A cylindrical shell of length L, radius R, and mass M is uniformly charged on its surface with total charge Q, and is rotating with an angular speed of about its axis which is oriented along the z-axis, as shown. a) Consider a horizontal thin ring-like region on the surface of shell with thickness dz, as shown and calculate the infinitesimal current di due to the rotating charge on this ring, in terms of dz. b) Calculate the infinitesimal magnetic moment du due to this current. c) Calculate the infinitesimal magnetic field dB, on the z-axis of cylinder at the point P at the distance of d above the center of cylinder (d>L/2). d) Integrate this dB over dz, to find the magnetic field B at point P. Also find B at very far away. e) Integrate du over dz, to find total magnetic momentu. f) Show that is related to the angular momentum of the rotating sphere. Show the relation. g) Use this u to calculate the B filed at point P when this point is very far away (d>> L & R).
A cylindrical shell of length L, radius R, and mass M is uniformly charged on its surface with total charge Q, and is rotating with an angular speed of about its axis which is oriented along the z-axis, as shown. a) Consider a horizontal thin ring-like region on the surface of shell with thickness dz, as shown and calculate the infinitesimal current di due to the rotating charge on this ring, in terms of dz. b) Calculate the infinitesimal magnetic moment du due to this current. c) Calculate the infinitesimal magnetic field dB, on the z-axis of cylinder at the point P at the distance of d above the center of cylinder (d>L/2). d) Integrate this dB over dz, to find the magnetic field B at point P. Also find B at very far away. e) Integrate du over dz, to find total magnetic momentu. f) Show that is related to the angular momentum of the rotating sphere. Show the relation. g) Use this u to calculate the B filed at point P when this point is very far away (d>> L & R).
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Transcribed Image Text:### Rotating Charged Cylindrical Shell Physics Problem
A cylindrical shell of length \( L \), radius \( R \), and mass \( M \) is uniformly charged on its surface with total charge \( Q \). It is rotating with an angular speed of \( \omega \) about its axis, which is oriented along the z-axis as shown in the accompanying diagram.
#### Problem and Steps:
**Diagram Explanation:**
The diagram depicts a cylindrical shell rotating around the z-axis. There is a horizontal thin ring-like region on the shell with a thickness \( dz \), which is highlighted to consider the infinitesimal current \( di \).
**a) Infinitesimal Current Calculation:**
- Consider a horizontal thin ring-like region on the surface of the shell with thickness \( dz \). Calculate the infinitesimal current \( di \) due to the rotating charge on this ring in terms of \( dz \).
**b) Infinitesimal Magnetic Moment Calculation:**
- Calculate the infinitesimal magnetic moment \( d\mu \) due to this current.
**c) Infinitesimal Magnetic Field Calculation:**
- Calculate the infinitesimal magnetic field \( dB \) at point \( P \), on the z-axis of the cylinder, at a distance \( d \) above the center of the cylinder (\( d > L/2 \)).
**d) Magnetic Field Integration:**
- Integrate this \( dB \) over \( dz \) to find the magnetic field \( B \) at point \( P \). Also, find \( B \) at a very far distance.
**e) Total Magnetic Moment Integration:**
- Integrate \( d\mu \) over \( dz \) to find the total magnetic moment \( \mu \).
**f) Relationship to Angular Momentum:**
- Show that \( \mu \) is related to the angular momentum of the rotating sphere. Demonstrate the relation.
**g) Calculation of Magnetic Field at Far Point:**
- Use this \( \mu \) to calculate the magnetic field \( B \) at point \( P \) when this point is very far away (\( d \gg L \) and \( R \)).
This problem is a classical exercise in electromagnetic theory and involves understanding the relationships between charge, magnetic fields, and angular momentum in a rotating system.
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