Gauss’s Law

1. Charge Density (Starting Point)

In real situations, charge is spread over a body, not concentrated at a single point. To describe this distribution, we define charge density.

(a) Linear Charge Density (λ)

• Charge distributed along a line (e.g., wire)

• Definition:

$$\lambda =\frac{dq}{dl}$$

• Unit: C m⁻¹

(b) Surface Charge Density (σ)

• Charge spread over a surface (e.g., spherical shell)

• Definition:

$$\sigma =\frac{dq}{dA}$$

• Unit: C m⁻²

(c) Volume Charge Density (ρ)

• Charge distributed throughout a volume

• Definition:

$$\rho =\frac{dq}{dV}$$

• Unit: C m⁻³

From volume charge density,

$$dq=\rho dV$$

This idea of distributed charge is essential for Gauss’s Law.

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2. Electric Flux (Key Idea Behind Gauss’s Law)

Electric flux measures how much electric field passes through a surface.

For a small area element dA:

$$d\mathrm{\Phi }=\vec{E}\cdot d\vec{A}$$

For a complete surface:

$$\mathrm{\Phi }=\oint \vec{E}\cdot d\vec{A}$$

• If field lines pass outward, flux is positive

• If field lines pass inward, flux is negative

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3. Statement of Gauss’s Law

The total electric flux through any closed surface is equal to

$\frac{1}{{\epsilon }_0}$

​ times the total charge enclosed by the surface.

Mathematically:

$$\boxed{{\displaystyle \oint \vec{E}\cdot d\vec{A}=\frac{Q_{\text{enclosed}}}{{\epsilon }_0}}}$$

where

${\epsilon }_0$ = permittivity of free space

$Q_{\text{enclosed}}$ = total charge inside the closed surface

This result follows directly from the inverse-square nature of Coulomb’s law and the concept of electric field and flux (see NCERT discussion in Chapter Electric Charges and Fields

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4. Gauss’s Law in Terms of Charge Density

If charge is distributed continuously:

$$Q_{\text{enclosed}}=\int \rho dV$$

So Gauss’s Law becomes:

$$\boxed{{\displaystyle \oint \vec{E}\cdot d\vec{A}=\frac{1}{{\epsilon }_0}\int \rho dV}}$$

This form is very important for theoretical understanding.

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5. Why Gauss’s Law Is Powerful

Gauss’s Law is useful only when symmetry is high, such as:

• Spherical symmetry

• Cylindrical symmetry

• Planar symmetry

In such cases, E is constant over the surface, making calculations easy.

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6. Applications of Gauss’s Law

(A) Electric Field Due to an Infinitely Long Straight Charged Wire

Charge density: λ
Gaussian surface: cylindrical

By symmetry:

• E is radial

• Same magnitude everywhere on curved surface

Flux:

$$\mathrm{\Phi }=E(2\pi rl)$$

Charge enclosed:

$$Q=\lambda l$$

Using Gauss’s Law:

$$E(2\pi rl)=\frac{\lambda l}{{\epsilon }_0}$$

$$\boxed{{\displaystyle E=\frac{\lambda }{2\pi {\epsilon }_{0}r}}}$$

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(B) Electric Field Due to an Infinite Plane Sheet of Charge

Surface charge density: σ
Gaussian surface: pillbox

Flux:

$$\mathrm{\Phi }=2EA$$

Charge enclosed:

$$Q=\sigma A$$

Applying Gauss’s Law:

$$2EA=\frac{\sigma A}{{\epsilon }_0}$$

$$\boxed{{\displaystyle E=\frac{\sigma }{2{\epsilon }_0}}}$$

Important result:
Electric field is independent of distance.

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(C) Electric Field Due to a Uniformly Charged Spherical Shell

(i) Outside the shell (r > R):

$$\boxed{{\displaystyle E=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}}}$$

Behaves like a point charge at the centre.

(ii) Inside the shell (r < R):

$$\boxed{{\displaystyle E=0}}$$

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7. Key Takeaways (Exam-Friendly)

• Gauss’s Law connects electric field and charge directly

• Works best for highly symmetric charge distributions

• Flux depends only on enclosed charge, not on shape

• Charges outside the surface do not affect net flux