Charge And Discharge Of Capacitors

Charge and Discharge of Capacitors A2 Unit 4 Charging a Capacitor If a potential difference,
, is applied across a capacitor, the capacitor charges until it has this same potential difference across it. However, as the capacitor charges, it will harder for the supply to “push” the charge onto (or “pull” it from) the plates. This means that the current flowing in the circuit will drop exponentially.

The potential difference across the capacitor increases and levels off in a similar way.

A capacitor is first charged by connecting the switch, S, to the ‘1’ position. It will charge until the potential difference across it is
.

Note that the charge on the capacitor plates follows a similar pattern to the potential difference, as  = 


Discharging a Capacitor

It can then be discharged through the resistor by putting the switch in the ‘2’ position.

We can then take this capacitor and discharge it through a resistor. Initially there will be a larger current, which will decrease gradually. This is because the potential difference across the capacitor will decrease as it loses charge.

The charge stored on the capacitor, and the current flowing through the circuit, will also both follow exponential decays.

Charge and Discharge of Capacitors A2 Unit 4 Why exponentials? How do we know that these rates of charging and discharging follow exponential patterns? We can observe this experimentally, and also derive it mathematically. Imagine at some time during discharge, when the capacitor has a charge Q still stored on the plates. A small time, Δ, later, the charge will be  − Δ. The current flowing in the circuit is then =

−Δ Δ

It is also =


=  

 
=  

So Δ  =− Δ  This type of result is known as a differential equation. The rate of change of Q is dependent on the value of Q itself. The solution to this equation is

 =   (You do not need to know how to solve this equation). The quantity  is known as the time constant, as is the time taken for the charge on the capacitor (or the potential difference across it, or the current in the circuit) to reach 1⁄ ≈ 0.37 of its original value of  (or
, or  ).

 = charge stored on capacitor (C)
= potential difference across capacitor (V)  = current in circuit  = initial charge (when discharging); final charge (when charging)
 = initial supply voltage (when charging); initial voltage across capacitor (when discharging).  = initial current flow (when charging or discharging)  = capacitance of capacitor (F)  = resistance of circuit (Ω)  = time (s)  = time constant (s) = exponential constant ≈ 2.718