6.1 capacitors

capacitor

an electrical component that stores charge on two metallic plates, with a dielectric (a type of insulator) placed between them to prevent charge from travelling between them

capacitance

the charge (coulombs) stored per unit potential difference (volt) across the two plates on a capacitor
C = Q/V
measured in Farads (F)

capacitors in series

kirchoff's voltage law states that the sum of emfs in any closed loop in a circuit is equal to the sum of the potential differences in the same loop
V = V1 + V2 + … + Vn
as V = Q/C this means
Q/CT = Q/C1 + Q/C2 + … + Q/CN
as Q is constant, this gives
1/CT = 1/C1 + 1/C2 + … + 1/CN
giving
CT = (1/C1 + 1/C2 + … + 1/CN)^-1

capacitors in parallel

kirchoff's current law states that the total current flowing into a node in a circuit is equal to the total current flowing out of it
IT = I1 + I2 + … + IN
as charge is It, and t is constant, this gives
QT = Q1 + Q2 + … + QN
as capacitance is Q/V and V is constant, this gives
CT = C1 + C2 + … + CN

how energy is stored in a capacitor

work is done by the power supply to deposit electrons onto the negative plate as like charges repel, and vice versa to remove electrons from the positive plate as opposite charges attract

work done in capacitors

voltage has a linear relationship with charge stored by a capacitor, and as voltage is the electrical potential energy per unit charge, the area under the graph must represent the work done in charging the capacitor and thus the energy stored in it
therefore
W = (1/2)QV
as Q = CV this gives
W = (1/2)V^2C and W = Q^2/2C

charging a capacitor

a power supply draws electrons from the plate connected to the positive terminal and deposits them onto the negatively connected plate, leaving them with equal and opposite charges
current will flow into the capacitor until the voltage between the plates is equal to the emf of the power supply

discharging a capacitor

the capacitor will release current until no more electrons can be pushed onto the negatively charged plate due to electrostatic repulsion from the electrons already present
current will flow out of the capacitor until there is no voltage between the plates

method of discharging capacitors

a charged capacitor can be discharged by disconnecting the power supply and connecting another electrical component, for example by flipping a switch
it is often discharged through a resistor to control the discharge rate

time constant of capacitors

the time taken in seconds for the charge, current or voltage of a discharging capacitor to decrease to 37% of its original value (or for it to rise to 63% for a charging capacitor)
37% is chosen as it's rouhgly equivalent to 1/e

time constant equation

τ = RC
where R is the resistance of the resistor being discharged through in ohms, and C is the capacitance of the capacitor in farads

capacitor time to half (half life)

τ(1/2) the time taken in seconds for the charge, current or voltage of a discharging capacitor to reach half of its initial value
equal to ln(2)τ or roughly 0.69τ

capacitor discharge equation

(inc picture)
where I is the current, I0 is the initial current, e as in the constant, t is the discharging time, R is the resistance of the resistor being discharged through, and C is the capacitance (RC is the time constant)
the current can be replaced with the charge or voltage of the capacitor

capacitor charge equation

(inc picture)
basically just the capacitor discharge equation but e^(-t/RC) is (1 - e^(-t/RC))
again the current can be replaced with the charge or voltage of the capacitor

graphical modelling of capacitor discharging

when a capacitor is discharges, the graphs of current, voltage or charge against time all show a pattern of exponential decay
(inc diagram)

investigating charge and discharge of a capacitor (method)

-set up a circuit so that a capacitor can be connected to either a power supply or a resistor depending on the position of the switch, and so that there is a voltmeter across the capacitor
-charge the capacitor fully (until the capacitor has the same voltage as the battery)
-discharge the capacitor through the resistor
-measure the voltage every ten seconds until the voltage becomes very small and changes minimally

investigating charge and discharge of a capacitor (results)

-plot a graph of voltage against time and observe its exponential decay-shaped line
-plot a graph of ln(voltage) against time and observe its straight shape
-determine the gradient of the line, equal to -1/RC (as log(e^(-t/RC))/t -> (-t/RC)/t -> -1/RC)
-you can use this to calculate the time constant (-1/grad) or capacitance (-1/gradR) (as C = τ/R)

constant ratio property

the nature of an exponential decay graph, such as the graph modelling a capacitor's discharge, to have a constant ratio of magnitude at any pair of points at equivalent distance
(e.x. any point on the graph will be x% of the magnitude of the point y seconds before it)

capacitor graphing equation

∆Q/∆t = -Q/RC
where ∆Q and ∆t are the change in charge and time, Q is the charge of the capacitor and CR are the capacitance of the capacitor and the resistance of the resistor being discharges through, aka the time constant

capacitor graphing equation additional information

used on graphs modelling a capacitor's discharge, and can be used to find the time constant of the capacitor