# Physics - Electromagnetism - Electric flux and Gauss's law

## Introduction

Hello it's a me again Drifter Programming!

Today's topic of **Physics **in the branch of **Electromagnetism **is **Electric flux and Gauss's law**! I will first get into what we call Electric flux and then will give some information about Gauss's law, a law that we will cover in-depth next time :)

So, without further do, let's dive straight into it!

## Getting into Gauss's law

Gauss's and Coulomb's law are two alternatives that define the correlation between a distribution of charge and the electric field it generates.

**Coulomb's law** of course defines the electric field at a point P generated from a point charge q. When having an extensive distribution of charge we suppose that this distribution is build up of many point charges using the Superposition principle and so "adding" all those field's together.

Using **Gauss's law **on the other hand (that we will cover in today's and the following two posts) we assume that an charge distribution is surrounded by a hypothetical surface. That way we only have to find the electric field at points on top of that surface. Gauss's law is a correlation between all the points of that surface and the total charge that is surrounded by that surface.

## Electric flux

To define Gauss's law we first have to define the **concept **of Electric flux.

The definition of Electric flux contains the **area A** of a surface and an** electric field E** that is applied on the points of the surface.

This surface of course doesn't have to be real! It's an **imaginery surface** in space!

Let's consider a small, flat surface of area A vertically across to an uniform electric field E:

We define that the **electric flux Φ** that passes through the surface A is equal to:

**Φ = EA**

and so the product of the area and electric field.

We can **represent Φ roughly through the elecric field lines** of E that pass through A.

- A larger area A means more lines pass through
- A stronger electric field => denser electric field lines

And so when these two things are true we have more lines/surface-unit.

But, what if the surface is not vertically across to the electric field?

Of course less electric field lines pass through the surface.

So, of importance is the projection of A to an hypothetical vertically across surface to E.

When **having an angle θ**,** the Electric Flux Φ** is:

**Φ = EAcosθ**

where Ecosθ is the component of E that is vertically across to the surface A.

This looks like this:

To make the definition even smaller and to make it more general **we determine the concept of a vector area A**, which is a vector unit with meter the area A of a surface and a direction vertically across to the surface we are describing.

The vector area A now defines the meter of the area and also the orientation in space.

The "new" equation is:

**Φ = EA**

where all variables are now vectors.

All those equations express the concept of Electric Flux in a different, but equivalent way!

By using any of those definions we have that **Electric Flux is calculated in Nm^2/C **in the **SI** metrics system.

So, what remains now?

Well, we covered the "simple" electric flux that is applied to a small, vertical (or with angle θ) surface A from an uniform electric field E.

What if the **electric field is non-uniform** and what if the **surface is part of a curved surface**?

Here we go again...integrals

We of course **split the curved surface into small elementary dA areas** and calculate the electric flux that passes through each of them.

That way **Electric Flux** is calculated through:

where in the first E, dA are scalar units, while in the second E, A are vector units.

Such an integral is of course a **surface integral**, like those that we covered in Mathematical analysis...

A lot of times we represent the direction of an **elementary surface A** using an **unit vector n** that is **vertically across** to that surface.

That way:

**dA = ndA **

where the A on the left is a vector and the A on the right is a scalar unit.

The vertically across vector n can go to any direction and so we have to choose between the "inside or outside" surface.

We always **choose the direction of n that goes to the outside space of the surface **and so talk about an **Flux that goes outwards**.

## Gauss's law

So, what is Gauss's law?

Gauss's lawstates that the total electric flux that goes through any closed surface (with some volume) is proportional to the total charge that this surface contains.

Let's start with an positive point charge q and it's electric field.

The **electric field lines stretch outwards to all the directions radially around** and so by placing the charge in the center of an** imaginary sphere surface of radius R** the meter of the **electric field E** at any point on that sphere is:

At every point of that surface, the electric field** E is vertically across** to the surface and the **meter **is also **equal**, because the distance is the same.

That way the** total electric flux** is simply the product of the electric field's meter and the total area of the sphere which of course if A = 4πR^2 and so:

We can clearly see that:

The electric flux isindependent of the radius Rof the sphere andonly depends on the charge that sphere surrounds.

Using that we can say that:

**What applies to the whole sphere also applies to a part of it**!

When having an **non-sphere surface** we just have to split it up into **small elementary "spheres" of dA** area which means that the total flux will of course be the** surface integral**:

This equation is true for any surface that surrounds the charge q and is closed!

The elementary areas dA and their corresponding unit vectors n of course "tend" outwards and so the electric flux is positive at points where the electric field lines go outside of the surface and negative when they direct inside. Of course the opposite applies for negative source charges!

For any **closed surface that doesn't contain charges** we have:

This is the mathematical equation that shows us that **a surface doesn't contain charges** or has t**he same amount of charges entering and leaving the surface**! So, the **electric field lines can start or end at a specific space area only when there is a charge at that area**.

One last thing remains now...

What about more charges that are being closed of the same (sphere or non-sphere) surface?

Let's suppose those are q1, q2, q3, ..., gn.

The total electric field is of course equal to the vector sum of the electric field's generated by each one of those charges if we look at them individually.

Let's also define the **total enclosed charge** as **Qencl = q1 + q2 + ... + qn**.

By also defining elementary areas dA and the vertically across components of the total electric field E to that areas, we then define the **general Gauss law**:

When **Qencl = 0** then the total flux that passes through the surface must also be zero, even when some areas have positive or negative flux.

All these different equations that we showed for Gauss's law define the same thing, but with different sizes an so **we will choose one instead of the other depending on the problem we have to solve.**

We will get into Applications next time...

## Previous posts about Physics

**Intro**

Physics Introduction -> what is physics?, Models, Measuring

Vector Math and Operations -> Vector mathematics and operations (actually mathematical analysis, but I don't got into that before-hand :P)

**Classical Mechanics**

Velocity and acceleration in a rectlinear motion -> velocity, accelaration and averages of those

Rectlinear motion with constant accelaration and free falling -> const accelaration motion and free fall

Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity

Rectlinear motion exercises -> examples and tasks in rectlinear motion

Position, velocity and acceleration vectors in a plane motion -> position, velocity and accelaration in plane motion

Projectile motion as a plane motion -> missile/bullet motion as a plane motion

Smooth Circular motion -> smooth circular motion theory

Plane motion exercises -> examples and tasks in plane motions

Force and Newton's first law -> force, 1st law

Mass and Newton's second law -> mass, 2nd law

Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction

Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications

Contact forces and friction -> contact force, friction

Dynamics of Circular motion -> circular motion dynamics, applications

Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications

Contact force and friction examples -> exercises in force and friction

Circular dynamic and vertical circle motion examples -> exercises in circular dynamics

Advanced Newton law examples -> advanced (more difficult) exercises

**Electromagnetism**

Getting into Electromagnetism -> electromagnetim, electric charge, conductors, insulators, quantization

Coulomb's law with examples -> Coulomb's law, superposition principle, Coulomb constant, how to solve problems, examples

Electric fields and field lines -> Electric fields, Solving problems around Electric fields and field lines

Electric dipoles -> Electric dipole, torque, potential and field

Electric charge and field Exercises -> examples in electric charges and fields

And this is actually it for today and I hope that you enjoyed it!

Next time we will get into Applications of Gauss's law

Bye!