A bar magnet with magnetic dipole moment 0.54 A ·m^2 lies on the negative x axis, as shown in Fi

A bar magnet with magnetic dipole moment 0.54 A ·m^2 lies on...

A bar magnet with magnetic dipole moment $0.54 \mathrm{A} \cdot \mathrm{m}^{2}$ lies on the negative $x$ axis, as shown in Figure $17.98 .$ A compass is located at the origin. Magnetic north is in the negative $z$ direction. Between the bar magnet and the compass is a coil of wire of radius $3.5 \mathrm{cm},$ connected to batteries not shown. The distance from the center of the coil to the center of the compass is $9.7 \mathrm{cm}$. The distance from the center of the bar magnet to the center of the compass is $22.5 \mathrm{cm} .$ A steady current of $0.96 \mathrm{A}$ runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the coil and the bar magnet, the compass points north. (a) Which end of the bar magnet is closer to the compass? (b) List all of the following statements that are correct: (1) At the location of the compass, the magnetic field due to the magnet is equal in magnitude to the magnetic field due to the coil. (2) It isn't necessary to take the magnetic field of the magnet into account, since the coil is in between the magnet and the compass. (3) Because not all parts of the magnet are the same distance from the compass, treating the magnet as a dipole located at the center of the magnet is an approximation. (4) The magnetic dipole moment of the coil is equal to the magnetic dipole moment of the compass. (5) At the location of the compass, the net magnetic field is equal to the magnetic field of the Earth. (c) How many turns of wire are in the coil?

A bar magnet with magnetic dipole moment $0.54 \mathrm{A} \cdot \mathrm{m}^{2}$ lies on the negative $x$ axis, as shown in Figure $17.98 .$ A compass is located at the origin. Magnetic north is in the negative $z$ direction. Between the bar magnet and the compass is a coil of wire of radius $3.5 \mathrm{cm},$ connected to batteries not shown. The distance from the center of the coil to the center of the compass is $9.7 \mathrm{cm}$. The distance from the center of the bar magnet to the center of the compass is $22.5 \mathrm{cm} .$ A steady current of $0.96 \mathrm{A}$ runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass.
Despite the presence of the coil and the bar magnet, the compass points north. (a) Which end of the bar magnet is closer to the compass? (b) List all of the following statements that are correct: (1) At the location of the compass, the magnetic field due to the magnet is equal in magnitude to the magnetic field due to the coil. (2) It isn't necessary to take the magnetic field of the magnet into account, since the coil is in between the magnet and the compass. (3) Because not all parts of the magnet are the same distance from the compass, treating the magnet as a dipole located at the center of the magnet is an approximation.
(4) The magnetic dipole moment of the coil is equal to the magnetic dipole moment of the compass. (5) At the location of the compass, the net magnetic field is equal to the magnetic field of the Earth. (c) How many turns of wire are in the coil?

Key Concepts

Dipole Approximation

The dipole approximation simplifies the analysis of magnetic fields by treating an extended object as if all its magnetic moment were concentrated at a single point. This approximation is valid when the observation point is sufficiently far from the magnet relative to its size, though it may introduce errors if the distances are comparable.

Compass Operation

A compass operates by aligning itself with the local magnetic field, indicating the direction of the net field. It responds to the combined effect of various magnetic fields, making it a sensitive tool for detecting and analyzing magnetic interactions.

Superposition of Magnetic Fields

When multiple sources of magnetic fields are present, the net magnetic field at any point is determined by the vector sum of the individual fields. This principle is essential in analyzing systems where different magnetic sources interact, such as when a magnet and a current loop both contribute to the field experienced by a compass.

Magnetic Dipole Moment

A magnetic dipole moment is a vector quantity that represents the strength and orientation of a magnet or current loop. It characterizes how an object interacts with external magnetic fields, both in terms of the field it produces and the torque it experiences when placed in another magnetic field.

Magnetic Field Due to a Dipole

The magnetic field produced by a dipole decreases with distance and is dependent on the alignment between the dipole moment and the point of observation. Its analytical form is often used to model the field of extended magnets when observations are made at points far from the magnet relative to its size.

Magnetic Field of a Current Loop

A current loop generates a magnetic field that is strongest at its center and along the axis perpendicular to the plane of the loop. The field strength is directly related to the number of turns, the current flowing through the loop, and its geometric dimensions such as the radius.

Transcript

00:01 Show that the energy of a dipole in the magnetic field b is given by u equals to minus m dot b for b part show that the interaction energy of two magnetic dipoles magnetic diapoles separated by the displacement r is given by u equals 1 nu not by 4 pi 1 by r cubed m1 .m2 minus 3m1 .r.

00:45 Rcaf m2 dot rcap m2 dot rcap.

00:50 Third part is express your answer to be in terms of angle theta 1 and theta 2 and use the result to find the stable configuration two diapoles would adopt.

01:04 And the part is suppose you had a large collection of compass needles mounted on the pins at regular intervals along a straight line, how would they point, assuming the earth's magnetic field can be neglected? the magnetic field is a vector and the direction of the magnetic field is from north pole to south.

01:33 The magnetic dipole is a combination of north and south pole separated by a small distance up.

01:43 If the dipole is placed in a uniform magnetic field, the south pole of dipole turns opposite to the field, opposite to field and the north of dipole is in the direction of field.

02:01 In field, the force experienced by both the poles equal, which constitutes the torque, the torque rotates the dipole, the dipole, made to align in the field direction of the magnetic field.

02:28 Hence the construction and magnetic dipole shows the direction of the magnetic field.

02:33 For first part, the torque n on the dipole of diapol moment m in the magnetic field b makes an angle of theta.

02:45 So the torque is given by m b sine theta.

02:48 The work done in rotating the dipole through a small angle d theta, then dw is tau d theta, which can be written as mb sine theta, d theta.

03:05 The total work done in rotating the diapol from its initial position, theta 1 to theta 2 is given by w.

03:18 Integration from theta 1 to theta 2 mb sine theta d theta which will be equals to mb cos theta 1 minus cos theta 2 if u theta 1 and u theta 2 be the potential energies potential energies of the dipole at position theta 1 and theta 2, then this work done accounts for the change in potential energy of the diapoline rotating it from theta 1 to theta 2.

04:01 So the work done w is given by u theta 2 minus u theta 1 which is equal to minus of mb cause theta 2 minus cos theta 1 which can also be written as minus of m dot b.

04:24 From this we can conclude that the potential energy of a dipole at position theta is minus m .b.

04:32 The diagram following shows two magnetic dipole separated by a distance are the diagram is shown...

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