Holding More Electrons

- Increasing the supply voltage (potential difference which charges the capacitor) means the capacitor can be charged up to a higher voltage, as more electrons can be crammed on to one plate
- Making the plates closer together (which will involve making the dielectric thinner) will allow their fields to interact more strongly, holding more electrons to the plate
- Making the plates larger (giving them a greater area) is another way to make the capacitor store more charge
- Consider a collection of many different capacitors, all connected to the same battery voltage
- The best capacitors will have thin, very insulating, dielectrics and large plates quite close together, as discussed above
- They will be able to hold more electrons than other capacitors, even though they’re at the same voltage
- The number of electrons which can be stored when at a certain voltage is set by the capacitance of the capacitor
- If you read the labels on the best capacitors from the aforementioned selection, you’d find that they were the ones with the highest capacitances

Charge and Capacitance

- To work out the capacitance of a capacitor, divide the charge it can store by the voltage it needs to charge it up to that charge
- The symbol for charge is Q, and it’s measured in Coulombs (or C, for short)
- Each electron has a charge of -1.6*10
^{-19}C, so the charge held by the capacitor is directly related to the number of electrons held (and an easier way of writing it, since the numbers aren’t so large) - The symbol for capacitance is C, and it’s measured in Farads (F, for short)
- A capacitor with a capacitance of 1F could hold 1C of charge, if charged at a potential difference of 1V
- That would actually make it massive by normal standards – most capacitors are measured in micro-Farads (μF), nano-Farads (nF) or even pico-Farads (pF)
- The charge a capacitor can store is its capacitance multiplied by the charging potential difference. In other words:
- Q = VC

Energy

- If you plotted a graph with the charging potential difference on the x axis and the charge held by the capacitor on the y axis, it’d have a straight line
- The gradient of the line would be the capacitance, and it would go through the origin
- (Imagine Q = VC as y = mx + c, with Q as y and V as x)
- Remember the definition of potential difference? Energy per unit charge…
- Multiplying potential difference by charge gives you the energy
- On a graph of V and Q, this is the area under the line
- Since it’s a straight line graph, this area is a triangle, with area (base * height) / 2
- E = 1/2 QV
- E is the energy the capacitor is storing; V is the potential difference; Q is the charge

Current and Stuff

- As you know, current is the rate of flow of charge, Q/t
- This means that charge is current multiplied by time
- Q = I t
- Q is charge (Coulombs), I is current (Amps) and t is time (seconds)
- You now have a lot of simple equations which you can substitute into each other to eliminate unknowns
- Substituting Q = VC into E = 1/2 VQ shows that E = 1/2 CV
^{2} - If you plotted potential difference against energy, you’d a parabola
- The graph of E against V
^{2}would be a straight line, with 1/2 C as its gradient - Don’t try to memorize all the possible graphs
- If you come across a graph you’re not familiar with, have a look at what’s on the axes, and see whether there’s an equation linking those two variables. Once you find one, you may be able to work out what the gradient and the area represent

Use in Timing

- Capacitors take different amounts of time to charge and discharge through different resistors
- Because of this, circuits containing capacitors and resistors can be used to create time delays in electronic devices
- Increasing the capacitance or the resistance makes the charging or discharging take longer
- Different combinations of resistors and capacitors (‘RC Circuits’) can be compared by comparing their time constants
- The time constant of an RC Circuit is RC. (The resistance of the resistor multiplied by the capacitance of the capacitor)
- The symbol for the time constant is τ (the Greek letter Tau)
- That means the equation is τ = RC
- The units of the time constant work out as seconds

Time Constant

- The time constant (RC) is also the time taken for the charge to fall to 0.37 times its original value
- That’s 0.37Q
_{0}if Q_{0}is the charge held by the capacitor before it starts to discharge - Another way of remembering it is Q
_{0}/e, as 0.37 = 1/e^{1} - If charging was linear and not exponential, the time constant would be the amount of time the capacitor takes to fully discharge
- In reality, the discharge rate decreases as the capacitor discharges (as explained in Part 1), so the capacitor only manages to discharge down to about 0.37Q
_{0}in that time - The equation for a capacitor discharging is exponential (as will be described in the next post), so capacitors never ought to fully discharge, but in practise they do
- (Just like how radioactive substances eventually defy their equation and fully decay)
- It’s important that the time constant is in SECONDS (not minutes, hours, days or anything. Those are fine for a radioactive decay constant, but not for an RC circuit time constant!

Data, Formulae and Relationships Booklet

- τ = RC is in the formula booklet
- You’ll just have to remember that 0.37Q
_{0}is reached after RC seconds - Q = VC is in the formula booklet, written as C = Q/V
- E = 1/2 QV is in the formula booklet (they’ve even added that E = 1/2 CV
^{2}) - It may help to remember Ohm’s Law (V = IR), which isn’t in there
- Q = It is in the formula booklet, written as I = Δq/Δt
- Combine to create more cool equations!
- (Batteries not included. Equations sold separately)

There’s a third part to this post, as well!

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