0
$\begingroup$

Imagine you have a disk with some number of charged metallic spheres spaced equally around a disk at some distance from the center of the disk and those spheres are mechanically attached to the disk. And if you have a changing magnetic field pass through through the center of the disk in a direction normal to the surface of the disk, there is a torque that is applied to the disk.

For the sake of simplifying a problem,

  • can you instead imagine a uniformly charged line (whose total charge is equal to that of all the spheres) at the same distance from the center of the disk without changing the value of the torque applied to the disk?

I do not believe the two systems would result in a different torque as you could imagine the entire charge is located at a point at the center of each of the spheres and I do not believe there would be a difference in torque between a discrete and a continuous charge distribution of the same charge as long as it is the same distance from the center.

$\endgroup$

2 Answers 2

1
$\begingroup$

Yes i think you are right. As the magnitude of induced electric field will be constant at a constant distance from centre, the magnitude of force on all these spheres would be same and so will be there contributing torque. Torque will not change if you assume those sphere as line distribution of charge if net charge and mass of distribution is same as net charge and mass of spheres. Even if we assume all of that charge is concentrated at a single point with same distance from centre, the torque will come out to be the same.

Edit: All of the above things are valid only if those spheres has negligible radius as compared to there distance from centre. Otherwise the moment of inertia of resultant systems will be different for both cases. That's because moment of inertia of those spheres about an axis passing through center of disk will be n(Md^2 + 2/5MR^2) where M is mass of one sphere, R is its radius,d is its distance from centre and n will be number of such spheres. But For the line distribution case moment of inertia about same axis will be Md^2 .

$\endgroup$
0
$\begingroup$

I do not think that would apply since the spheres are continuous objects and the wire is just a line.

If your spheres are hollow, they are 2-dimensional surfaces extended to three dimensions. So they are continuous objects.

IF your spheres are solid, then the problem becomes more complex as they are now three-dimensional continuous bodies. It's just like assuming the moment of inertia to be a point particle instead of a continuous object. Suppose you have a sphere swinging about a pivot point and connected to that point by a mass-less rod. You can't assume the sphere is a point particle like you do in introductory physics. In smaller cases that might work and give a very good approximation, but if your object got larger the moment of inertia about the center would matter. Same thing here. For smaller spheres, the torque might not be affected if you approximate with a point charge or a zero-diameter wire, but if your spheres are large then you will have to do some calculus.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.