Gauss Law Class 12: Find Detailed Notes

Gauss’ Law is the most precise mathematical way to understand the relationship between electric charge and the net electric flux (Φ) this charge produces through a surface that is closed. 

Gauss Law Statement in NCERT 

"Electric flux through a closed surface S = q/ε_0

q = total charge enclosed by S.

The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface” (p.30, Electric Charges and Fields).

Think in this way:

The law is quite versatile but always applies to closed surfaces. Their shapes and sizes do not matter though in this context. 

We cannot know how many electric field lines are passing through a closed surface without calculating the electric flux. 

But what we know is that:

• The flux going out the closed surface is positive flux. The flux going inward is negative flux. The first one has a positive sign. The second one has a negative sign. 
• If the number of field lines entering is the same to the ones going out, total flux will be zero.

Gauss Law gives this calculation mathematical form. In fact, two forms: one algebraic and the integral is another. 

Algebraic Form of Gauss Law

Φ = q/ε_0

Integral Form of Gauss Law

${\mathrm{\Phi }}_E=\text{∮}\mathrm{}\mathrm{}\vec{E}\cdot d\vec{A}=\frac{q_{\mathrm{e}\mathrm{n}\mathrm{c}}}{{ϵ}_0}$

where:

${\mathrm{\Phi }}_E$ = electric flux through the closed surface

$\vec{E}$ = electric field

$d\vec{A}$ = area vector on the surface

$q_{\text{enc}}$ = charge enclosed

${ϵ}_0\approx 8.854\times {10}^{-12}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^2$ permittivity of free space.

The Gaussian surface is a construct instead of a tangible one, which makes the calculation of electric flux in relation to the electric charge simple and useful. 

But this non-physical Gaussian surface make calculations easier for these reasons.

• It must be a closed surface that sits on the periphery of a three-dimensional object that has volume. This is how we can understand the calculation whether the point charges are on the inside, outside, or on the surface. 
• This surface must have the electric field perpendicular to it. The area vector that is dA must also always point radially outwards from the surface. 
• The surface must never pass through any discrete charge. It only works for continuous distribution of charges.
• The magnitude of the electric field must have symmetry. 

Types of Symmetry Gaussian Surfaces Must Have

Charges with continuous distributions will have different electric field calculations depending on the symmetries.

If charges are in spherical order, the Gaussian surface will be concentric. 

In cylindrical symmetry, we have a surface that’s coaxial. 

With planar symmetry, there’s a uniform charge distribution, the Gaussian surface like a cylinder that’s perpendicular to the plane. 

To build some more conceptual awareness around Gauss' Law Class 12, we’re listing some properties so that you apply the formulas correctly. 

• The shape and size of the surface does not matter. It must be a closed surface for you to apply Gauss' Law. This is one misconception many have. 
• Charges outside the Gaussian surface do not affect the total electric flux passing through it. 
• When the total electric flux is zero, it does not necessarily imply that the electric field is zero everywhere on the surface. It just means the total charge in the closed surface is zero.

The Gauss Law replaces the Coulomb’s Law to make calculations on electric fields much easier. 

It is especially difficult to use Coulomb’s law formula for electrostatic force when there is a large number of charges spread across any closed surface. 

So if the charge distribution shows any of the three symmetries - spherical, cylindrical, or planar - replace Coulomb’s Law with Gauss' Law.
What happens with this approach is that you will not have to do complex integrals. Instead, you will be using the simpler approach to finding the total electric flux as a product of surface area and electric field strength. 

Gauss' Law is so simple yet a super elegant mathematical concept in physics established by the maths genius Carl Friedrich Gauss over a century ago. But there lies the trickiness, too, especially when approaching competitive tests like JEE Main. 

We get it, the time-pressure to finish and get good marks is really challenging when you’re caught off-guard. Try solving practice problems of old JEE Main papers related to the ones below.

• Calculating electric flux that’s placed at the corner of a cube. 

Logic you must use: Don’t forget to use the logic of charge sharing and symmetry. 

• Finding net electric flux through a sphere when an electric dipole is at its centre.

Logic you must use: Do not confuse zero flux with zero electric field.