A circular loop of wire with area A lies in the x y -plane....

A circular loop of wire with area $A$ lies in the $x y$ -plane. As viewed along the $z$ -axis looking in the $- z$ -direction toward the origin, a current $I$ is circulating clockwise around the loop. The torque produced by an external magnetic field $\vec { B }$ is given by $\vec { \tau } = D ( 4 \hat { \imath } - 3 \hat { \jmath } ) ,$ where $D$ is a positive constant, and for this orientation of the loop the magnetic potential energy $U = - \vec { \mu } \cdot B$ is negative. The magnitude of the magnetic field is $B _ { 0 } = 13 D / I A$ . (a) Determine the vector magnetic moment of the current loop. (b) Determine the components $B _ { x } , B _ { y }$ and $B _ { z }$ of $\vec { B } .$

A circular loop of wire with area $A$ lies in the $x y$ -plane. As viewed along the $z$ -axis looking in the $- z$ -direction toward the
origin, a current $I$ is circulating clockwise around the loop. The
torque produced by an external magnetic field $\vec { B }$ is given by $\vec { \tau } = D ( 4 \hat { \imath } - 3 \hat { \jmath } ) ,$ where $D$ is a positive constant, and for this orientation of the loop the magnetic potential energy $U = - \vec { \mu } \cdot B$ is
negative. The magnitude of the magnetic field is $B _ { 0 } = 13 D / I A$ .
(a) Determine the vector magnetic moment of the current loop.
(b) Determine the components $B _ { x } , B _ { y }$ and $B _ { z }$ of $\vec { B } .$

Key Concepts

Magnetic Potential Energy

The potential energy associated with a magnetic dipole in a magnetic field is given by U = -? · B. This expression reflects that the dipole has minimum potential energy when it is aligned with the field, which is important for understanding the stability of the dipole’s orientation.

Vector Decomposition in Cartesian Coordinates

Decomposing vectors into their Cartesian components (x, y, and z) is essential for analyzing physical systems in multiple dimensions. This allows for the application of vector operations, such as cross products, to determine relationships between various physical quantities like torque, magnetic moment, and magnetic field.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that characterizes the magnetic strength and orientation of a current loop. It is defined as ? = I A n, where I is the current, A is the area of the loop, and n is a unit vector perpendicular to the loop’s plane determined by the right?hand rule. This concept underpins the interaction between current loops and magnetic fields.

Magnetic Torque on a Dipole

A magnetic dipole in a magnetic field experiences a torque given by the vector cross product ? = ? × B. This torque tends to align the dipole with the magnetic field and its magnitude and direction are determined by the magnitudes of ? and B as well as the sine of the angle between them.

Transcript

00:01 Okay, so we're going to look ahead at this problem.

00:05 I'm just drawn the circular loop with the current clockwise as given.

00:11 So it says we're looking down the negative x -axis.

00:15 So that means that positive z -axis is pointing out of the page, and we have the x and y quartets.

00:22 These are somewhat arbitrary, but i just drew them anyway.

00:25 Okay, so the first thing we need to know is what is the vector magnetic moment of this? well, so we know that the magnitude of magnetic moment is equal to the flow of current times the area of the loop.

00:44 But of course we need to know the direction for the vector.

00:49 So because the loop points in the x, y plane, the normal vector to this points along the z axis.

00:58 So it will either be plus or minus the k axis.

01:03 For this.

01:04 We just need to determine which one now.

01:09 And so actually that's kind of simple.

01:11 Because if you know your right -hand rule, you know that in order to determine the direction, if i curl my fingers in the direction of the current, my thumb will point in the direction of the magnetic moment vector.

01:26 So although i can't do this on the recording, if i curl my fingers in the direction of i, that means that this points into the page.

01:38 In other words, it points in the minus z direction.

01:42 So for part a, it's just equal to i times a times the minus k hat vector.

01:50 So that will be my solution for part a.

01:55 Okay, so using the result from part a, we can now use some of the other information given in the problem.

02:01 So for example, we know that the torque which is provided by the magnetic field to this magnetic loop, has the following form.

02:16 It has the form some constant d times 4i minus 3j.

02:27 Well, we also know that we can write the torque on a magnetic moment due to a magnetic field as the vector magnetic moment cross with the vector magnetic field.

02:44 And of course, while we know what this first part is, we know what this is, and so that will allow us to solve for information for the magnetic field.

02:53 Now we don't know what the individual components are, but that's fine.

02:56 Let's just write this out.

02:58 So i have, in this case, i have minus i -a -k as my vector for the magnetic moment.

03:10 And i'm just going to write this out as bxi plus b, y, j, plus b, z, k.

03:21 And now if we remember the vector identities, if i take k cross i, that will give me j.

03:29 So if i'm going to write this out, this will give me minus i -a -b -x, and that will give me the j direction.

03:42 And when i do k cross j, that will give me minus ihat.

03:47 So i have minus and minus...