Dipole moment of a rotating disk

In summary, the question is about predicting the motion of a thin circular loop placed in the same plane as a rotating disk, at a distance d>>R from the disk and constrained to spin about the axis joining the two centres. The loop is assumed to be a wire carrying a current and at an initial angle relative to the plane of the disk. The loop can be seen as interacting dipoles with the disk. To predict the subsequent motion of the loop, one must calculate the magnetic field and torque acting on the loop.
  • #1
VVS2000
150
17
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
 

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  • #2
VVS2000 said:
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?
 
  • #3
Steve4Physics said:
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?
 

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  • #4
VVS2000 said:
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?
 
  • #5
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.
 
  • #6
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?
 
  • #7
kuruman said:
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??
 
  • #8
VVS2000 said:
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.

VVS2000 said:
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.
 
  • #9
Steve4Physics said:
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??
 
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  • #11
VVS2000 said:
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.
 
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  • #12
kuruman said:
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
 
  • #13
Thanks steve4physics and kuruman for all your help and inputs! Really appreciate it
 
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  • #14
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??
 
  • #15
VVS2000 said:
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.
 
  • #16
VVS2000 said:
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).
 

1. What is a dipole moment of a rotating disk?

The dipole moment of a rotating disk is a measure of the distribution of electric charge within the disk as it rotates. It is a vector quantity that describes the strength and direction of the electric dipole created by the disk's rotation.

2. How is the dipole moment of a rotating disk calculated?

The dipole moment of a rotating disk can be calculated by multiplying the magnitude of the charge on the disk by the distance between the center of the disk and the axis of rotation. This is represented by the equation μ = qd, where μ is the dipole moment, q is the charge, and d is the distance.

3. What factors affect the dipole moment of a rotating disk?

The dipole moment of a rotating disk is affected by the magnitude of the charge on the disk, the distance between the center of the disk and the axis of rotation, and the speed of rotation. Additionally, the shape and size of the disk can also impact the dipole moment.

4. How does the dipole moment of a rotating disk affect its behavior?

The dipole moment of a rotating disk plays a crucial role in determining the behavior of the disk. It affects the strength of the electric field created by the disk, which in turn can influence the motion of charged particles around the disk and the interactions between the disk and other charged objects.

5. What are some real-world applications of the dipole moment of a rotating disk?

The dipole moment of a rotating disk has various applications in different fields, including electromagnetics, electronics, and fluid dynamics. It is commonly used in electric motors, generators, and turbines to convert mechanical energy into electrical energy. It is also used in particle accelerators and mass spectrometers to manipulate and analyze charged particles.

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