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 )

• Answer to Question 1 (Conceptual outcome)
• Answer to Question 2 (Electric potential )
• Answer to Question 3 (Potential and field near the center)
• Answer to Question 4 (Electron's equation of motion)
• Answer to Question 5 (Explain physical results)

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