Physics - Electromagnetism - Electric field energy and density



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


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 


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!


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