A thin dielectric disk with radius a has a total charge +Q distributed uniformly over it...

A thin dielectric disk with radius a has a total charge +Q distributed uniformly over its surface (Fig. 28.30). It rotates n times per second about an axis perpendicular to the surface of the disk and passing through its center. Find the magnetic field at the center of the disk.

Solution Guide

Identify and Set Up

1. Think of the rotating disk as a series of concentric rotating rings. Each ring acts as a circular current loop that produces a magnetic field at the center of the disk.

2. Use the results of Section 28.5 to find the magnetic field due to a single ring. Then integrate over all rings to find the total field.

Execute

3. Find the charge on a ring with inner radius r and outer radius r + dr (Fig. 28.30).

4. How long does it take the charge found in step 3 to make a complete trip around the rotating ring? Use this to find the current of the rotating ring.

5. Use a result from Section 28.5 to determine the magnetic field that this ring produces at the center of the disk.

6. Integrate your result from step 5 to find the total magnetic field from all rings with radii from r = 0 to r = a.

Evaluate

7. Does your answer have the correct units? 8. Suppose all of the charge were concentrated at the rim of the disk (at r = a). Would this increase or decrease the field at the center of the disk?