In general, the contribution to the total electric field `E` from an infinitesimal piece of charge `dE` is
The total field is then calculated by integrating over all the charge:
Where `k=1/(4pi epsilon_0)` and `r` is the distance from `dq` to the point `P` .
For this problem, if the point `P` is a distance `z` above the ring then we can set up a triangle to get a relationship between `z` and the distance `r`.
The vector `dE` pointing at point `P` from a chunk of charge, can be broken up into two components, one in the radial direction and one in the vertical direction.
When considering the total electric field contribution from the ring, the radial contribution of every piece of charge will sum to zero and only the `z` contribution will be left due to the symmetry of the ring.
`lambda` is the piece of charge divided by a piece of length along the ring.
From the fact that `r*theta` is equal to the arc length around a circle we can say that
`E_z=int (k*z*lambda dl)/r^3`
`E_z=int (k*z*lambda R d(theta))/r^3`
Now substitute for `r` and integrate theta from `[0,2pi]` .
`E_z=int_0^(2pi) (k*z*lambda R d(theta))/(sqrt(z^2+R^2))^3`
`E_z=(k*z*lambda R)/(sqrt(z^2+R^2))^3 int_0^(2pi) d(theta) `
`E_z=(2pi k*z*lambda*R )/(sqrt(z^2+R^2))^3`
Therefore the electric field at point `P` is entirely in the `z` -direction with magnitude: