# Intro - The Capacitor and the Circuit $q = CV$ Note that when the main power supply is unplugged, the electrons stored in a capacitor are going to travel backwards - acting as a battery! When we have two capacitors in series, the total capacitance can be found as a reciprocal summation %%diagram of 2 capacitors in series%%: $\frac{1}{C_{T}} = \frac{1}{C_{1}} + \frac{1}{C_{2}}$ When the two capacitors are in parallel, the total capacitance can be found as a summation: $C_{T} = C_{1}+C_{2}$ ## Discharging a Capacitor We can find the rate of discharge of a capacitor by measuring the voltage across a resistor (which can be represented as any device, such as your CPU!) %%diagram%% $V = V_{0}(1-e^{-\frac{t}{CR}})$ $Q = Q_{0}(1-e^-{\frac{t}{CR}})$ $I = I_{0}(1-e^{-\frac{t}{CR}})$ therefore we can isolate the time it takes for the capacitance to reach a certain level for whatever quantity as: $t = CR\ln \frac{V_{0}}{V}$ >[!Success]- Finding the Time - The Maths >$\frac{V}{V_{0}} = 1 - e^{t/CR}$ ## Charging a Capacitor TBA Capacitors charge - that's how batteries work! To find the rate of charging, just $$ ## The Time Interval The denominator of the coefficient $CR$ is known as the *time interval*, or $\tau$ (tau). It's a cool way to simplify things, giving us: $V = V_{0}(1-e^{t/{\tau}})$ If this 'time constant' is equal to the time (so $t/\tau$ is 1) then we get a value for the voltage as **37%** of the original voltage $V_0$. Delighting! # Permittivity as a function of Capacitance # Case Study - The Exploding Capacitor This is why your macbook battery tends to get a little hot - when a capacitor gets overloaded and electron density is high, extra electrons end up going into the wire, making it get hot. TBA desc # Example Questions # What's Next?