# 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 [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/f5Iimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/f5Iimage.png) ... 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: [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/XDKimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/XDKimage.png) ... 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: [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/lswimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/lswimage.png) In parallel, it's the opposite - the capacitances are just added to each other: [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/6myimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/6myimage.png) ## Capacitor Energy Capacitor energy can be given as: [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/TEIimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/TEIimage.png) ... 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: [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/U7aimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/U7aimage.png) 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 x0 is the initial value of the variable, C is capacitance, R is resistance and t is elapsed time:
**Current** **Charge** **PD**
**Charging** [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/3CFimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/3CFimage.png)[![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/1nJimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/1nJimage.png) [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/126image.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/126image.png)
**Discharging** [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/3CFimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/3CFimage.png) [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/9jHimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/9jHimage.png) [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/gmqimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/gmqimage.png)
(... 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. [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/KcKimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/KcKimage.png) 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: [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/h6timage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/h6timage.png) In here, the y-intercept is "ln(V0)" 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(V0)". ### 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: [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/kpGimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/kpGimage.png) 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](https://www.ocr.org.uk/Images/170416-modelling-decay-of-charge-activity-teacher-instructions.pdf): [![image.png](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/scaled-1680-/akdimage.png)](https://bookstack.asadhussain.net/uploads/images/gallery/2026-01/akdimage.png) ... with a time increment of +0.1.