Consider N point charges located at the vertices of a regular N-sided polygon

 Consider N point charges located at the vertices of a regular N-sided polygon. The radius of the circle that passes through all of these charges is R. The total charge on all of the points is Q, and each point has charge Q/N. Take the center of the circle to be the origin of your coordinate system, one of the
vertices to be along the x-axis, and the z-axis to be perpendicular to the plane of the polygon.

(1) Write down the expression for the electric field at any point in space produced by one of the point charges.
(2) Write down the expression for the electric field at the center of the circle produced by one of the point charges.
(3) Write down the expression for the electric field at any point in space produced by all of the point charges.
(4) Write down the expression for the electric field at the center of the circle produced by all of the point charges.
(5) Is the electric field at the center of the circle zero when N is even ? Is the electric field at the center of the circle zero when N is odd ?
(6) Write down the expression for the electric potential at any point in space produced by one of the point charges.
(7) Write down the expression for the electric potential at the center of the circle produced by one of the point charges.
(8) Write down the expression for the electric potential at any point in space produced by all of the point charges.
(9) Write down the expression for the electric potential at the center of the circle produced by all of the point charges.
(10) Is the electric potential at the center of the circle zero when N is even ? Is the electric potential at the center of the circle zero when N is odd ?
(11) Show that the gradient of your potential is equal to your electric field.

Part B. Consider the continuous generalization of the previous problem: Total charge Q is uniformly distributed around a circle of radius R. Take the center of the circle to be the origin of your coordinate system and the z-axis to be perpendicular to the plane of the circle.

(1) Write down the expression for the electric field at the center of the circle.
(2) Write down the expression for the electric potential at the center of the circle.
(3) Show, in the limit N goes to infinity, that your results for the polygon become equal to your results for the charged ring.