To study the exponential decay of current in a circuit containing Resistance and Capacitance and to determine the R.C. time constant.
Source of e.m.f (battery eliminator), Resistors - 10 K, 30K, 5K - ohms, Electrolytic Capacitors – 470, 1000, 2000 microfarads, Galvanometer (50- 0-50), Stop clock, tap key, connecting wires.
The ability of a conductor to hold electric charge is called the capacity of a conductor. Different conductors at the same potential have different capacities for holding charge, depending o their size and shape. The capacity or capacitance of a conductor may be defined as the charge required to raise its potential by unity. The unit of capacity is a farad. Capacitors serve as devices for storing electrical energy. Capacitors are classified into different types depending upon the type of dielectric medium present between the plates; types of charges induced at the plates.
Resistance is the characteristic of a particular specimen of the material. Its unit is an Ohm. Resistivity is the characteristic of the nature of the material.
In an electric circuit, when a capacitor and a resistor are connected as shown in figure 1, electric charges build up across the capacitor. The discharge of the condenser takes place when the circuit is disconnected from the source of e.m.f.
When the tap key (K) in figure 1 is pressed, a constant e.m.f. works in the circuit, the condenser plates receive the charge till the potential differences across them becomes equal to E. When the key is released, the discharge of the condenser takes place. Let 'q' be the charge on the capacitor at a time 't' after the key is released. The instantaneous value of the p.d. across the capacitor is given by q/c and E=0; i.e.,
q/c - IR = 0 --------------------------(1)
But I = -dq/dt
q/c + R dq/dt = 0
or dq/dt = - q/RC
Integrating
Log _{e}q = -t/RC + B------------------------(2)
Where B is the constant of integration.
At t = 0; q = q _{0}; hence B = log _{e}q _{0}
Substituting this value of B in equation (2), we get
Log _{e}q / q _{0}= -t/RC ------------------------(3)
Or
q = q _{0}e _{-t/RC}------------------------------(4)
∴Rate of discharge is
dq/dt = - q _{0}e^{-t/rc/RC}
The circuit is connected as shown in figure 1, taking one set of R and C. The capacitor C is charged for a short time till the deflection in the galvanometer is maximum, but within the scale. The tap key is then released. The capacitor now starts discharging through the resistor R. The deflection decreases steadily.
The stop clock is started at a suitable initial point (need not be maximum) and the deflection is noted at suitable intervals of time. It is continued till the deflection falls below 0.36 of starting value. The experiment is repeated for the other sets of R and C and the observations are tabulated in Table 1. The time constant is calculated theoretically from the values of R & C used, and also from the graphs; as shown in figure. 2.
Table.1
S.No | Time/sec | Set 1 R _{1}=Ω C _{1}=µf/Voltage or current | Time/Sec | Set 2 R _{2}=Ω C _{2}=µf/Voltage or current | Time/Sec | Set 3 R _{3}=Ω C _{3}=µf/Voltage or current |
1 | ||||||
2 | ||||||
3 | ||||||
4 | ||||||
5 | ||||||
5 | ||||||
6 |
I = I _{0exp}(-1)
I = 0.36I _{0}
Thus, it is to be observed.
(i) Smaller is the time constant; more rapid is the discharge of the capacitor
(ii) The current in a R.C. Circuit falls exponentially with time.
RC time constant | T _{theoretical= }RXC | P _{practical}(from Graph) |
R _{1}=----------Ω C _{1} =----------µf | ||
R _{2}=----------Ω C _{2} =----------µf | ||
R _{3}=----------Ω C _{3} =----------µf |
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