Question

A nonconducting disk has a radius R, carries a uniform surface charge density s, and rotates with angular speed w.

(a) Consider an annular strip that has radius ?, width ??, and charge ??. Show that the

current produced by this strip is ?? = ?????.

(b) Show that the net magnetic field at the center of the disk is ?)???⁄2.

(c) Find the magnetic field on the axis of the disk, a distance z from the center.

Answer 1

(a) Consider an annular strip that has radius ?, width ??, and charge ??. Show that the current produced by this strip is ?? = ?????.

• Let us imagine that the strip has a radius = r
• and the width = dr
• If the surface charge density is σ,
• then the ring contains a small charge = dq

which is the area enclosed in the ring, times the surface charge density.

Now, the ring has an area

• The current is :

where dt = time it takes for the ring to go around in an orbit.

In terms of the orbital frequency dt = 1/f,we kow the dq and dt values so substitute in dI

we know that

Therefore current produced by this strip is ?? = ?????.

b.) Show that the magnetic field strength at the center of the disk is (1/2)????.

Now, as we have seen, the magnetic field at a distance z away from a ring of radius r is

we know the dI value so substitute in above equation

Interegrate the above equation with respect to "z " from r = 0 to r = R

at the center, we simply set z = 0 in our above expression

(c) Find the magnetic field on the axis of the disk, a distance z from the center

To find the field at the center, simply set z = 0 in our above equation,

so we already done in part-(b) is