5. Electrostatic effects
Discharging a capacitor P.24
 Picture 5.10. Circuit of RC discharge.
 A capacitor will not hold its charge forever. This is because the air around it conducts very slightly. So a small current flows from the positive plate to the negative plate, discharging the capacitor. But how long will this take and can we speed it up?
 A simple discharge circuit Picture 5.10 shows a circuit with a charged capacitor and a 40 W resistor. When the switch is closed, the capacitor will discharge through the resistor. Let's see what happens to the current in the circuit. We will follow the discharge in small steps of time and build up the values in table 1. (Roll your cursor over the steps to build up the table and the graph.) Click here to start.
Picture 5.11. Table of results for RC discharge
Picture 5.12. Exponential decay of charge on a capacitor.
 Step 1a To begin with, the charge on the 2 mF capacitor is: Q0 = 20 mC Step 1b This means the voltage is: V0 = Q/C = 20 x 10-3 ÷ 2 x 10-3 = 10V Step 1c When the switch is closed, the voltage on the capacitor will push a current through the resistor. The size of the current is: I0 = 10/40 = 0.25 A Step 1d In the first 10 milliseconds, the amount of charge that will flow is: DQ = 0.25 x 0.01 = 2.5 mC Step 2a This means that the charge on the capacitor is less. It is now: Q1 = 20 – 2.5 = 17.5 mC Step 2b This means that the voltage on the capacitor will fall to . V1 = 8.75 V Step 2c Now that there’s a smaller voltage, the current will be less. This will mean that the amount of charge that flows off in the next 10 milliseconds will be less. Step 3 And therefore, although the voltage will drop again, it won’t drop by as much. Similarly, the current is smaller (but not by as much as in the first step). Step 4+ Starting at 0.03, roll over the times in the table to fill in the table.
 The discharge curve (picture 5.12) is exponential. Let's see what this means.
 Picture 5.13. Exponential changes.
 Exponential curves We can always recognise an exponential curve because the y-axis changes by an equal proportion for equal divisions on the x-axis. For example, bacterial growth is often exponential – it doubles every 3 seconds. Radioactive decay is exponential – the activity of carbon-14, for example, halves every 5700 years. Now we can see that our capacitor discharging through a resistor is exponential – it drops to 87.5% of its starting value every 10 ms.
 Does it reach zero? Although, in theory, exponential curves never reach the origin, in reality a capacitor will always discharge. Here are two views as to why: The engineer: it will eventually get so small that we can’t measure it. The physicist: it will eventually get down to a small number of individual electrons on the negative plate. These will, randomly, leave the plate. Even so, we can’t measure the time it takes to fully discharge. So we have to measure how long it takes to reach a certain proportion (rather like half lives for radioactive isotopes). We’ll see how on the next page.

 Question 19 Imagine we set up a circuit like picture 5.10 with a charged capacitor connected to a 60 W resistor. The 4 mF capacitor is charged to 10 V. a) What is the current as the capacitor starts to discharge? b) At this rate, how much charge leaves the capacitor in the first 10 ms? c) What is the new charge (after the first 10 ms)? d) What is the new voltage (after the first 10 ms)? e) What is this voltage as a proportion of the starting voltage? f) How long will it take for the voltage to drop to half of its starting value? (Hint – use a calculator to keep multiplying the proportion you found in part e by itself, until you get to a half).

 Summary                                           Close the charge on a capacitor decays exponentially it drops by the same proportion in the same time