (galvanometer resistance) is constant, then V_{s} \propto I_{s}$

$So percentage change in V_{s} is also 25 % .$

Topic Question Set

Q 1 :

The magnetic moment of a bar magnet is 0.5 $A m^{2}$ . It is suspended in a uniform magnetic field of $8 \times 10^{- 2}$ T. The work done in rotating it from its most stable to most unstable position is [2024]

Q 2 :

A current of 200μA deflects the coil of a moving coil galvanometer through 60°. The current to cause deflection through $π$ /10 radian is [2024]

Q 3 :

A coil having 100 turns, area of $5 \times 10^{- 3}$ $m^{2}$ , carrying current of 1 mA is placed in a uniform magnetic field of 0.20 T in such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through 90° is ____ μJ. [2024]

Q 4 :

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is $1.5 \times 10^{- 5}$ Tm. The magnetic moment of the dipole is _____. (Given: $μ_{0} / 4 π = 10^{- 7}$ $T m A^{- 1}$ ) [2024]

Q 6 :

A uniform magnetic field of $2 \times 10^{- 3} T$ acts along positive Y-direction. A rectangular loop of sides 20 cm and 10 cm with current of 5A is in Y-Z plane. The current is in anticlockwise sense with reference to negative X axis. Magnitude and direction of the torque is [2024]

Q 8 :

A solenoid having area A and length ' $l$ ' is filled with a material having relative permeability 2. The magnetic energy stored in the solenoid is:: [2025]

Q 9 :

A magnetic dipole experiences a torque of $80 \sqrt{3} Nm$ when placed in uniform magnetic field in such a way that dipole moment makes angle of 60° with magnetic field. The potential energy of the dipole is: [2025]

Q 12 :

The magnetic moments associated with two closely wound circular coils A and B of radius $r_{A} = 10$ cm and $r_{B} = 20$ cm respectively are equal if (where $N_{A}, I_{A}$ and $N_{B}, I_{B}$ are number of turns and current of A and B respectively). [2023]

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Topic Question Set

Q 1 : The magnetic moment of a bar magnet is 0.5 Am2. It is suspended in a uniform magnetic field of 8×10-2 T. The work done in rotating it from its most stable to most unstable position is [2024]

Q 2 : A current of 200μA deflects the coil of a moving coil galvanometer through 60°. The current to cause deflection through π/10 radian is [2024]

Q 3 : A coil having 100 turns, area of 5×10-3 m2, carrying current of 1 mA is placed in a uniform magnetic field of 0.20 T in such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through 90° is ____ μJ. [2024]

Q 4 : The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is 1.5×10-5 Tm. The magnetic moment of the dipole is _____. (Given: μ0/4π=10-7 TmA-1) [2024]

Q 6 : A uniform magnetic field of 2×10-3T acts along positive Y-direction. A rectangular loop of sides 20 cm and 10 cm with current of 5A is in Y-Z plane. The current is in anticlockwise sense with reference to negative X axis. Magnitude and direction of the torque is [2024]

Q 8 : A solenoid having area A and length 'l' is filled with a material having relative permeability 2. The magnetic energy stored in the solenoid is:: [2025]

Q 9 : A magnetic dipole experiences a torque of 803 Nm when placed in uniform magnetic field in such a way that dipole moment makes angle of 60° with magnetic field. The potential energy of the dipole is: [2025]

Q 11 : A square loop of area 25 cm2 has a resistance of 10 Ω. The loop is placed in uniform magnetic field of magnitude 40.0 T. The plane of loop is perpendicular to the magnetic field. The work done in pulling the loop out of the magnetic field slowly and uniformly in 1.0 sec, will be [2023]

Q 12 : The magnetic moments associated with two closely wound circular coils A and B of radius rA=10 cm and rB=20 cm respectively are equal if (where NA,IA and NB,IB are number of turns and current of A and B respectively). [2023]

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Q 1 :

The magnetic moment of a bar magnet is 0.5 $A m^{2}$ . It is suspended in a uniform magnetic field of $8 \times 10^{- 2}$ T. The work done in rotating it from its most stable to most unstable position is [2024]

Q 2 :

A current of 200μA deflects the coil of a moving coil galvanometer through 60°. The current to cause deflection through $π$ /10 radian is [2024]

Q 3 :

A coil having 100 turns, area of $5 \times 10^{- 3}$ $m^{2}$ , carrying current of 1 mA is placed in a uniform magnetic field of 0.20 T in such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through 90° is ____ μJ. [2024]

Q 4 :

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is $1.5 \times 10^{- 5}$ Tm. The magnetic moment of the dipole is _____. (Given: $μ_{0} / 4 π = 10^{- 7}$ $T m A^{- 1}$ ) [2024]

Q 6 :

A uniform magnetic field of $2 \times 10^{- 3} T$ acts along positive Y-direction. A rectangular loop of sides 20 cm and 10 cm with current of 5A is in Y-Z plane. The current is in anticlockwise sense with reference to negative X axis. Magnitude and direction of the torque is [2024]

Q 8 :

A solenoid having area A and length ' $l$ ' is filled with a material having relative permeability 2. The magnetic energy stored in the solenoid is:: [2025]

Q 9 :

A magnetic dipole experiences a torque of $80 \sqrt{3} Nm$ when placed in uniform magnetic field in such a way that dipole moment makes angle of 60° with magnetic field. The potential energy of the dipole is: [2025]

Q 12 :

The magnetic moments associated with two closely wound circular coils A and B of radius $r_{A} = 10$ cm and $r_{B} = 20$ cm respectively are equal if (where $N_{A}, I_{A}$ and $N_{B}, I_{B}$ are number of turns and current of A and B respectively). [2023]