A square loop of wire of side 3.0 cm carries 3.0 A of current. A uniform magnetic field of mag

A square loop of wire of side 3.0 cm carries 3.0 A of...

A square loop of wire of side $3.0 \mathrm{~cm}$ carries $3.0 \mathrm{~A}$ of current. A uniform magnetic field of magnitude $0.67 \mathrm{~T}$ makes an angle of $37^{\circ}$ with the plane of the loop. (a) What is the magnitude of the torque on the loop? (b) What is the net magnetic force on the loop?

A square loop of wire of side $3.0 \mathrm{~cm}$ carries $3.0 \mathrm{~A}$ of current. A uniform magnetic field of magnitude $0.67 \mathrm{~T}$ makes an angle of $37^{\circ}$ with the plane of the loop.
(a) What is the magnitude of the torque on the loop?
(b) What is the net magnetic force on the loop?

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Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity defined for a current loop and is proportional to the current and the area of the loop. It is given by ? = I · A, where I is the current through the loop and A is the area vector perpendicular to the loop's plane. This concept is fundamental in understanding how current distributions interact with external magnetic fields.

Torque on a Magnetic Dipole

When a magnetic dipole is placed in a magnetic field, it experiences a torque that tends to align the dipole’s magnetic moment with the field. This torque is given by ? = ? × B, with a magnitude of ? = ?B sin?, where ? is the angle between the magnetic moment and the magnetic field. This effect explains how engines and devices like electric motors operate by converting electrical energy into mechanical work.

Net Force on a Current Loop in a Uniform Magnetic Field

A key principle in magnetism is that a current loop in a uniform magnetic field experiences forces on its individual segments, but these forces are equal and opposite, resulting in a net force of zero. This concept is crucial in distinguishing between the translational force and the rotational (torque) effects on magnetic dipoles in uniform fields.

Transcript

00:01 All right, we have a square loop of wire that is going to have a current in it, three amps, and it's placed into a magnetic field of 0 .67.

00:11 It is not perpendicular.

00:14 The angle with the plane of the loop, the angle that the magnetic field makes with the plane of the loop is 37 degrees.

00:27 But when we take the cross product, we don't want to.

00:31 To know that angle.

00:32 When we take the cross product, we are taking the sign of theta, and theta is supposed to be perpendicular to the loop.

00:44 So therefore, if we have the loop here in the magnetic field, let's see, if we have the loop here, we drop a normal.

00:53 So that would be perpendicular to the loop.

00:56 And then the magnetic field would be, let's see, 37 from the plane of.

01:02 Of the loop, 53, so it would be more like this.

01:07 This would be the magnetic field, this would be 37 degrees.

01:11 But inside the equation, we want the one to the normal, which is 53 degrees.

01:20 Alright, so to find the torque, we take the number of loops, which there's just one, multiply by the area.

01:29 It is a square with 0 .03 meters as a.

01:32 Side, so 0 .03 squared times the current, which 3 .0 times the magnetic field, which is 0 .67, times the sign of theta, which is 53...

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