E7. Capacitors

Capacitors are crucial in many aspects of day to day life, from Defibrillators to Energy Storage. Here, we'll briefly explore the physics concepts involved in Capacitors in a digestible manner.

1. Capacitance

1.1. What is Capacitance

Capacitance is a measure of an object’s ability to store an electric charge, based on the Potential Difference acting through the plate. It’s a property specific to each material and will vary between materials. The higher the Capacitance, the greater the amount of energy that the material can store.

1.2. How to Calculate Capacitance

1.2.1. C = q/V → Capacitance = Electrical Charge / Potential Difference

We’ve already explained previously that Potential Difference is the amount of work done by some charge to move through an Electric Field. If more energy is applied to our plates, then they’ll break free from the plates, which means less Charge is stored in our plates. Likewise, if our charge increases then we have more electrical energy which would be stored in our material.

1.2.2. C = ε0 (A / d) → Capacitance = Permittivity of Free Space * Area of Plate / Distance between the Plates

The Permittivity of Free Space is a constant that we previously explored, which tells us how an Electric Field travels through a Vacuum. As for our other values: If our plates get larger, then more charge can be stored in each plate, which increases our Capacitance; And if our distance increases, then the charged particles have to move further to move between plates for use, and so fewer charged particles will be available on each plate.

• To explain this, we can use the equation for calculating an Electric Field is Fq (Force * Charge), which can be rewritten as qEp / d (Charge * Electric Potential Energy / Distance), meaning as our distance grows, our Electric Field weakens. Since our electric field is directly tied to the amount of Charge, a weaker Electric Field means that we have less charge stored between our plates.

1.2.3. E = 0.5C * V^2 → Potential Energy = Half of the Capacitance multiplied by the Potential Difference Squared

This equation relates our Electrical Potential Energy to our Capacitance and Potential Difference, and we can rearrange this to be: C = 2E / (V^2).

• Here, we can see that as our Potential Energy increases, for a given Potential Difference, our Capacitance will increase.

• On the other hand, if we have a greater Potential Difference with a constant amount of Electric Energy between our plates, then the Capacitance will be lower.

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2. The Composition of a Capacitor

The Capacitor, like all other circuit components, requires a closed/complete circuit so that current can flow, and so that charge can be stored in the conducting plates.

The Conducting Plates are responsible for storing the Electric Charge and maintaining a difference in Electric Potential. One plate, for example, will have a very Negative Electric Charge, whilst the other side will have a very positive Electric Charge.

These plates will have the same charge as the Tuble (The +/- Terminal of a Cell) to which they’re connected.

• These plates will have different Electric Potentials, with one having a very High Electric Potential due to it having a large amount of + Charge, whilst the other plate has a lot of - Charge.

This can be calculated as W = qEd (Work Done to move the Charged Particle = Charge * Electric Field * Distance traveled by charge). The distance our charge travels is the same as the distance between the plates.

• This loss in energy, as a result of doing work, will be the same as the change in Electric Potential energy, as any changes in our system will mean that energy is transferred to other systems → The total energy of our system must be conserved.

This means that the change in our Electric Potential Energy is: Ep = -qEd

And because Electric Potential Energy is the same as our Electric Potential multiplied by our charge, we can reorganize to get: V = -Ed

However, in between both plates is an insulative material known as a Dielectric.

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3. Dielectrics

Dielectric Materials are a form of insulate material, placed between the two plates of opposing electric charge to prevent the area between them from growing too hot.

This is important for the Capacitor to work properly and helps with increasing our Capacitance, which means that more energy can be stored between our plates.

• As a very high current passes, there’s an Electric Field present, which can cause electrons to leave the plate, and in turn, remain in this gap in between our Plates, which would reduce its Electric Potential Energy.

• The Dielectric prevents this by reducing the strength of the Electric Field, which helps prevent electrons from jumping.

Recall that Electric Potential Energy is just the amount of energy an object has due to its position in an Electric Field → If our electrons leave the plate, then the amount of Charge stored in our plates will end up decreasing, and as such there will be a lower Capacitance.

The presence of this Dielectric also allows us to put our plates closer together without the electrons jumping, both of which help with increasing our Capacitance as our distance has decreased, and our stored charge has NOT decreased.

Note that Dielectric Materials are Polar → One side has a very slight negative charge, and the other side has a very slight positive charge. This means that the Dielectric Material produces its weak electric field, which works against the Electric Field of the plates, whilst keeping the charge the same.

• In this case, the positive end of our Dielectric Material would line up with the Negative Plate of the Capacitor, and vice versa.

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Recap:

1. Capacitance: A measure of an object’s ability to store Electric Charge, and thus Electric Potential Energy, for a given Potential Difference.

   1. Calculating Capacitance:

      1. C = q/V

      2. C = ε0 (A / d)

      3. E = 0.5C * V^2 → C = 2E / (V^2)

2. Dielectrics: Insulators that are put between plates to reduce the Electric Field between our plates, allowing for plates to be put closer without Charge (Electrons) leaving the plate, which helps increase our Capacitance.

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Additional Resources:

https://byjus.com/physics/dielectric-properties/

https://byjus.com/physics/capacitor-and-capacitance/