8 min read•june 18, 2024
Krish Gupta
Daniella Garcia-Loos
Krish Gupta
Daniella Garcia-Loos
A is a device that can be used to store charge, and therefore, . They are used in a wide range of electrical devices including the flash on your cell phone camera. There are several different ways to construct a capacitor, but we're going to focus on the parallel-plate version.
A capacitor is a device that stores electric charge and energy in an . It consists of two conductors, called plates, separated by an insulating material called the .
Here are some key points about capacitors:
The parallel plate capacitor is created by taking two conductive plates and separating them by a small distance. A dielectric is often added to increase the amount of charge a capacitor can store. We'll discuss more about dielectrics in the next section.
Let's create a simple capacitor using two metal plates and connect them to a battery to charge them up. Recall from Unit 1, that the strength of the electric field is proportional to the amount charge. E=kQ/r^2
We also know that the potential difference (V) between the plates is related to the electric field through ΔV=Ed. Following this thought process, we can see that V∝Q as well. The more charge that gets stored on each plate, the stronger the field, and the higher the voltage between the plates will be. We'll define a new quantity, capacitance(C), as the constant of proportionality between V and Q such that:
The unit for capacitance is the Farad (F), where 1F =1C/1V
Let's derive what capacitance actually is. This derivation is beyond the course but will give you a deeper understanding of circuits.
We can also define capacitance in terms of the physical dimensions of the capacitor. Recall that σ=Q/A (area charge density) for a sheet of charge, and E=σ/ϵ0 for a conductive plate.
This equation is for a capacitor where the plates are separated by air. We're going to tweak this equation a bit in the dielectric section when we discuss different materials to place between the plates.
From here, we can see that capacitance is directly proportional to the area of the plates (A) and inversely proportional to the distance between them. This should make sense since a larger plate has more room for the charge to occupy and, therefore, more should be able to fit on it.
Because the capacitor stores charge, it also stores electric potential energy (UC). The amount of energy stored can be determined through a derivation. However, the derivation requires understanding of integral calculus we will just work with the final product.
Energy in a capacitor is the energy stored in the electric field between the capacitor plates. It is a measure of the potential energy of the electric charges stored on the capacitor plates.
Here are some key points about energy in a capacitor:
The capacitance depends on the area of the plate, which for a circular plate is 2πr^2
Dielectrics are insulating materials that are often used in capacitors to increase their capacitance. They help solve the problem of how to get more charge into a capacitor without having the voltage decrease. C=Aϵo/d. Modifying the equation to include a dielectric involves adding a new term κ, which is the dielectric constant. In general, the easier a material is to polarize, the higher it's dielectric constant is. Values for common dielectrics are shown below:
Material | Dielectric Constant |
Vacuum | 1 |
Air | 1.00059 |
Bakelite | 4.9 |
Fused Quartz | 3.78 |
Neoprene Rubber | 6.7 |
Nylon | 3.4 |
Paper | 3.7 |
Polystyrene | 2.56 |
Pyrex Glass | 5.6 |
Silicon Oil | 2.5 |
Strontium Titanate | 233 |
Teflon | 2.1 |
Water | 80 |
Great question! It's because a dielectric becomes polarized easily. In fact, the easier the dielectric becomes polarized, the greater its κ becomes. Let's look at an image to understand why the polarization helps increase the capacitance.
In image (a), we can see that the molecules of the dielectric become polarized and align opposing the charge on the plates. This produces a layer of opposite charge on the surface of the dielectric that attracts more charge onto the plate, because of Coulomb's Law, increasing its capacitance.
Another way to understand how a dielectric increases capacitance is to look at how it changes the electric field inside the capacitor. Image (b) shows the electric field lines with a dielectric in place. Since some of the field lines end on charges in the dielectric (because the polarity of the dielectric is opposite that of the plates), the overall field between the plates is weaker than if there were a vacuum between the plates, even though the same charge is on the plates.
The voltage between the plates is V=Ed, so it is also reduced by the dielectric. This means there is a smaller V for the same charge Q and since C = Q/V, the capacitance is greater.
** After looking back at 20+ years of FRQs, if you see an FRQ on circuits, there's a really good chance it will have a capacitor in it. You've been warned 🙂**
Capacitors have their own special equations for determining equivalent resistance in series or parallel, just like resistors.
For a , individual capacitors act as one large capacitor storing a large charge (Qtotal = Q1 + Q2 +Q3) resulting in a total capacitance that is simply the sum of the individual values.
A is a bit trickier since the charge is split up along each of the capacitors, but we can derive an expression for this by using the KVL (sum of voltage drops needs to be equal to the battery voltage)
In a DC circuit, an initially uncharged capacitor will begin storing charge on its plates, increasing its potential difference until the voltage of the capacitor is equal to the voltage of the battery or other supply source. At this point, there is no current passing through the capacitor and it acts as an or a break in the wire.
For example, in the circuit below, the current initially flows through both branches, but as the voltage of C1 approaches the battery voltage, less and less current passes through R1. When steady state is reached, the circuit will appear to be a series circuit with only R2 in it.
The is a very common type of capacitor where a and capacitor are connected in series with each other. A switch is used to allow the capacitor to charge (position a) or discharge (position b).
An RC circuit is a type of electrical circuit that contains a resistor and a capacitor connected in series or parallel. RC circuits are used to filter signals, smooth out voltage fluctuations, and discharge stored energy.
Here are some key points about RC circuits:
The cool thing about RC circuits is that the charging and discharging time can be tweaked by changing the values of C and R.
Knowing the exact equations or graphs of RC Circuit equations in not on the exam but will help enhance your understanding. Graphing these functions shows us the changes in V and I as the capacitor charges. Notice how when the steady-state is reached, the current in the capacitor is basically zero.
Now we can make the same sort of graphs as we did for the charging segment.