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

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

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

$Work done, W = Δ U = 2 M B$

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

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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(C) $i \propto θ (angle of deflection)$

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

$\Rightarrow i_{2} = 60 μ A$

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]

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(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]

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(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]

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(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]

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

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]

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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]

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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]

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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]

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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]

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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]

0%

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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]