Introduction

What this section does

• This is a recall section. It re-states Gauss' law, studied in Std XI, so that the rest of the chapter can use it.
• Gauss' law relates the electric field on a closed surface to the total charge enclosed by that surface, and so gives an alternative route to $E$ that avoids adding up Coulomb forces.

The law

$$\phi = \oint \vec{E}\cdot\vec{ds} = \dfrac{q}{\varepsilon_0}$$

• $\phi$ is the total electric flux coming out of a closed surface and $q$ is the total charge inside that closed surface. The imaginary closed surface used is called a Gaussian surface.
• It is a powerful tool only when the charge distribution has enough symmetry — linear, cylindrical or spherical — for $E$ to have the same magnitude everywhere on a well-chosen Gaussian surface.

The five steps the board expects

• Identify the charge distribution and its density: linear ($\lambda$), surface ($\sigma$) or volume.
• Visualise a Gaussian surface justified by the symmetry of that distribution.
• Obtain the flux from Gauss' law — call this equation (A).
• Obtain the flux again by direct calculation, treating $E$ as the unknown and using the geometry of the Gaussian surface — call this equation (B).
• Equate the right-hand sides of (A) and (B) and solve for $E$.