The exponential decay of a discharging capacitor.

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.

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.

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

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

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)

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

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.

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

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

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.

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

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

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