Numb3rs: Capacitance

Capacitance is the ability of a Capacitor to store charge.

Capacitor

A Capacitor is an electrical component that can store energy in an electric field. They consist at least of two electrical conductors separated by a dielectric (insulator), although many variations can be found.

A potential voltage difference between the two fields will create a static electric field develops across the dielectric. Positive charge will collect on one of the plates and negative charge on the other one.

Charge

the charge stored equals to the Capacitance of the object and the Voltage across the object.

Energy

To derive an expression for how much energy is stored in a capacitor we consider the infinitesimal small voltage due to the infinitesimal little charge \(y\)

Energy is defined as:

The Energy for every small amount of voltage can therefore be expressed as:

At this point we can take our Capacitance expression and plug it in.

Then we can add all these infinitesimal small Energies up to get the total amount of energy:

Capacitance

The Capacitance between two plates can be derived as followed

Electric Field

The Electric Field between two plates:

Voltage

The Voltage between two plates:

Capacitance

The Capacitance between two plates:

Examples

In these examples we are trying to find:

  • \(E\), the Electric Field
  • \(\Delta V = \int_A^B E(r) dr\), the difference in Voltage
  • \(C = \frac{q}{V}\), the Capacitance

Cylinder

__IMAGE_PLACEHOLDER__

Electric Field

The \(\cos\) term will fall away since the Field is perpendicular to the area.

The Area will just add up to the total Area since again all the small Area’s are at the same angle to the Electric Field.

Therefore the Electric Field will be:

Voltage Difference

Voltage is defined as:

Therefore we can define the change in Voltage as

We can plug in our result from earlier to get

We can factor out all the constants to get

Capacitance

Capacitance is defined as

Plug and play

Solving Magic

Sphere

__IMAGE_PLACEHOLDER__

Electric Field

Therefore the Electric Field will be:

Voltage Difference

Same steps as for the cylinder can be applied:

The integral can simply be solved by usage of the power rule on \(r^{-2}\).

Capacitance

Capacitance is defined as

Plug and play

Sources

  • My Notes

The Numb3rs Series

Numb3rs used to be its own subsection on this site, where I wrote math & science tutorials. Unfortunately this never came to frution so I integrated ths articles with the rest of the blog. If you want to check out the other Numb3rs posts, you can get an overview here.

By Cecil Woebker


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