## Introduction

Hello it's a me again Drifter Programming!

Today we continue with **Electromagnetism **to get into **other ways of calculating Electric field energy**, now that we covered Capacitance, and we will also get into **Energy density**!

All the mathematical equations will be drawn using quicklatex!

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

## Electric field energy

In the previous posts we defined a new concept, the concept of Capacitance, where the capacity was defined equal to:

where the second part is true for parallel conductive plates!

A way of calculating the potential energy U was using the Work of the electric force F of an electric field E. But, when having a capacitor we can also use the equation *Q = CV* and change the integral for that a little bit!

Let's suppose u and q are elementary (small) changes of potential difference and charge respectively while "charging". Of course: u = q/C.

The Work dW that is needed to move the charge dq is:

The **total Work W** needed to increase the charge q from zero to Q is:

This is of course equal to the total Work produced from the electric field onto the charge q when discharging it from Q to zero.

Supposing the **potential energy of an uncharged capacitor is zero **then** the potential energy and Work of the a charged capacitor are equal**!

The potential difference between the conductive plates is: *V = Q/C* and so **U and W can be expressed as**:

Of course Q is in Coulomb (C), C in Farad (F = C/V), V in Volt (V = J/C) and U in Joule (J).

The last of those equations tells us:

The total work W is equal to the total charge Q that "moved" times half the potential difference V/2 during the duration of a charge.

## Energy density

The stored energy U of a capacitor is of course strongly connected with the electric field E in between of it's conductive plates. We can clearly say that the potential energy is "stored" in the electric field between it's plates.

This is why we define the concept of **energy density u** as:

which calculates the** energy per volume** of the electric field of two parallel conductive plates with Area A and distance d.

We also know that the capacitance of such a capacitor can be calculated using:

The potential difference V and electric field E are binded by:* V = Ed*.

Using those two equation we end up with:

which gives us the **energy density** of this type of capacitor.

It's proven that **this equation applies to any capacitor and electric field E that is in a vacuum space**. During this series we will see that this also applies for non-vacuum areas!

## How all this is useful

**Potential energy equations:**

The equations for electric potential energy give us another way of calculating the energy or other "variables" in this equations, which means that we can calculate 1 of them if we know 2 of the others.

Mostly, we will just use them to calculate the potential energy when knowing two of the variables: charge Q, potential difference V and capacitance C.

**Energy density:**

The equation of energy density and the ones that we get from it are useful when we know how much energy we want to store per cubic-meter in a vacuum and want to calculate the electric field magnitude E that is needed to be able to store that much per cubic-meter.

Sometimes we might even want to calculate the energy density "change" when increasing or decreasing the electric field magnitude E in some way.

Examples around that and the previous stuff will come in the final post of this "chapter" and so after we cover Dielectrics!

## 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

Electric flux and Gauss's law -> Electric flux, Gauss's law

Applications of Gauss's law (part 1) -> applying Gauss's law, Gauss applications

Applications of Gauss's law (part 2) -> more Gauss applications

Electric flux exercises -> examples in electric flux and Gauss's law

Electric potential energy -> explanation of work-energy, electric potential energy

Calculating electric potentials -> more stuff about potential energy, potential, calculating potentials

Equipotential surfaces and potential gradient -> Equipotential surface, potential gradient

Millikan's Oil Drop Experiment -> Millikan's experiment, electronvolt

Cathode ray tubes explained using electric potential -> cathode ray tube explanation

Electric potential exercises (part 1) -> applications of potential

Electric potential exercises (part 2) -> applications of potential gradient, advanced examples

Capacitors (Condensers) and Capacitance -> Capacitors, capacitance, calculating capacitance

How to solve problems around Capacitors -> combination, solving problems, simple example

And this is actually it for today!

Next time we will talk about Dielectric materials!

Bye!