CAPACITORS IN CIRCUITS

1. Charging and discharging capacitors

Consider the circuit shown on the left. When the switch S is open, there is no current flowing through the curcuit, and there is no charge on the capacitor, C.



As soon as the switch is closed, a current starts to flow through the resistor R, and a charge accumulates on the capacitor plates. Intinially, this current has the value E/R, but this value decays with time as the capacitor becomes charged. After a certain time, the capacitor is fully charged, and the potential difference across the plates equals the e.m.f. of the battery. No current flows through the circuit - the charged capacitor acts as an insulator. The behaviours of the current, charge and PD across C are shown in the graphs above (A, B, and C).

The greater the resistance of R and the greater the capacitance of C, the longer it will take to charge the capacitor. (For educators.)

As soon as the switch S is opened, current flows across the resistor R and the ammeter A, (graph E, above left), but in the opposite direction to what was observed when the switch was closed (graph A). This current decays to a value of zero. At the same time, both the charge on C and the PD across C decay to zero (graphs F and G above).

The greater the resistance of R and the greater the capacitance of C, the longer it will take to discharge the capacitor. (For educators.)

2. Capacitors in series

If two or more capacitors having capacitances C1, C2, ... Cn, are placed in series, then the combined capacitance C is given by

(For educators: See how this formula is derived.)

3. Capacitors in parallel

If two or more capacitors having capacitances C1, C2, ... Cn, are placed in parallel, then the combined capacitance C is given by

How can we use capacitors in a circuit?

(Click here for a familiar example.)

4. Additional questions


Time taken to charge and discharge a capacitor

The time, t, taken to charge a capacitor with capacitance, C farads, in series with a resistor with resistance R Ω may be calculated from

v = V(1 - e-t/RC)

where v is the potential difference across the capacitor at time t, V is the final potential difference, and e is the base of natural logarithms, (2.718). The product R x C is has the dimensions of time, and is known as the TIME CONSTANT. It is the time required for the potential difference across the capacitor to increase from zero to 63.2% of its final value, V.

The time, t, taken to discharge a capacitor with capacitance, C farads, in series with a resistor with resistance R Ω may be calculated from

v = V e-t/RC


Capacitors in series: derivation of formula

Consider the diagram on the left: Let the top plate of capacitor C1 acquire a charge +Q. This causes electrons to be drawn to the bottom plate of C1, causing it to acquire a charge -Q. These electrons can only come from the top plate of C2, which then gets a charge +Q. The bottom plate of C2 then becomes charged to a value of -Q.

The negative charge on the bottom plate of C1 exactly balances the positive charge on the top plate of C2, since these two plates are connected to each other and nothing else. This argument can be extended to any number of plates connected in series, leading us to the conclusion that when capacitors are connected in series, the MAGNITUDE of the charges on each plate is the same, regardless of the capacitance of the individual capacitors.

Again, have a good look at the diagram above. The potential difference beween the two points a and c, Vac is given by:

Similarly, the potential difference , Vcb between the points c and b is given by:


But the potential difference applied to the circuit by the battery, Vab is the sum of the two potential differences Vac and Vcb, which can be related to the charge on the capacitor plates, Q, and the individual capacitances of the two capacitors:

It follows therefore that the total capacitance CTot is given by: