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explain charging of a capacitor alongwith necessary derivation when the capacitor is connected across a dc voltage source

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The figure below shows a capacitor, ( C ) in series with a resistor, ( R ) forming a RC Charging Circuit connected across a DC battery supply ( Vs ) via a mechanical switch. at time zero, when the switch is first closed, the capacitor gradually charges up through the resistor until the voltage across it reaches the supply voltage of the battery. The manner in which the capacitor charges up is shown below.

RC Charging Circuit

Let us assume above, that the capacitor, C is fully “discharged” and the switch (S) is fully open. These are the initial conditions of the circuit, then t = 0, i = 0 and q = 0. When the switch is closed the time begins at t = 0 and current begins to flow into the capacitor via the resistor.

Since the initial voltage across the capacitor is zero, ( Vc = 0 ) at t = 0 the capacitor appears to be a short circuit to the external circuit and the maximum current flows through the circuit restricted only by the resistor R. Then by using Kirchhoff’s voltage law (KVL), the voltage drops around the circuit are given as:

Vs-R x i(t)-Vc(t)=0

The current now flowing around the circuit is called the Charging Current and is found by using Ohms law as: i = Vs/R.

RC Charging Circuit Curves

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