A square loop of wire with side 0.60 m carries a current of 9.0 A as shown in the side-view diag

A square loop of wire with side 0.60 m carries a current of...

A square loop of wire with side $0.60 \mathrm{m}$ carries a current of $9.0 \mathrm{A}$ as shown in the side-view diagram. When there is no applied magnetic field, the plane of the loop is horizontal and the nonconducting, nonmagnetic spring $(k=550 \mathrm{N} / \mathrm{m})$ is unstretched. A horizontal magnetic field of magnitude $1.3 \mathrm{T}$ is now applied. At what angle $\theta$ is the wire loop's new equilibrium position? Assume the spring remains vertical because $\theta$ is small. [Hint: Set the sum of the torques from the spring and the magnetic field equal to 0.]

Key Concepts

Equilibrium Condition (Torque Balance)

In static equilibrium, the net torque acting on a system must be zero. This means that any torque produced by external influences, such as a magnetic field, is exactly balanced by internal or restoring torques, such as those from a spring. Setting the sum of torques equal to zero allows one to solve for the equilibrium position, typically characterized by a small angle of deviation.

Small-Angle Approximation

The small-angle approximation is a mathematical simplification used when dealing with angles that are very small (measured in radians). In this approximation, trigonometric functions can be approximated by their first-term Taylor series expansions (for example, sin(?) ? ?). This simplification allows for easier analytical solutions in problems involving rotational equilibrium and oscillatory behavior.

Restoring Torque from Elastic Forces

A restoring torque arises in mechanical systems when an elastic element, such as a spring, is deformed from its equilibrium position. According to Hooke’s law, the force (and consequently the torque, when applied at a distance) is proportional to the displacement from equilibrium. This restoring torque acts in the opposite direction to any applied torque, providing a counteracting force that tends to return the system to its original state.

Magnetic Dipole Moment

A magnetic dipole moment is a vector quantity that characterizes a current-carrying loop by the product of the current and the area of the loop. It points perpendicular to the plane of the loop and is a measure of the loop's ability to interact with an external magnetic field, thereby experiencing a torque that tends to align the dipole with the field.

Magnetic Torque on a Current Loop

When a current loop is placed in a magnetic field, it experiences a magnetic torque given by the cross product of the magnetic dipole moment and the magnetic field. This torque tends to rotate the loop so that its dipole moment becomes aligned with the field. The magnitude of the torque is proportional to the product of the dipole moment, the field strength, and the sine of the angle between them.

Transcript

00:01 In the given problem, first of all here this is the side view of a square coil, current carrying coil which is in square shape.

00:19 This coil is perpendicular, is kept perpendicular to the plane of paper.

00:25 In such a manner, the current is coming out of the plane of paper in this left arm and it is getting into the plane of paper.

00:34 In this right arm and then it is kept in a magnetic field in a horizontal magnetic field this is a horizontal magnetic field here this coil starts rotating because it is carrying a current also so it will experience a door and it will start rotating about its center here this is the center now there is a center now there is a soft spring also attached with it like this.

01:18 So we consider, if we consider only the small angular displacement, so this will be the extension in the length of the spring.

01:31 So this torque, suppose the clockwise torque acting on this coil, it will start rotating the coil in clockwise direction, but then a restoring force will be developed in this spring here.

01:48 So this restoring force developed in the spring f, it will create a counterclockwise star on this coil due to which this coil will not be able to rotate further and it will come to an equilibrium.

02:08 Now the side length of this square coil is given as let it be l and that is given as 0 .60 centimeter.

02:27 So first of all we find clockwise torque.

02:31 Clock was taught and this is the maximum clockwise torque experienced by the coil because initially the coil was kept such that its plane was parallel to the magnetic field.

02:47 So it should be experiencing maximum torque.

02:50 So clockwise taught experienced by the coil squared shaped coil due to magnetoid.

03:05 Magnetic field that is given as tau cw clockwise b -i -n -a the magnetic field i current passing through this coil and number of turns in it and a is its area of cross -section and the counterclockwise torque exerted by the screen that will be given as c cw is is equal to force, restoring force developed in the spring will take by its perpendicular distance from the axis of the coil means this l by 2.

04:04 So here it will be l by 2.

04:08 And for the restoring force we know this is given as spring factor k multiplied by the displacement which is delta x into l by 2...

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