Circuits Containing Resistors & Capacitors (RC Circuits) Chapter Opener. Caption: Newton’s laws are fundamental in physics. These photos show two situations of using Newton’s laws which involve some new elements in addition to those discussed in the previous Chapter. The downhill skier illustrates friction on an incline, although at this moment she is not touching the snow, and so is retarded only by air resistance which is a velocity-dependent force (an optional topic in this Chapter). The people on the rotating amusement park ride below illustrate the dynamics of circular motion.

RC Circuits When the switch is closed, the capacitor will begin to charge. As it does, the voltage across it increases, and the current through the resistor decreases. Figure 26-17. After the switch S closes in the RC circuit shown in (a), the voltage across the capacitor increases with time as shown in (b), and the current through the resistor decreases with time as shown in (c).

To find the voltage as a function of time, use the Loop Rule to write the equation for the voltage changes around the loop: I = dQ/dt, so integrate to find the charge as a function of time: The voltage across the capacitor is VC = Q/C. The current I at time t is found by differentiating the charge: The quantity RC that appears in the exponent is called the time constant of the circuit:

Example RC Circuit, with EMF. The capacitance in the circuit shown is C = 0.30 μF, the total resistance is R = 20 kΩ, the battery emf is E = 12 V. Calculate: (a) the time constant, (b) the maximum charge the capacitor could acquire, (c) the time it takes for the charge to reach 99% of this value, (d) the current I when the charge Q is half its maximum value, (e) the maximum current, (f) the charge Q when the current I is 0.20 of its maximum value. Solution: a. The time constant is RC = 6.0 x 10-3 s. b. The maximum charge is the emf x C = 3.6 μC. c. Set Q(t) = 0.99 Qmax and solve for t: t = 28 ms. d. When Q = 1.8 μC, I = 300 μA. e. The maximum current is the emf/R = 600 μA. f. When I = 120 μA, Q = 2.9 μC.

If an isolated charged capacitor is connected across a resistor, it discharges: The voltage & current as functions of time can be found from the charge: Figure 26-18. For the RC circuit shown in (a), the voltage VC across the capacitor decreases with time, as shown in (b), after the switch S is closed at t = 0. The charge on the capacitor follows the same curve since VC α Q. and

Example: Discharging RC circuit. In the RC circuit shown, the battery has fully charged the capacitor, so Q0 = C E. Then at t = 0 the switch is thrown from position a to b. The battery emf is 20.0 V, and the capacitance C = 1.02 μF. The current I is observed to decrease to 0.50 of its initial value in 40 μs. (a) What is the value of Q, the charge on the capacitor, at t = 0? (b) What is the value of R? (c) What is Q at t = 60 μs? Solution: a. At t = 0, Q = CE = 20.4 μC. b. At t = 40 μs, I = 0.5 I0. Substituting in the exponential decay equation gives R = 57 Ω. c. Q = 7.3 μC.

Conceptual Example: Bulb in RC circuit. In the circuit shown, the capacitor is originally uncharged. Describe the behavior of the lightbulb from the instant switch S is closed until a long time later. Solution: When the switch is closed, the current is large and the bulb is bright. As the capacitor charges, the bulb dims; once the capacitor is fully charged the bulb is dark.

Resistor in a turn signal. Example Resistor in a turn signal. Estimate the order of magnitude of the resistor in a turn-signal circuit. Figure 26-21: (a) An RC circuit, coupled with a gas-filled tube as a switch, can produce a repeating “sawtooth” voltage, as shown in (b). Solution: A turn signal flashes about twice per second; if we use a 1 μF capacitor, we need a 500 kΩ resistor to get an 0.5-second time constant.

Electric Hazards Most people can “feel” a current of 1 mA; a few mA of current begins to be painful. Currents above 10 mA may cause uncontrollable muscle contractions, making rescue difficult. Currents around 100 mA passing through the torso can cause death by ventricular fibrillation. Higher currents may not cause fibrillation, but can cause severe burns. Household voltage can be lethal if you are wet and in good contact with the ground. Be careful!

A person receiving a shock has become part of a complete circuit. Figure 26-23. A person receives an electric shock when the circuit is completed.

Faulty wiring and improper grounding can be hazardous Faulty wiring and improper grounding can be hazardous. Make sure electrical work is done by a professional. Figure 26-24. (a) An electric oven operating normally with a 2-prong plug. (b) Short to the case with ungrounded case: shock. (c) Short to the case with the case grounded by a 3-prong plug.

The safest plugs are those with three prongs; they have a separate ground line. Here is an example of household wiring – colors can vary, though! Be sure you know which is the hot wire before you do anything. Figure 26-26. Four wires entering a typical house. The color codes for wires are not always as shown here—be careful!