# Circuit with Resistors and Capacitors

A resistor may be connected either in series or in parallel with the capacitor as shown in below.

**Series**

*RC*circuit**Fig. 1**

In this combination, capacitor takes longer time to charge. |

The charging current is maximum in the beginning; it decreases with time and becomes zero after a long time. |

**Parallel**

*RC*circuit**Fig. 2**

Resistor has no effect on the charging of capacitor. |

Resistor provides an alternative path for the electric current. |

# Three states of RC circuits

*Initial state:*Just after closing the switch or just after opening the switch.*Transient state:*Instantaneous state, i.e., any time after closing or opening the switch.*Steady state:*A long time after closing or opening the switch. In the steady state condition, the capacitor is charged or discharged.

# Charging and discharging of capacitor in series RC circuit

As shown in Fig. 3(a), when switch

*S*is closed, the capacitor starts charging. In this transient state, potential difference appears across capacitor as well as resistor. When capacitor gets fully charged the entire potential difference appeared across the capacitor and nothing is left for the resistor, as shown in Fig. 3(b).**(a) Transient state**

**(b) Steady state**

**Fig. 3**

# Charging

In transient state of charging, charge on the capacitor at any instant and potential difference across the capacitor at any instant

**Fig. 4**

# Discharging

After the completion of charging, if battery is removed, capacitor starts discharging. In transient state, charge on the capacitor at any instant and potential difference cross the capacitor at any instant (Fig. 5).

**Fig. 5**

# Time constant (Ï„)

The quantity

*RC*is called the time constant, i.e.,*Ï„*=*RC*.*In charging:*It is defined as the time during which charge on the capacitor rises to 0.63 times (63%) the maximum value. That is when

*t*=

*Ï„*=

*RC*,

*Q*=

*Q*

_{0}(1 â€“

*e*

^{â€“1}) = 0.639

*Q*

_{0}.

*In discharging:*It is defined as the time during which charge on a capacitor falls to 0.37 times (37%) of the initial charge on the capacitor. That is when

*t*=

*Ï„*=

*RC*,

*Q*=

*Q*

_{0}(

*e*

^{â€“1}) = 0.37

*Q*

_{0}.

# Mixed *RC* circuit

In a mixed

*RC*circuit, as shown in Fig. 6, when switch*S*is closed, current flows through the branch containing resistor as well as through the branch contains capacitor and resistor (because capacitor is in the process of charging) [Fig. 6(a)].**Fig. 6 (a) Transient state**

When capacitor gets fully charged (steady state), no current flows through the line in which capacitor is connected [Fig. 6(b)].

**Fig. 6 (b) Steady state**

Therefore, the current through resistor

*R*_{1}is 3, hence potential difference across resistance will be equal to . The same potential difference will appear across the capacitor, hence charge on capacitor in steady state .