Charging a Capacitor
Whenever we connect an uncharged capacitor with a voltage source it is exponentially charged until the voltage developed across the capacitor becomes equal and opposite to voltage of the source.
Let us connect one capacitor
C in series with a resistor
R. We also connect this series combination of capacitor
with a battery
V through a push switch S.
When we push the switch, the capacitor
is uncharged, hence no voltage developed across the capacitor, thus the capacitor will behave as short circuit. The current
through the circuit will only be limited by resistance
R. That initial current
is V / R. Now slowly the voltage is being developed across the capacitor, and this developed voltage is in opposite of the polarity of battery
. As a result the current
in the circuit gradually decreased. It will be gradually decreased to zero, when the voltage across the capacitor becomes equal and opposite of the voltage of battery
The voltage is gradually increased across the capacitor during charging. Let us consider the rate of increase of voltage across the capacitor is dv/dt at any instant t. The current though the capacitor at that instant is
Applying, Kirchhoff’s Voltage Law, in the circuit at that instant, we can write,
Integrating both side we get,
Now, at the time of switching on the circuit, voltage across the capacitor was zero. That means, v = 0 at t = 0.
Putting these values in above equation, we get
After getting the value of A, we can rewrite the above equation as,
Now, we know that,
This is the expression of charging current I, during process of charging.
The current and voltage of the capacitor during charging is shown below.