Question

A thin disk with a circular hole at its center, called an annulus, has inner radius R1 and outer radius R2. The disk has a uniform positive surface charge density σ on its surface. (Figure 1)

A)The annulus lies in the yz-plane, with its center at the origin. For an arbitrary point on the x-axis (the axis of the annulus), find the magnitude of the electric field E⃗ . Consider points above the annulus in the figure.

Express your answer in terms of the variables R1, R2, σ, x and appropriate constants.

B)A point particle with mass m and negative charge −q is free to move along the x-axis (but cannot move off the axis). The particle is originally placed at rest at x=0.01R1 and released. Find the frequency of oscillation of the particle.

Express your answer in terms of the variables R1, R2, σ, m, q, and appropriate constants.

please make a clear statement while all constants used

2 2

Answer 1

A thin disc with a circular hole at its center, called an annulus has inner radius R_1 and outer radius R_2 The disk has a uniform positive surface charge density sigma on its surface. (A) The annulus lies in the y - z plane with its center at the origin. Fore an arbitrary point on the x - axis we are to find the magnitude of the electric field (E). step 1: Disk can be considered as a combination of many rings so, lets firstly derive the electric field due to a ring. Lets, determine the field at point P on the axis of the ring. The field is along the x - axis. The perpendicular fields sum to zero, while the differential x - component of the field is - d E_x = k dq/mu^2 cos theta cos theta = base/hypotenuse = x/mu = x/ squareroot x^2 + a^2 = k dq/mu^2 x/(x^2 + a^2)^1/2 = x k dq/(x^2 + a^2)^1 + 1/2