Electromagnetic Waves, Antennas, and RF: Gauss’s Law

EM06 introduced electric flux through a closed surface,

(1)

For a sphere centered on a point charge, the Coulomb field gave the striking result

(2)

independent of the sphere’s radius. Gauss’s Law generalizes that result to every closed surface and every charge distribution:

(3)

Here Q_enc means the net charge enclosed by the closed surface. Positive and negative enclosed charges contribute algebraically.

Gauss’s law is one of Maxwell’s equations. Its great practical power comes from combining a universal physical law with symmetry. The law itself is always true, but solving for the Electric Field from it is easy only when the geometry lets the flux integral simplify [1, 2, 3, 5].

1 Closed surfaces and enclosed charge

A Gaussian surface is not a physical membrane. It is an imaginary closed surface chosen to help analyze an electric field.

For every closed surface, the area vector points outward:

(4)

Gauss’s law relates the net outward electric flux to the total charge enclosed by that surface:

(5)

Charge outside the surface can change the electric field at points on the surface, but it contributes zero net flux through the complete closed surface.

1.1 What does not matter to the right-hand side

The right-hand side depends only on enclosed charge. It does not depend on:

• the size of the Gaussian surface,
• the shape of the Gaussian surface,
• where an enclosed charge sits inside the surface,
• charges located outside the surface.

These facts concern the total flux. They do not imply that the field magnitude is constant over an arbitrary surface.

2 The centered point charge revisited

Consider a point charge q and choose a spherical Gaussian surface of radius r centered on the charge.

The electric field is radial:

(6)

The outward area vector is also radial:

(7)

Therefore,

(8)

Because E is constant everywhere on the sphere,

(9)

The area of a sphere is 4πr², so

ΦE	= E(4πr2)	(10)
	= (4πr2)	(11)
	= .	(12)

Thus,

(13)

3 Gauss’s law is always true; symmetry makes it useful

Gauss’s law does not require symmetry. However, suppose we try to use it to solve for E:

(14)

To pull E outside the integral, we usually need to know that its magnitude is constant over the relevant part of the surface. We also need to know the angle between E and dA.

The most useful symmetry classes are:

1. spherical symmetry,
2. cylindrical symmetry,
3. planar symmetry.

Each symmetry suggests a Gaussian surface whose shape matches the field geometry.

4 Spherical symmetry

For a spherically symmetric charge distribution, the field at distance r from the center must point radially and its magnitude can depend only on r:

(15)

Choose a sphere of radius r. Then

(16)

Gauss’s law gives

(17)

Therefore,

(18)

The important quantity is the charge enclosed within radius r.

5 Example 1: field outside a uniformly charged sphere

Suppose a sphere of radius R contains total charge Q distributed spherically symmetrically. Find the field for r > R.

A Gaussian sphere of radius r > R encloses the full charge:

(19)

Therefore,

(20)

Outside the distribution, the field has the same form as that of a point charge Q at the center.

6 Example 2: field inside a uniformly charged solid sphere

Let a solid sphere of radius R carry total charge Q uniformly throughout its volume. Its volume charge density is

(21)

For r < R, the Gaussian sphere encloses only the charge inside radius r:

(22)

Substituting the density,

(23)

Gauss’s law gives

(24)

Hence

(25)

Inside a uniformly charged solid sphere, the field grows linearly with radius.

7 Cylindrical symmetry

Now consider an ideal infinite line of charge with constant linear charge density λ:

(26)

By cylindrical symmetry, the electric field must point radially away from the line and depend only on the perpendicular distance s from the line:

(27)

Choose a cylindrical Gaussian surface of radius s and length L, coaxial with the line charge.

On the curved side, E is parallel to the outward normal, so

(28)

On the two flat end caps, the normals point along the cylinder axis while the field points radially sideways. Therefore,

(29)

on the caps.

The curved area is

(30)

The enclosed charge is

(31)

Gauss’s law becomes

(32)

Therefore,

(33)

8 Example 3: numerical field of an ideal line charge

Let

(34)

and evaluate the field at

(35)

Using

(36)

we obtain approximately

(37)

For positive λ, the field points radially away from the line.

9 Planar symmetry

Consider an ideal infinite sheet with uniform surface charge density σ:

(38)

Symmetry requires the electric field to be perpendicular to the sheet. The magnitude must be the same on both sides.

Choose a thin cylindrical pillbox whose flat faces have area A and lie parallel to the sheet.

There is no flux through the curved side because the field is tangent to that surface. The field is perpendicular to both flat caps, so each cap contributes

(39)

The total flux is therefore

(40)

The enclosed charge is

(41)

Gauss’s law gives

(42)

Thus,

(43)

Unlike the point-charge field and the line-charge field, the ideal infinite-sheet field does not decrease with distance.

10 Example 4: field of two oppositely charged infinite sheets

Suppose two parallel infinite sheets carry equal and opposite surface charge densities +σ and −σ.

Each sheet produces magnitude

(44)

Between the sheets, the two fields point in the same direction and add:

(45)

Outside the pair, the two fields point in opposite directions and cancel:

(46)

This idealized result is the basis for the approximately uniform field between large parallel capacitor plates away from their edges.

11 Choosing a useful Gaussian surface

A Gaussian surface is useful when symmetry lets us answer two questions in advance:

1. What direction must E point?
2. On what parts of the surface is the field magnitude constant?

A useful rule of thumb is:

12 A Gaussian surface does not create symmetry

One of the most common mistakes is to choose a convenient sphere or cylinder around an asymmetric charge distribution and then assume the field must be constant on that surface.

The logic is the opposite:

(47)

Choosing a sphere does not make an asymmetric electric field spherically symmetric.

13 Example 5: why Gauss’s law may not solve for the field

Suppose a single point charge lies off-center inside a spherical Gaussian surface.

Gauss’s law still gives

(48)

However, the field magnitude is not constant over the sphere and the field is not everywhere normal to the spherical surface. Therefore the integral cannot be simplified to

(49)

Gauss’s law gives the total flux immediately, but it does not by itself give the field at each point on that sphere.

14 External charges and zero net flux

Consider a closed surface containing no charge:

(50)

Gauss’s law says

(51)

This does not mean

(52)

on the surface. External charges may produce a substantial electric field. Their field lines enter the surface somewhere and leave elsewhere so that the signed flux cancels.

15 Example 6: uniform external field through a closed box

Place a closed rectangular box in a uniform electric field

(53)

The right face has outward normal +x and contributes positive flux

(54)

The left face has outward normal −x and contributes

(55)

The other four faces have zero flux. Thus

(56)

This is consistent with

(57)

16 From the integral law to the differential law

EM03 introduced divergence as local outward flux per unit volume. The divergence theorem states

(58)

Gauss’s law gives

(59)

If the charge density is ρ(r), then

(60)

Therefore,

(61)

Because this must hold for arbitrary volumes,

(62)

This is the differential form of Gauss’s law.

17 Physical meaning of the differential form

The equation

(63)

says that electric charge is a local source or sink of the electric field.

Where

(64)

the divergence is positive. Where

(65)

the divergence is negative. In a charge-free region,

(66)

so

(67)

This last equation will become especially important when electromagnetic waves are studied in source-free space.

18 Example 7: infer charge density from divergence

Suppose an electric field in a region is

(68)

where a is a constant.

The divergence is

∇⋅ E	= + +	(69)
	= a + a + a	(70)
	= 3a.	(71)

Using differential Gauss law,

(72)

Therefore,

(73)

19 Integral and differential forms describe the same law

The two forms are

(74)

and

(75)

The integral form relates a complete closed surface to the total charge inside it. The differential form describes the same physics locally, point by point.

The divergence theorem connects them.

20 Common mistakes

20.1 Assuming Gauss’s law requires symmetry

It does not. Gauss’s law is always true. Symmetry is required only if we want the flux integral to simplify enough to solve easily for the field.

20.2 Using total charge instead of enclosed charge

The right-hand side is

(76)

not the total charge everywhere in the problem.

20.3 Assuming zero flux means zero field

A closed surface can have

(77)

while the field is nonzero everywhere on the surface.

20.4 Forgetting the outward normal

For closed surfaces, dA always points outward.

20.5 Pulling E outside the integral without justification

The step

(78)

requires E to be constant over the relevant surface region.

20.6 Choosing a Gaussian surface first and inventing symmetry afterward

The charge distribution determines the symmetry. The Gaussian surface should be chosen to exploit that existing symmetry.

21 Why Gauss’s law matters later in this series

Gauss’s law is not merely a shortcut for electrostatics. It is one of Maxwell’s equations:

(79)

Later, in source-free space,

(80)

so electromagnetic waves satisfy

(81)

This condition helps explain why freely propagating plane electromagnetic waves are transverse.

The same surface-integral language also reappears in magnetic flux, Faraday’s law, electromagnetic energy flow, antenna apertures, and radiated power.

Summary

The essential results of EM07 are:

• Gauss’s law is

• It is valid for every closed surface.
• Symmetry is not required for the law to be true, but it is usually required to solve easily for E.
• Spherical symmetry suggests a spherical Gaussian surface.
• Cylindrical symmetry suggests a coaxial cylindrical Gaussian surface.
• Planar symmetry suggests a pillbox Gaussian surface.
• Charge outside a Gaussian surface contributes zero net flux through that complete surface.
• The differential form is

• The divergence theorem connects the integral and differential forms.

The next lesson, EM08, introduces electric current and current density, moving from static charge toward the sources that will eventually generate magnetic fields and electromagnetic radiation.

References