Question

How to choose a Gaussian surface?

Answer 1

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Answer 2

Mathematically the Gaussian surface should be chosen so that |E⃗ ||E→| is constant on the surface so that

∮E⃗ ⋅dS⃗ =∫|E⃗ |dScosθ=|E⃗ |∫dScosθ∮E→⋅dS→=∫|E→|dScos⁡θ=|E→|∫dScos⁡θ

so that evaluate the magnitude of E⃗ E→ from the charged enclosed by your surface.

In practice, choosing dS⃗ dS→ so that |E⃗ ||E→| is constant on the surface amounts to choosing a surface that has the same symmetry as E⃗ E→. Thus, if you can argue that E⃗ E→ will be spherically symmetric, then your surface should be a sphere; if E⃗ E→ has cylindrical symmetry, then the surface should be a cylinder etc.

The symmetry of E⃗ E→ is often dictated by the symmetries of the charge distribution, i.e. a spherically-symmetric charge distribution will produce a spherically symmetric 

Choose the Gaussian surface in such that the electric field at every point on it is constant. Ultimately, you should be looking for symmetries, since it would simplify calculations a lot.

For example, consider finding the magnitude of the electric field due to an infinite thin sheet of charge, having a uniform positive charge density σσ. We choose the Gaussian surface to be a cylinder going into the sheet and out of it. Finding the flux coming out of the Gaussian cylinder and splitting the integral for the flux coming out of the end faces, we have

∮SE⃗ ⋅dS→=∮endfaceE⃗ ⋅dS→+∮endfaceE⃗ ⋅dS→∮SE→⋅dS→=∮endfaceE→⋅dS→+∮endfaceE→⋅dS→

∮SE⃗ ⋅dS→=2∮endfaceEdS=2ES(1)(1)∮SE→⋅dS→=2∮endfaceEdS=2ES

Since E⃗ E→ and dS

$hope \: it \: \: helps \: you$