Dipole moment of a rotating disk

[VVS2000]

Homework Statement

This question has two parts:
1. Find the dipole moment of a rotating disk if radius r, having uniform surface charge density and angular velocity w(omega)
2. Plot the magnetic field lines. If a thin circular loop is placed in the same plane as the disk at a distance d>>R from the disk and constrained to spin about the axis corresponding to the line joining the two centres of disk and loop, predict the subsequent motion of the loop

Relevant Equations

dipole moment u=IA
Angular velocity w=2*pi/T

I could do the first part of the question with ease but second part I am not sure how to proceed. Should we calculate the magnetic field at d(where the loop is) and infer something from that for it's motion?? Plz help me out
Thanks in advance

 

[Steve4Physics]

2. Plot the magnetic field lines. If a thin circular loop is placed in the same plane as the disk at a distance d>>R from the disk and constrained to spin about the axis corresponding to the line joining the two centres of disk and loop, predict the subsequent motion of the loop
.
Should we calculate the magnetic field at d(where the loop is) and infer something from that for it's motion?? Plz help me out

Are you sure the question is complete/accurate? For example, we are not told if the 'loop' is charged/uncharged/insulator/conductor. Also, it is not 100% clear if the loop's centre is able to move w.r.t. the disc - though I guess a fixed position for the loop's centre is implied by use of the term 'constrained'.

If the only degree of freedom for the loop is rotating about the specified axis (which is a diameter of the loop), I guess your first decision is to choose: rotation or no rotation. What do you think?

 

[VVS2000]

Are you sure the question is complete/accurate? For example, we are not told if the 'loop' is charged/uncharged/insulator/conductor. Also, it is not 100% clear if the loop's centre is able to move w.r.t. the disc - though I guess a fixed position for the loop's centre is implied by use of the term 'constrained'.

If the only degree of freedom for the loop is rotating about the specified axis (which is a diameter of the loop), I guess your first decision is to choose: rotation or no rotation. What do you think?

This is the question if it clarifies some of the questions you had.
And yes, it has to either rotate or not rotate. But how would I explain it's final motion? Like what will be the reason behind it?
Also, how would one proceed to plotting a magnetic field of a rotating disk?

 

[Steve4Physics]

This is the question if it clarifies some of the questions you had.
And yes, it has to either rotate or not rotate. But how would I explain it's final motion? Like what will be the reason behind it?
Also, how would one proceed to plotting a magnetic field of a rotating disk?

Aha! There is critical additional information in the full question which you didn’t supply:
- the loop is wire (conducting);
- the loop is carrying a current (I);
- the loop is at an initial angle ($\phi$) relative to the plane of disc.

These are important points! Why did you leave them out?!

Do you know the underlying principle of how a simple DC motor works? If not, it will help if you read-up on this.

As for “plotting a magnetic field of a rotating [charged] disk”, well, you’ve already calculated the dipole moment of the rotating disc. What does this tell you about the shape of the magnetic field around the disc?

 

[kuruman]

Another important clue is that the second loop is placed at distance d >> R. To me this says that the two loops can be seen as interacting dipoles. One can find the B-field at the loop due to the rotating disk. Knowing that, one can find the force and restoring torque for small angular displacements.

 

[VVS2000]

Sorry about the incomplete question.
Yes I know how dc motors and generators work. How would that link here?
So what I got is magnetic dipole moment in the direction if angular velocity, so the field lines are in the opposite direction to this right?

 

[VVS2000]

Another important clue is that the second loop is placed at distance d >> R. To me this says that the two loops can be seen as interacting dipoles. One can find the B-field at the loop due to the rotating disk. Knowing that, one can find the force and restoring torque for small angular displacements.

So If I find B at a distance d I can find torque from it right??

 

[Steve4Physics]

Yes I know how dc motors and generators work. How would that link here?

The coil in a DC motor experiences a torque because the coil carries a current and is in a magnetic field. The torque makes the coil rotate. In this question, the loop will experience a torque for the same reasons. (But it's not the same as a DC motor as there is no commutator.) It's up to you to work out what affect the torque will have on the loop.

So what I got is magnetic dipole moment in the direction if angular velocity, so the field lines are in the opposite direction to this right?

No. Each field line is a loop (the term 'loop' should not to be confused with the wire loop). The inner part of each loop is in the same direction as $\vec {\omega}$ and the outer part of each loop is in the opposite direction. The point is, that the magnetic field produced by the rotating charged disc is simply the field of a magnetic dipole. Look up the field of a magnetic dipole using Google images if you don't know how to what it looks like.

 

[VVS2000]

Look up the field of a magnetic dipole using Google images if you don't know how to what it looks like.

Yeah I think I got it
The upper and lower faces of the disc act as the poles right??

 

[Steve4Physics]

Yeah I think I got it
The upper and lower faces of the disc act as the poles right??

It's best not to think in terms of 'poles'. Magnetic field lines don't start and end anywhere; they are continuous loops. (Unlike electric field lines - which start and end on positive and negative charges.)

Look at this diagram for a simple bar-magnet for example: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/bar.gif

And this for a coil: https://www.pngwing.com/en/free-png-sxzwh

 

[kuruman]

Yeah I think I got it
The upper and lower faces of the disc act as the poles right??

I don't know how you did part (a) but the standard procedure would be to find the dipole moment of a charged disk of radius $r$ spinning with angular speed $omega$ and then supeimpose the dipole moments of all such rings from zero to $R$. Given that the distance of the other ring is $d>>R$, you can treat the disk as a point magnetic dipole. There is no benefit to thinkng of upper and lower faces because they are so far away that they coalesce. There is no "upper" and "lower". "All you have to work with is the dipolar field.

 

[VVS2000]

I don't know how you did part (a) but the standard procedure would be to find the dipole moment of a charged disk of radius $r$ spinning with angular speed $omega$ and then supeimpose the dipole moments of all such rings from zero to $R$. Given that the distance of the other ring is $d>>R$, you can treat the disk as a point magnetic dipole. There is no benefit to thinkng of upper and lower faces because they are so far away that they coalesce. There is no "upper" and "lower". "All you have to work with is the dipolar field.

Yeah that's how I did it. Please excuse usage of physics terms. I am trying to not sound so misleading when it comes to describing similar physics problems. I will try not to sound like that and be a bit more specific in the future

 

[VVS2000]

Thanks steve4physics and kuruman for all your help and inputs! Really appreciate it

 

[VVS2000]

Sorry to bother you fellas again but While I was working out the torque, for the given condtion d>>r for a rotating disk, the magnetic field goes to zero.
B= (r^2+2d^2)/sqrt(r^2+d^2) - 2d
Clearly for d>>r the magnetic field is almost zero. So the current loop won't experience a restoring torque right??

 

[kuruman]

Clearly for d>>r the magnetic field is almost zero.

It's not that clear. I am not sure you have an expression that agrees with mine mainly because I have not done part (a) yet. Whatever the correct expression, I would define variable $\epsilon=R/d$ first, then do a Taylor expansion and keep the leading term. The two are far away from each other but mathematically infinitely far away. That will give you the "almost" in "almost zero."

There is a quicker way: Look up the magnetic dipolar field due to magnetic moment $\vec m$ which you know from part (a) and plug it in.

 

[Steve4Physics]

Sorry to bother you fellas again but While I was working out the torque, for the given condtion d>>r for a rotating disk, the magnetic field goes to zero.
B= (r^2+2d^2)/sqrt(r^2+d^2) - 2d
Clearly for d>>r the magnetic field is almost zero. So the current loop won't experience a restoring torque right??

The torque is not necessarily small. The charge and/or $\omega$ could be very large.
I guess the ‘d>>R’ condition is so that:
a) you can use a simple approximate expression for the field strength a distance d from the dipole;
b) you can treat the field through the wire loop as uniform.
The question should also say that a<<d (‘a’ being the radius of the wire loop).