A uniform magnetic field of 3000 G is established along the positive z-direction. A rectangular loop of sides 10 cm and 5 cm carries a current of 12 A. What is the torque on the loop in the different cases shown in Fig. 4.28? What is the force on each case? Which case corresponds to stable equilibrium?
Magnetic field strength, B = 3000 G = 3000 × 10−4 T = 0.3 T
Length of the rectangular loop, l = 10 cm
Width of the rectangular loop, b = 5 cm
Area of the loop,
A = l × b = 10 × 5 = 50 cm2 = 50 × 10−4 m2
Current in the loop, I = 12 A
Now, taking the anti-clockwise direction of the current as positive and vise-versa:
(a) Torque, $\vec{\tau}=I \vec{A} \times \vec{B}$
From the given figure, it can be observed that A is normal to the y-z plane and B is directed along the z-axis.
$\therefore \tau=12 \times\left(50 \times 10^{-4}\right) \hat{i} \times 0.3 \hat{k}$
$=-1.8 \times 10^{-2} \hat{j} \mathrm{Nm}$
The torque is $1.8 \times 10^{-2} \mathrm{~N} \mathrm{~m}$ along the negative $y$-direction. The force on the loop is zero because the angle between $A$ and $B$ is zero.
(b) This case is similar to case (a). Hence, the answer is the same as (a).
(c) Torque $\tau=I \vec{A} \times \vec{B}$
From the given figure, it can be observed that A is normal to the x-z plane and B is directed along the z-axis.
$\therefore \tau=-12 \times\left(50 \times 10^{-4}\right) \hat{j} \times 0.3 \hat{k}$
$=-1.8 \times 10^{-2} \hat{i} \mathrm{Nm}$
The torque is $1.8 \times 10^{-2} \mathrm{~N} \mathrm{~m}$ along the negative $x$ direction and the force is zero.
(d) Magnitude of torque is given as:
$|\tau|=I A B$
$=12 \times 50 \times 10^{-4} \times 0.3$
$=1.8 \times 10^{-2} \mathrm{~N} \mathrm{~m}$
Torque is $1.8 \times 10^{-2} \mathrm{~N} \mathrm{~m}$ at an angle of $240^{\circ}$ with positive $x$ direction. The force is zero.
(e) Torque $\tau=I \vec{A} \times \vec{B}$
$=\left(50 \times 10^{-4} \times 12\right) \hat{k} \times 0.3 \hat{k}$
= 0
Hence, the torque is zero. The force is also zero.
(f) Torque $\tau=I \vec{A} \times \vec{B}$
$=\left(50 \times 10^{-4} \times 12\right) \hat{k} \times 0.3 \hat{k}$
= 0
Hence, the torque is zero. The force is also zero.
In case (e), the direction of $I \vec{A}$ and $\vec{B}$ is the same and the angle between them is zero. If displaced, they come back to an equilibrium. Hence, its equilibrium is stable.
Whereas, in case (f), the direction of $I \vec{A}$ and $\vec{B}$ is opposite. The angle between them is $180^{\circ}$. If disturbed, it does not come back to its original position. Hence, its equilibrium is unstable.
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- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,