A square loop of wire with side 0.60 m carries a current of 9.0 A as shown in the figure. When

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 figure. 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

Hooke's Law in Rotational Systems

Hooke's law, which states that the force (or in this context, the torque) is proportional to the displacement (or rotational deflection), is extended in rotational systems. When a spring is involved, the restoring torque it produces is proportional to the angular displacement, providing the counterbalance to the magnetic torque in the equilibrium analysis.

Small Angle Approximation

The small angle approximation simplifies trigonometric functions by approximating sine and tangent of small angles as being nearly equal to the angle measured in radians. This simplification is useful in equilibrium calculations, allowing complex torque relationships to be linearized and more easily solved.

Torque Equilibrium in Rotational Systems

For a system to be in equilibrium, the sum of all torques acting on it must be zero. In problems like this one, the balancing of the torque due to the magnetic field and the restoring torque from another source (such as a spring) allows the determination of the equilibrium angle, ensuring the net rotational effects cancel out.

Magnetic Moment of a Current Loop

A current loop acts like a magnetic dipole with a magnetic moment whose magnitude is the product of the current and the area enclosed by the loop. This concept is crucial because the interaction between the magnetic moment and an external magnetic field produces a torque that tends to align the loop with the field direction.

Magnetic Torque on a Dipole

When placed in a magnetic field, the magnetic dipole experiences a torque that is given by the cross product of its magnetic moment and the magnetic field. This torque tries to rotate the loop such that its magnetic moment aligns with the field, and it plays a fundamental role in determining the equilibrium orientation of the loop.

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 the 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.

01:45 In this spring here.

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

02:08 Now a 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 clock by the coil.

02:41 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...

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