Have you read parts 1 and 2 of this post?

Exponential Discharging

- Capacitors discharge quickly to begin with, but the discharge rate drops as they discharge
- The charge stored by a capacitor decays exponentially when it is discharging
- In an exponential relationship, the rate of change is proportional to the quantity present
- The more electrons on the plate, the more electrons leave the plate per unit time
- Since the number of electrons is hard to measure, the charge, Q, held by the capacitor is used in the graphs and equations

Equations for Discharging

- Using the same format as for radioactive decay, but with electric charge instead of un-decayed nuclei, you get:
- Q = Q
_{0}e^{-t/RC} - Where RC is the time constant of the discharging circuit
- From Ohm’s Law, V=IR you can see that V (potential difference) and I (current) are proportional when R (resistance) is constant
- Because of this, the voltage across the capacitor also decays exponentially as it discharges:
- V = V
_{0}e^{-t/RC} - Finally, the current decays exponentially as well, because the current is the rate of flow of charge, and this rate decreases as the capacitor discharges (the charge flows more slowly)
- I = I
_{0}e^{-t/RC}

Exponential Charging Equations

- Capacitors initially charge up quickly, but the charge rate drops as they charge (because it gets increasingly harder for the battery to force more electrons on to the same plate)
- The current starts off large, but as the charge builds up, it starts to flow more slowly and the current decreases
- The same goes for the charging current as the discharging current:
- I = I
_{0}e^{-t/RC} - But remember: The discharging current is in the opposite direction from the charging current, so one of them will have to be a negative current
- The form of the charge and voltage curves are different, though. They start off increasing quickly and increase exponentially slower
- It’s
**not**y = e^{-x}(starts off decreasing quickly and decreases exponentially slower) - …and it’s
**not**y = e^{x}(starts off increasing slowly and increases exponentially quicker) - It’s actually: Q = Q
_{0}(1-e^{-t/RC}) and V = V_{0}(1-e^{-t/RC})

In Summary

- Charging isn’t just a regular exponential curve – it has the (1 – e
^{to the power of stuff}) bit - Discharging equations are slightly more simple
- The equation for current is the same for both, BUT IN OPPOSITE DIRECTIONS
- Here’s a table to make that clearer:

Charging | Discharging | |

Charge, Q /Coulombs | Q = Q_{0}(1-e^{-t/RC}) |
Q = Q_{0}e^{-t/RC} |

Potential Difference, V /Volts | V = V_{0}(1-e^{-t/RC}) |
V = V_{0}e^{-t/RC} |

Current, I /Amps | I = I_{0}e^{-t/RC} |
I = I_{0}e^{-t/RC} (the other direction) |

The Gradient

- On these charging and discharging graphs of charge against time, the gradient is the change in charge divided by the change in time
- dQ / dt… sound familiar?
- Q/t = I = current
- The gradient of a capacitor charging or discharging graph is the current

Data, Formulae and Relationships Booklet

- Q = Q
_{0}e^{-t/RC}appears to be in the booklet - The rest of the charging and discharging exponential equations don’t seem to be in there
- Learn the blue ones in the table

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