1) a thin ring made of uniformly charged insulating material has total charge q and radius r. the ring is positioned along the x-y plane of a 3d coordinate system such that the center of the ring is at the origin of the coordinate system. (a) determine an expression for the potential at an arbitrary location along the z-axis in terms of q, r, and z. (b) use this expression to determine an expression for the magnitude of the electric field at an arbitrary location along the z-axis in terms of q, r, and z. hint: apply the technique of charge integration in part (a) and poten

(A) considering the charge "q" evenly distributed, applying the technique of charge integration for finite charges, you obtain the expression for the potential along any point in the Z-axis:

With been the vacuum permittivity

(B) The expression for the magnitude of the E(z) electric field along the Z-axis is:

Explanation:

(A) Considering a uniform linear density on the ring, then:

(1)⇒(2)⇒(3)

Applying the technique of charge integration for finite charges:

(4)

Been r' the distance between the charge and the observation point and a, b limits of integration of the charge. In this case a=2π and b=0.

Using cylindrical coordinates, the distance between a point of the Z-axis and a point of a ring with R radius is:

(5)

Using the expressions (1),(4) and (5) you obtain:

Integrating results:

  (S_a)

(B) For the expression of the magnitude of the field E(z), is important to remember:

(6)

But in this case you only work in the z variable, soo the expression (6) can be rewritten as:

(7)

Using expression (7) and (S_a), you get the expression of the magnitude of the field E(z):

(S_b)