# Electric Field and Gauss's Law

Physics I · Principles of Electromagnetism and Optics · https://tryals.app/learn/physics-i/electric-field-and-gausss-law

## Electric Fields and Gauss's Law

Instead of tracking forces charge by charge, imagine every charge filling space with an invisible **electric field**: a ready-made instruction at every point saying which way a test charge would be pushed.

The **electric field** $\vec{E}$ is the force per unit charge at a point in space:

$$\vec{E} = \frac{\vec{F}}{q_0} = k_e\frac{Q}{r^2}\hat{r}$$

**Electric field lines** start on positive charges, end on negative charges, never cross, and have a density proportional to field strength.

**Gauss's law** is the field's conservation statement, relating the electric flux through any closed surface to the enclosed charge:

$$\oint_S \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$

Gauss's law is one of **Maxwell's four equations** and is equivalent to Coulomb's law for electrostatics.

## Gaussian Surfaces and Differential Form

**Choosing a Gaussian surface**: Exploit symmetry so that $\vec{E}$ is constant on the surface and easy to pull out of the integral.

| Charge distribution | Gaussian surface | Result |
| --- | --- | --- |
| Point charge | Sphere | $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$ |
| Infinite line ($\lambda$) | Cylinder | $E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$ |
| Infinite plane ($\sigma$) | Pillbox | $E = \dfrac{\sigma}{2\varepsilon_0}$ |

The **differential form of Gauss's law** links the field's divergence to the local charge density $\rho$:

$$\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0}$$

> **Common pitfall**: Zero flux through a closed surface does **not** mean zero field on it. An external charge sends lines in and out; they cancel in the count, yet the local field is non-zero.

## Practice questions

4 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort each Gaussian surface by the symmetry it exploits.

**Answer:**

- Spherical: Point charge or spherical shell, Uniformly charged sphere
- Cylindrical: Infinite line charge, Long charged wire
- Planar (pillbox): Infinite plane of charge, Parallel plate capacitor (between plates)

**Why:** Choose a Gaussian surface that matches the symmetry of the charge distribution.

Page: https://tryals.app/practice/physics-i/electric-field-and-gausss-law/sort-each-gaussian-surface-by-the-symmetry-it-exploits

### 2. A balloon is charged and its charge stays fixed. If you **double the radius** of an imaginary sphere drawn around it, what happens to the total electric **flux** through the sphere?

A. It doubles, because the surface is farther out
B. Nothing, flux depends only on the enclosed charge
C. It quadruples with the surface area
D. It drops to one quarter, like the field strength

**Answer:** B. Nothing, flux depends only on the enclosed charge

**Why:** Flux counts field lines crossing the surface, and every line from the charge crosses any enclosing sphere exactly once. The $1/r^{2}$ field and $r^{2}$ area cancel perfectly, that cancellation **is** Gauss’s law.

Page: https://tryals.app/practice/physics-i/electric-field-and-gausss-law/a-balloon-is-charged-and-its-charge-stays-fixed-if-you-double-the

### 3. Gauss’s law holds for any closed surface, but it only helps compute a field when the surface is chosen well. What makes a good choice?

A. The smallest surface that encloses all the charge
B. A spherical surface, which works for any distribution
C. A surface passing through the point where the field is wanted, whatever its shape
D. One matching the symmetry of the charge distribution

**Answer:** D. One matching the symmetry of the charge distribution

**Why:** Symmetry is what makes the integral trivial. Without it Gauss’s law is still correct and computationally useless, which is why it solves spheres, cylinders and planes and little else.

Page: https://tryals.app/practice/physics-i/electric-field-and-gausss-law/gausss-law-holds-for-any-closed-surface-but-it-only-helps-compute-a

### 4. Order the steps for using Gauss’s law to find the field of an infinite charged line.

**Answer:**

1. Use symmetry: the field must point radially outward, same at equal radii
2. Choose a coaxial cylinder as the Gaussian surface
3. Argue the flat end-caps contribute zero flux
4. Write the side-wall flux as $E \cdot 2\pi r L$
5. Set the flux equal to $\lambda L/\varepsilon_0$ and solve for $E$

**Why:** Symmetry first, surface second, algebra last. The result, $E = \lambda/2\pi\varepsilon_0 r$, falls off as $1/r$, slower than a point charge’s $1/r^{2}$, because fresh charge keeps "coming into view" as you back away from an infinite line.

Page: https://tryals.app/practice/physics-i/electric-field-and-gausss-law/order-the-steps-for-using-gausss-law-to-find-the-field-of-an