# Lecture 3: Gauss's Law

## Physics II — Electromagnetism, Optics & Modern Physics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Define electric flux and calculate it for uniform and non-uniform electric fields
2. State Gauss's law and explain its physical meaning
3. Choose appropriate Gaussian surfaces based on the symmetry of charge distributions
4. Apply Gauss's law to calculate electric fields for spherical, cylindrical, and planar symmetry
5. Describe the behavior of electric fields in and around conductors in electrostatic equilibrium

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## Lecture Content

### I. Electric Flux

Electric flux (Phi_E) measures the "flow" of the electric field through a surface. For a uniform field E passing through a flat surface of area A, the flux is Phi_E = E dot A = EA cos(theta), where theta is the angle between E and the outward normal to the surface, and A = A n_hat is the area vector whose magnitude is A and whose direction is the outward normal.

The flux is at its maximum when E is perpendicular to the surface (theta = 0), giving Phi_E = EA. It is zero when E is parallel to the surface (theta = 90 degrees), because no field lines pass through the surface. For a non-uniform field or a curved surface, the flux must be calculated using the integral form: Phi_E = integral of E dot dA over the surface. The SI units of electric flux are N m^2/C (equivalently V m). Flux can be positive, indicating field lines exiting through the surface, or negative, indicating field lines entering.

<image>Three panels illustrating electric flux. Panel A: A uniform electric field E passing perpendicularly through a flat rectangular surface, with the area vector A parallel to E; flux is maximum (Phi = EA). Panel B: The same field passing through the surface tilted at angle theta to the field; flux is EA cos(theta). Panel C: The surface is parallel to E so the area vector is perpendicular to E; flux is zero. Each panel shows the field lines, the surface, the normal vector n_hat, and the angle theta clearly labeled.</image>

### II. Gauss's Law

**Gauss's law** states that the total electric flux through any closed surface equals the net charge enclosed divided by epsilon_0: Phi_E = closed integral of E dot dA = Q_enclosed / epsilon_0. A **Gaussian surface** is any hypothetical closed surface used to apply this law. It is a mathematical construct, not a physical object, and its shape is chosen to exploit the symmetry of the charge distribution.

Several key points are essential for applying Gauss's law correctly. Only the charge **inside** the Gaussian surface contributes to the total flux through that surface. Charges outside the surface contribute zero net flux because every field line that enters must also exit. However, the electric field E appearing in the integral is the **total** field from all charges, both inside and outside the surface. Gauss's law is always true, but it is only practical for calculating E when the charge distribution has enough symmetry that E can be pulled out of the integral. Gauss's law is one of Maxwell's four equations and stands as a fundamental law of electromagnetism.

### III. Applying Gauss's Law: Spherical Symmetry

For a **point charge** or a uniformly charged sphere viewed from outside, the natural Gaussian surface is a concentric sphere of radius r. By symmetry, E is constant in magnitude and radially directed over the entire Gaussian surface. The flux is therefore E(4 pi r^2) = Q / epsilon_0, which gives E = Q / (4 pi epsilon_0 r^2) = kQ / r^2. This result recovers Coulomb's law and proves that the field outside a uniform sphere is identical to that of a point charge at the center.

For a **uniformly charged solid insulating sphere** of charge Q and radius R, the field outside (r > R) is E = kQ / r^2, the same as a point charge. Inside the sphere (r < R), the enclosed charge is Q_enclosed = Q(r^3/R^3), since the charge is proportional to the volume enclosed. The field inside is therefore E = kQr / R^3, which increases linearly with r and equals zero at the center.

For a **thin spherical shell** of charge Q and radius R, the field outside (r > R) is E = kQ / r^2, while the field inside (r < R) is exactly zero because the shell encloses no charge when the Gaussian surface is inside it. All of the field "lives" outside the shell.

<image>A graph of electric field magnitude E versus distance r from the center for three cases, stacked vertically. Top: A point charge — E falls off as 1/r^2 everywhere. Middle: A uniformly charged insulating solid sphere of radius R — E increases linearly from 0 to kQ/R^2 for r < R, then falls off as 1/r^2 for r > R, with a clear peak at r = R. Bottom: A thin conducting spherical shell of radius R — E = 0 for r < R, then jumps to kQ/R^2 at r = R and falls off as 1/r^2 for r > R. Each plot is clearly labeled with expressions for E in each region.</image>

### IV. Applying Gauss's Law: Cylindrical Symmetry

For an **infinite line of charge** with linear charge density lambda, the appropriate Gaussian surface is a coaxial cylinder of radius r and length L. The flux through the curved surface is E(2 pi r L), while the flux through the end caps is zero because E is perpendicular to their outward normals. Setting E(2 pi r L) = lambda L / epsilon_0 gives E = lambda / (2 pi epsilon_0 r) = 2k lambda / r. The field points radially outward and falls off as 1/r.

For an **infinite cylindrical conductor** of radius R carrying charge per unit length lambda, the field outside (r > R) is E = lambda / (2 pi epsilon_0 r), identical to a line charge. Inside the conductor (r < R), the field is zero because all the charge resides on the surface.

For a **coaxial cable** with charge per unit length +lambda on the inner conductor and -lambda on the outer conductor, the field between the conductors is E = lambda / (2 pi epsilon_0 r). Both inside the inner conductor and outside the outer conductor, the field is zero.

### V. Applying Gauss's Law: Planar Symmetry

For an **infinite plane of charge** with surface charge density sigma, the Gaussian surface is a "pillbox" cylinder that straddles the plane. The flux passes through the two end caps, giving 2EA = sigma A / epsilon_0, so E = sigma / (2 epsilon_0). The field is uniform, directed away from a positive sheet on both sides, and independent of distance from the plane.

For **two parallel planes** carrying charges +sigma and -sigma, the fields from each sheet add between the planes to give E = sigma / epsilon_0, and cancel outside the planes to give E = 0. This is the ideal parallel plate capacitor configuration and plays a central role in the study of capacitance.

### VI. Conductors in Electrostatic Equilibrium

A conductor in electrostatic equilibrium exhibits several important properties. First, **E = 0 everywhere inside** the conductor; if the field were nonzero, the free charges would move, contradicting the assumption of equilibrium. Second, **all excess charge resides on the surface**, a consequence that follows from E = 0 inside and Gauss's law. Third, the **electric field at the surface is perpendicular** to the surface, because any tangential component would drive surface currents. Fourth, the field just outside the surface has magnitude E = sigma_local / epsilon_0, where sigma_local is the local surface charge density. Fifth, **charge accumulates at points of high curvature** (sharp points), leading to very high fields at sharp tips. This is the operating principle behind lightning rods.

**Electrostatic shielding**, also known as the Faraday cage effect, is a direct consequence of these properties. A hollow conductor shields its interior from external electric fields: the interior field is zero regardless of what external charges are present. This principle is used in shielded cables, microwave oven screens, and MRI rooms to protect sensitive equipment or prevent radiation leakage.

<image>A cross-section diagram of an irregularly shaped conductor in electrostatic equilibrium. The interior is labeled E = 0. Plus signs (+) are distributed on the outer surface, with higher density at regions of greater curvature (sharper points). Short arrows at several points on the surface show the electric field vectors perpendicular to the surface. A Gaussian surface (dashed line) is drawn just inside the conductor surface, enclosing no net charge, illustrating why the interior field must be zero.</image>