Chapter 23 Circuits Topics: Sample question: Circuits containing multiple elements Series and parallel combinations Complex Multi-loop Circuits RC circuits Sample question: An electric eel can develop a potential difference of over 600 V. How do the cells of the electric eel’s body generate such a large potential difference? Slide 23-1

Series and Parallel Series Parallel Circuit elements in a chain between 2 points Same current flows through circuit elements I1 = I2 Electric Potentials add => Delta Vtotal = Delta V1 + Delta V2 Parallel Circuit elements on multiple paths connecting the same points Since paths connect the same points, Delta V’s are the same Currents Add => Itotal = I1 + I2 Answer: i) B, C ii) A, D iii) E Slide 22-25

QuickCheck The battery current I is 3 A 2 A 1 A 2/3 A 1/2 A Answer: D

QuickCheck The battery current I is 3 A 2 A 1 A 2/3 A 1/2 A

QuickCheck The diagram below shows a circuit with two batteries and three resistors. What is the potential difference across the 200 Ω resistor? 2.0 V 3.0 V 4.5 V 7.5 V There is not enough information to decide. Answer: A

QuickCheck The diagram below shows a circuit with two batteries and three resistors. What is the potential difference across the 200 Ω resistor? 2.0 V 3.0 V 4.5 V 7.5 V There is not enough information to decide.

QuickCheck 23.7 The current through the 3 ohm resistor is 9 A 6 A 5 A Answer: E

Why is one bulb Brighter, 40 W vs. 100 W Which has the greatest resistance? In parallel, which carries the greatest current? Voltage Divider Slide 22-35

Kirchhoff’s Laws Slide 23-11

Series and Parallel Circuits There are two possible ways that you can connect the circuit. Series and parallel circuits have very different properties. We say two bulbs are connected in series if they are connected directly to each other with no junction in between.

Calculating Equivalent Resistance Slide 22-35

Example Problem A resistor connected to a power supply works as a heater. Which of the following two circuits will provide more power? Answer: The circuit on the left has an equivalent resistance of 18+18 = 36 Ohms. The power is V^2/R = 81/36. For the circuit on the right, the equivalent resistance is 1/(1/18+1/18) = 1/(1/9) = 9 Ohms, so the power is 81/9. The power is greater for the circuit on the right because the resistance is lower leading to a larger current with the same voltage.

Equivalent Resistance Examples By using only two resistors singly, in series, or in parallel you are able to obtain resistances of 18.0, 24.0, 72, and 96 . What are the two resistances? (Make sure your answer is consistent.) Find the equivalent resistance of the following circuit: Be sure to show each step. Find the heat energy generated by the 3 Ohm and the 12 Ohm resistors in the circuit in problem 2 Slide 22-35

Circuit Analysis with Equivalent Resistance For the following circuit, calculate the equivalent resistance for this system of resistors: Now find the Equivalent resistance for this system for the following case: R1 = R2 = R3= R4 = R5 = 10 Ohms Find Delta V, I, and Power dissipated by R1 Slide 22-35

QuickCheck 23.13 The battery current I is 3 A 2 A 1 A 2/3 A 1/2 A Answer: D

The battery current I is 3 A 2 A 1 A 2/3 A 1/2 A QuickCheck 23.13 The battery current I is 3 A 2 A 1 A 2/3 A 1/2 A

QuickCheck 23.5 Which is the correct circuit diagram for the circuit shown? Answer: A

QuickCheck 23.5 Which is the correct circuit diagram for the circuit shown? A

Circuit Analysis using Kirchoff’s rules (Loop + Junction) Slide 23-11

Circuit from End of Circuit Activity 1 Circuit Analysis using Kirchoff’s rules (Loop + Junction) Circuit from End of Circuit Activity 1 Slide 22-35

Circuit Analysis using Kirchoff’s rules (Loop + Junction) Slide 22-35

Capacitors in Parallel and Series Parallel or series capacitors can be represented by a single equivalent capacitance.

Capacitor Combinations Slide 19-24

QuickCheck Which of the following combinations of capacitors has the highest capacitance? Answer: B

Equivalent Capacitance  = RC Slide 23-29

RC Circuits RC circuits are circuits containing resistors and capacitors. In RC circuits, the current varies with time. The values of the resistance and the capacitance in an RC circuit determine the time it takes the capacitor to charge or discharge.

RC Circuit Demonstrations

RC Circuits The figure shows an RC circuit consisting of a charged capacitor, an open switch, and a resistor before the switch closes. The switch will close at t = 0.

RC Circuits

RC Circuits The initial potential difference is (ΔVC)0 = Q0/C. The initial current—the initial rate at which the capacitor begins to discharge—is

RC Circuits

RC Circuits After some time, both the charge on the capacitor (and thus the potential difference) and the current in the circuit have decreased. When the capacitor voltage has decreased to ΔVC, the current has decreased to The current and the voltage decrease until the capacitor is fully discharged and the current is zero.

RC Circuits The current and the capacitor voltage decay to zero after the switch closes, but not linearly.

RC Circuits The decays of the voltage and the current are exponential decays:

RC Circuits The current and voltage in a circuit do not drop to zero after one time constant. Each increase in time by one time constant causes the voltage and current to decrease by a factor of e−1 = 0.37.

RC Circuits

RC Circuits The time constant has the form τ = RC. A large resistance opposes the flow of charge, so increasing R increases the decay time. A larger capacitance stores more charge, so increasing C also increases the decay time.

QuickCheck The following circuits contain capacitors that are charged to 5.0 V. All of the switches are closed at the same time. After 1 second has passed, which capacitor is charged to the highest voltage? Answer: C

Charging a Capacitor In a circuit that charges a capacitor, once the switch is closed, the potential difference of the battery causes a current in the circuit, and the capacitor begins to charge. As the capacitor charges, it develops a potential difference that opposes the current, so the current decreases, and so does the rate of charging. The capacitor charges until ΔVC = ℇ.

Charging a Capacitor The equations that describe the capacitor voltage and the current as a function of time are

QuickCheck 23.26 The red curve shows how the capacitor charges after the switch is closed at t = 0. Which curve shows the capacitor charging if the value of the resistor is reduced? B Smaller time constant. Same ultimate amount of charge. © 2015 Pearson Education, Inc.

Example Problem In the circuit shown below, rank in order, from most to least bright, the brightness of bulbs A–E. Explain. Answer: A, D&E (tie), B&C (tie) Slide 23-42

In the circuit shown below: Example Problem In the circuit shown below: Rank in order, from most to least bright, the brightness of bulbs A–D. Explain. Describe what, if anything, happens to the brightness of bulbs A, B, and D if bulb C is removed from its socket. Explain. Answer: A&D (tie), B&C (tie) Slide 23-41

Example Problem In the circuit shown below, use Kirchhoff's rules to write equations to find the currents. Answer: A, D&E (tie), B&C (tie) Slide 23-42

Capacitor Combinations Slide 19-24

Equivalent Capacitance  = RC Slide 23-29

Equivalent Capacitance  = RC Slide 23-29

Slide 23-29

Slide 23-29

RC Circuits  = RC Slide 23-29

RC Circuits  = RC Slide 23-29

Slide 23-29

 = RC Slide 23-29

Clicker Question 2 The following circuits contain capacitors that are charged to 5.0 V. All of the switches are closed at the same time. After 1 second has passed, which capacitor is charged to the highest voltage? Answer: C Slide 23-30

Answer The following circuits contain capacitors that are charged to 5.0 V. All of the switches are closed at the same time. After 1 second has passed, which capacitor is charged to the highest voltage? Answer: C Slide 23-31

Additional Clicker Questions In the circuit shown below, the switch is initially closed and the bulb glows brightly. When the switch is opened, what happens to the brightness of the bulb? The brightness of the bulb is not affected. B. The bulb gets dimmer. C. The bulb gets brighter. The bulb initially brightens, then dims. E. The bulb initially dims, then brightens. Answer: B Slide 23-39

Answer In the circuit shown below, the switch is initially closed and the bulb glows brightly. When the switch is opened, what happens to the brightness of the bulb? B. The bulb gets dimmer. Answer: B Slide 23-40