1. The exponential decay of a discharging capacitor.

2. In this section we will learn how:
To derive a general equation for capacitor decay of charge using an analogy, or MODEL.
Solve this exponential equation so we can work out the charge on capacitor plates at any time.
Generate a picture of charge decay on an exponential curve.
Understand what the TIME CONSTANT is.

3. This may sound a little confusing at first, but we can solve the exponential decay of a capacitor by comparing it to a water clock!
First, revise your work on exponentials earlier in the course.
We are also going to SOLVE an exponential equation.Don’t worry about the maths – you only have to remember a simple equation for this.

4. The exponential water clock
Vol of water V
Pressure difference p across tube. h
Flow rate = dV/dt
Fine tube to restrict flow
Pressure diff is proportional to height.
Xsection is constant, so h is proportional to vol V.
So p  V.
Flow rate is prop to pressure difference.
f = dV/dt  p
dV/dt  - V

5. Water level drops exponentially if dV/dt  p
Vol
Time

6. V = Q/C I = dQ/dt = V/R dQ/dt = - Q/RC (or - 1/RC x Q)
Exponential decay of charge.
Capacitance C
Charge Q
Resistance R
Current I = dQ/dt
p.d = V
V = Q/C
I = dQ/dt = V/R
dQ/dt = - Q/RC
(or - 1/RC x Q)

7. Charge decays exponentially if current is proportional to p. d
Charge decays exponentially if current is proportional to p.d. and capacitance is constant.
Charge Q
Time

8. Use the Excel spreadsheet (save this file and open it separately to give full functionality of spreadsheet) model of capacitor decay to see how R and C affect the decay of different capacitors. This will help you understand the importance of this constant.

9. Using our analogy of the water clock, we now have a general relationship to describe how charge decays from the plates of a capacitor with capacitance C and in a circuit with resistance R.
This means decay, not growth
Constant dQ 1 = - Q dt RC

10. Q e-t/RC Qo Solving this to find Q AT ANY TIME: This means decay
This is the power of e Q
e-t/RC = Qo
This is the charge at the beginning
This is a number like . It’s value is about 2.7

11. Log to the base e; it’s on your calculator
Take logs (to the base e) of both sides.
ln (Q/Qo) = -t/RC
Log to the base e; it’s on your calculator
ln Q - ln Qo = -t/RC
We can now find charge Q at any time t, as long as we know R and C, along with original charge Qo.

12. Why RC is so important.
What are its units??

13. Q e-t/RC Qo seconds This must be in seconds! A number= Qo
A number
No units: a ratio

14. Qo Charge Q (or current I.)
Summary graph for charging a capacitor. Qo
Q = Qo/e
= Qo x 0.37
Charge Q (or current I.)
Q = Qo/e2
= Qo x 0.372 ? RC
2RC
3RC
4RC
Time