Question 3: Can I approximate the potential and the electric field at a point p near the center ?
(Yes, use Taylor Series Expansion and the Gradient of the potential).
Answer to Question 3
Taylor Series Expansion . f[y]=f[y0]+f'[y0] (y-y0) / 1! + f[y]'' (y-y0)^2 / 2! + .......
From the answer to Q2, the electric potential at any given point on the y-axis is
k Q Vp = ------------- 2 2 Sqrt[R + y ]
From Vp, the approximate electric potential near the center at y0=0 ,using Taylor Series Expansion up to a second order term is
2 2 k Q (2 R - y ) Vp = ----------------- 3 2 R
The approximate electric field near y=0 is the negative derivative of the electric potential.
k Q y Ep = - dVp/dy = --------- 3 R
(End of Answer to Question 3 )
Websites related to this problem
Expand
your Knowledge (Taylor series, gradient , "Mathematica"
solutions )