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RC circuits are circuits that contain both resistors and capacitors. In a DC circuit (or, steady-state circuit, or constant current circuit), a capacitor acts like an open switch. Its steady-state voltage will be V = Q/C and will be equal to the voltage of the battery. However, when the switch is first closed for the circuit, charge must flow until the capacitor is charged. This may only take a fraction of a second, or a few seconds. This is called a transient current. Consider a simple loop circuit which has a battery with terminal voltage Vbattery, a resistor R, a capacitor C, and a switch. Just before the switch is closed the current is zero. Just after it is closed a current flows and the capacitor starts to charge. Its voltage will be V = Q/C. By the time Q grows until Q/C = Vbattery, the current will be zero. This early current is the transient current. Let us write Kirchhoff's Loop Law for the circuit: Vbattery - I R - Q/C = 0Let us note that the current I can be written as I = dQ/dt. The expression then becomes: Vbattery - (dQ/dt)R - Q/C = 0The solution to this equation is: Q = (C Vbattery)(1 - exp(-t/RC))Note that aftera long time exp(-t/RC) becomes equal to 0, and the charge is constant. If we take the derivative of Q with respect to t, we get: I = dQ/dt = ((Vbattery)/R)exp(-t/RC)We see that after a long time, the current is zero. The quantity RC is called the time constant. It has units of time, and determines how fast a capacitor charges and discharges. If we start with a charged capacitor, and have only a resistor and an open switch in the curcuit, we can then discharge the capacitor by closing the switch. The Kirchhoff's Loop Law equation becomes: - I R - Q/C = 0Again, using I = dQ/dt, we have : - (dQ/dt)R - Q/C = 0The solution is: Q = Qo exp(-t/RC)Here Qo is C Vbattery. Again, using I = dQ/dt, we have: I = (Q0/RC)exp(-t/RC) = (Vbattery)/R)exp(-t/RC)Both the charge and the current become zero after a time as the capacitor is discharged. For a measure of the time we use the time constant t = R C. When t = R C, the exp(-t/RC) = exp(-1) = 0.37, or the charge on the capacitor is down to just over one-third of the original value.
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