Apparent disagreement between Coulomb's Law and Gauss' Law

[shj]

This is probably my misunderstanding, so please clarify.

In a region of empty space, there are two point charges with the charges+Q and -Q. Exactly in the middle of the two charges (distance r from both charges) is point P, colinear with the centers of both charges. A Gaussian surface that includes point P is drawn above.

Using Coulomb's Law, we can find the electric field at point P:
E=2*((1/4πε₀)Q/r²)=(1/2πε₀)Q/r²)
(since the electric field vectors caused by both charges have the same magnitude and add at point P)

However, if I try to use Gauss's Law to calculate the electric field at point P, I get:
E*4πr²=Q_enclosed/ε₀, or
E=(1/4πε₀)Q/r²)
(since the Gaussian surface is symmetric to the electric field, I simplified the surface integral to E*4πr²)

The two calculations differ! Can someone please clarify the error?!

 

[Stavros Kiri]

Simple. The total field is not constant across the Gauss surface. You would really need to do the integral.

In other words, here is your error:

(since the Gaussian surface is symmetric to the electric field, I simplified the surface integral to E*4πr2)

Not symmetric for the total field!

Note: Welcome to PF!

 

[shj]

Simple. The total field is not constant across the Gauss surface. You would really need to do the integral.

In other words, here is your error:

Not symmetric for the total field!

Note: Welcome to PF!

Oh alright. Thank you.

 

[Stavros Kiri]

Oh alright. Thank you.

You're welcome!

 

[Dale]

since the Gaussian surface is symmetric to the electric field,

The electric field is not spherically symmetric

Edit: oops, I am too late. Good job @Stavros Kiri