Question

Which Gaussian surface would you prefer to choose if you had to calculate electric field due to a charge distribution such that the integration complexity is reduced?

• A. the one which is a sphere around the charged distribution
• B. The one where electric field and area vector has constant angle and electric field magnitude has constant value
• C. the one which has the same direction for area vectors of all infinitesimal elements
• D. the one that has a the same direction of electric field at all infinitesimal elements

Answer 1

The correct option is B

The one where electric field and area vector has constant angle and electric field magnitude has constant value

Flux $\varphi =\oint \overrightarrow{E}.\overrightarrow{ds}$

We need to simplify this integral in order to we Gauss’ law to calculate electric field due to a distribution. It would be nice to get rid of the vector notation.

As we know $\overrightarrow{E}.\overrightarrow{ds}=Edscos\varnothing$and if for all infinitesimal elements the angle between electric field and area vector is constant (say c) then

$\varphi =c\oint Eds$

Now if magnitude of electric field i.e. E also happens to be constant we can take that out of the integration and it would be awesome.

$\varphi =c.E.\oint ds$

Calculating $\oint$is not tough, it basically gives the area of the Gaussian surface.

We should therefore “prefer” a Gaussian surface where the angle between electric field and area vector remains a constant; and electric field magnitude has the same value everywhere.

Or at least a Gaussian surface which can be broken down into parts where these conditions are true.