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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

76013.png

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
76023.png
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).
 
76035.png
(a) Transient state
 
76041.png
(b) Steady state
Fig. 3

Charging

In transient state of charging, charge on the capacitor at any instant 75236.png and potential difference across the capacitor at any instant 75243.png
 
76076.png
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 75249.png and potential difference cross the capacitor at any instant 75255.png (Fig. 5).
 
76106.png
Fig. 5

Time constant (τ)

The quantity RC is called the time constant, i.e., τ = 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 = τ = RC, Q = Q0(1 – e–1) = 0.639Q0.
 
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 = τ = RCQ = Q0(e–1) = 0.37Q0.

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)].
 
77087.png
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)].
 
77099.png
Fig. 6 (b) Steady state
 
Therefore, the current through resistor R1 is 375261.png, hence potential difference across resistance will be equal to 75270.png. The same potential difference will appear across the capacitor, hence charge on capacitor in steady state 75276.png.




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