Answer:
(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:
![V(z)=\frac{Q}{4\pi (\epsilon_{0}) \sqrt{R^{2} +z^{2}} }](https://tex.z-dn.net/?f=V%28z%29%3D%5Cfrac%7BQ%7D%7B4%5Cpi%20%28%5Cepsilon_%7B0%7D%29%20%5Csqrt%7BR%5E%7B2%7D%20%2Bz%5E%7B2%7D%7D%20%20%7D)
With
been the vacuum permittivity
(B) The expression for the magnitude of the E(z) electric field along the Z-axis is:
![E(z)=\frac{QZ}{4\pi (\epsilon_{0}) (R^{2} +z^{2})^{\frac{3}{2} } }](https://tex.z-dn.net/?f=E%28z%29%3D%5Cfrac%7BQZ%7D%7B4%5Cpi%20%28%5Cepsilon_%7B0%7D%29%20%28R%5E%7B2%7D%20%2Bz%5E%7B2%7D%29%5E%7B%5Cfrac%7B3%7D%7B2%7D%20%7D%20%20%20%20%7D)
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:
![V(z)= 4\pi (\epsilon_{0})\int\limits^a_b {\frac{\lambda_{0}R}{ \sqrt{R^{2} +Z^{2}} }} \, d\phi](https://tex.z-dn.net/?f=V%28z%29%3D%204%5Cpi%20%28%5Cepsilon_%7B0%7D%29%5Cint%5Climits%5Ea_b%20%7B%5Cfrac%7B%5Clambda_%7B0%7DR%7D%7B%20%5Csqrt%7BR%5E%7B2%7D%20%2BZ%5E%7B2%7D%7D%20%20%7D%7D%20%5C%2C%20d%5Cphi)
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)