6.1 - Capacitors

Core Info and Definitions 
 A capacitor is a circuit component that stores charge in a circuit by separating equal and opposite charges onto two electrical conductors (plates) with an insulator in between them.  
 Capacitance, C, defines the quantity of charge Q which can be stored per unit potential difference across the plates. Measured in farads 
 Capacitance 
 
 ... where epsilon 0 is the permittivity of free space, epsilon r is relative permittivity of the material, A is the surface area of a plate, and d is the separation between the plates. 
 Capacitance of an Isolated Conducting Sphere: 
 
 ... where R is the radius of the sphere. 
 Total Circuit Capacitance 
 In series, capacitance sum is determined in a similar way to how resistance is determined with resistors in parallel: 
 
 In parallel, it's the opposite - the capacitances are just added to each other: 
 
 Capacitor Energy 
 Capacitor energy can be given as: 
 
 ... and you can use Q=CV to substitute values in to determine energy with charge. 
 (You do not need to know the integral derivation. It is only for explanation.) 
 The work done in moving a charge Q from one plate to another through a constant potential difference V is: 
 
 The reason the two energy equations are different is because V is constant in the latter , but not the former . Capacitance is a constant for each capacitor, however. 
 A capacitor can leak charge . This is because the insulator between the plates is not perfect, so there is a tiny current that passes through them. This charge leakage is more clearly observable when disconnecting a capacitor from the source of emf. 
 Charging and Discharging 
 Where x 0 is the initial value of the variable, C is capacitance, R is resistance and t is elapsed time: 
 
 
 
 
 Current 
 Charge 
 PD 
 
 
 Charging 
 
 
 
 
 
 
 
 
 
 Discharging 
 
 
 
 
 
 
 
 
 
 
 
 
 (... yes, current has the same equation for charging and discharging.) 
 The time taken for the charge of the capacitor to fall to 1/e (~37%) of its original charge is known as the time constant. 
 
 As more charge is added to a capacitor, it gets harder to add more charge to it. This concept is analogous to pumping a car tyre - it is initially easy to add air to it, but the increase of internal pressure makes adding more harder and harder. 
 Graphical Methods 
 Take V in discharging for an example. You can apply the natural logarithm "ln" to both sides to separate variables and obtain a straight line in a graph: 
 
 In here, the y-intercept is "ln(V 0 )" and the gradient is "-1/CR". You can use the gradient to find the capacitance of the circuit, and you can use the y-intercept to find the initial pd of the capacitor by raising e to the power of "ln(V 0 )". 
 Spreadsheet Modelling 
 You can model the discharge of a capacitor using a spreadsheet method without using experimental data. This is known as iterative modelling. You can do this with the following equation: 
 
 This gives the decrease of charge, so the output of this should be subtracted from the last value of charge. The table looks like this 1 : 
 
 ... with a time increment of +0.1.