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Chapter 18

Chapter 18. Direct Current Circuits. DC Circuits. Sources of emf. The source that maintains the current in a closed circuit is called a source of emf Any devices that increase the potential energy of charges circulating in circuits are sources of emf

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Chapter 18

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  1. Chapter 18 Direct Current Circuits

  2. DC Circuits

  3. Sources of emf • The source that maintains the current in a closed circuit is called a source of emf • Any devices that increase the potential energy of charges circulating in circuits are sources of emf • Examples include batteries and generators • SI units are Volts • The emf is the work done per unit charge

  4. EMF and Terminal Voltage This resistance behaves as though it were in series with the emf.

  5. emf and Internal Resistance • A real battery has some internal resistance • Therefore, the terminal voltage is not equal to the emf

  6. More About Internal Resistance • The schematic shows the internal resistance, r • The terminal voltage is ΔV = Vb-Va • ΔV = ε – Ir • For the entire circuit, ε = IR + Ir

  7. Internal Resistance and emf, cont • ε is equal to the terminal voltage when the current is zero • Also called the open-circuit voltage • R is called the load resistance • The current depends on both the resistance external to the battery and the internal resistance

  8. Internal Resistance and emf, final • When R >> r, r can be ignored • Generally assumed in problems • Power relationship • I e = I2 R + I2 r • When R >> r, most of the power delivered by the battery is transferred to the load resistor

  9. Resistors in Series • When two or more resistors are connected end-to-end, they are said to be in series • The current is the same in all resistors because any charge that flows through one resistor flows through the other • The sum of the potential differences across the resistors is equal to the total potential difference across the combination

  10. A series connection has a single path from the battery, through each circuit element in turn, then back to the battery.

  11. Resistors in Series, cont • Potentials add • ΔV = IR1 + IR2 = I (R1+R2) • Consequence of Conservation of Energy • The equivalent resistance has the effect on the circuit as the original combination of resistors

  12. Equivalent Resistance – Series • Req = R1 + R2 + R3 + … • The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistors

  13. Equivalent Resistance – Series: An Example • Four resistors are replaced with their equivalent resistance

  14. The current through each resistor is the same. • The voltage depends on the resistance. • The sum of the voltage drops across the resistors equals the battery voltage.

  15. 9 V 1)12 V 2)zero 3)3 V 4) 4 V 5) you need to know the actual value of R Assume that the voltage of the battery is 9 V and that the three resistors are identical. What is the potential difference across each resistor?

  16. 9 V 1)12 V 2)zero 3)3 V 4) 4 V 5) you need to know the actual value of R Assume that the voltage of the battery is 9 V and that the three resistors are identical. What is the potential difference across each resistor? Since the resistors are all equal, the voltage will drop evenly across the 3 resistors, with 1/3 of 9 V across each one. So we get a 3 V drop across each.

  17. R1= 4 W R2= 2 W 12 V 1)12 V 2)zero 3)6 V 4) 8 V 5) 4 V In the circuit below, what is the voltage across R1?

  18. R1= 4 W R2= 2 W 12 V 1)12 V 2)zero 3)6 V 4) 8 V 5) 4 V In the circuit below, what is the voltage across R1? The voltage drop across R1 has to be twice as big as the drop across R2. This means that V1 = 8 V and V2 = 4 V. Or else you could find the current I = V/R = (12 V)/(6 W) = 2 A, then use Ohm’s Law to get voltages.

  19. Resistors in Parallel • The potential difference across each resistor is the same because each is connected directly across the battery terminals • The current, I, that enters a point must be equal to the total current leaving that point • I = I1 + I2 • The currents are generally not the same • Consequence of Conservation of Charge

  20. A parallel connection splits the current; the voltage across each resistor is the same:

  21. Equivalent Resistance – Parallel, Example • Equivalent resistance replaces the two original resistances • Household circuits are wired so the electrical devices are connected in parallel • Circuit breakers may be used in series with other circuit elements for safety purposes

  22. Equivalent Resistance – Parallel • Equivalent Resistance • The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance • The equivalent is always less than the smallest resistor in the group

  23. R2= 2 W R1= 5 W 10 V 1)10 A 2)zero 3)5 A 4) 2 A 5) 7 A In the circuit to the right, what is the current through R1?

  24. R2= 2 W R1= 5 W 10 V 1)10 A 2)zero 3)5 A 4) 2 A 5) 7 A In the circuit below, what is the current through R1? The voltage is the same (10 V) across each resistor because they are in parallel. Thus, we can use Ohm’s Law, V1 = I1 R1 to find the current I1 = 2 A.

  25. 1)increases 2)remains the same 3)decreases 4) drops to zero Points P and Q are connected to a battery of fixed voltage. As more resistors R are added to the parallel circuit, what happens to the total current in the circuit?

  26. 1)increases 2)remains the same 3)decreases 4) drops to zero Points P and Q are connected to a battery of fixed voltage. As more resistors R are added to the parallel circuit, what happens to the total current in the circuit? As we add parallel resistors, the overall resistance of the circuit drops. Since V = IR, and V is held constant by the battery, when resistance decreases, the current must increase.

  27. 1) all the current continues to flow through the bulb 2) half the current flows through the wire, the other half continues through the bulb 3)all the current flows through the wire 4) none of the above Current flows through a lightbulb. If a wire is now connected across the bulb, what happens?

  28. 1) all the current continues to flow through the bulb 2) half the current flows through the wire, the other half continues through the bulb 3)all the current flows through the wire 4) none of the above Current flows through a lightbulb. If a wire is now connected across the bulb, what happens? The current divides based on the ratio of the resistances. If one of the resistances is zero, then ALL of the current will flow through that path.

  29. 1) glow brighter than before 2) glow just the same as before 3) glow dimmer than before 4) go out completely 5) explode Two lightbulbs A and B are connected in series to a constant voltage source. When a wire is connected across B, bulb A will:

  30. 1) glow brighter than before 2) glow just the same as before 3) glow dimmer than before 4) go out completely 5) explode Two lightbulbs A and B are connected in series to a constant voltage source. When a wire is connected across B, bulb A will: Since bulb B is bypassed by the wire, the total resistance of the circuit decreases. This means that the current through bulb A increases.

  31. 1) circuit 1 2) circuit 2 3) both the same 4) it depends on R The lightbulbs in the circuit below are identical with the same resistance R. Which circuit produces more light? (brightness  power)

  32. 1) circuit 1 2) circuit 2 3) both the same 4) it depends on R The lightbulbs in the circuit below are identical with the same resistance R. Which circuit produces more light? (brightness  power) In #1, the bulbs are in parallel, lowering the total resistance of the circuit. Thus, circuit #1 willdraw a higher current, which leads to more light, because P = I V.

  33. A C B 10 V 1) twice as much 2) the same 3) 1/2 as much 4) 1/4 as much 5) 4 times as much The three lightbulbs in the circuit all have the same resistance of1 W . By how much is the brightness of bulb B greater or smaller than the brightness of bulb A? (brightness  power)

  34. A C B 10 V 1) twice as much 2) the same 3) 1/2 as much 4) 1/4 as much 5) 4 times as much The three light bulbs in the circuit all have the same resistance of1 W . By how much is the brightness of bulb B greater or smaller than the brightness of bulb A? (brightness  power) We can use P = V2/R to compare the power: PA = (VA)2/RA = (10 V) 2/1 W= 100 W PB = (VB)2/RB = (5 V) 2/1 W= 25 W

  35. R1 S R3 V R2 1) increase 2) decrease 3) stay the same What happens to the voltage across the resistor R1 when the switch is closed? The voltage will:

  36. R1 S R3 V R2 1) increase 2) decrease 3) stay the same What happens to the voltage across the resistor R1 when the switch is closed? The voltage will: With the switch closed, the addition of R2 to R3decreases the equivalent resistance, so the current from the battery increases. This will cause an increase in the voltage across R1 .

  37. R1 S R3 R4 V R2 1) increases 2) decreases 3) stays the same What happens to the voltage across the resistor R4 when the switch is closed?

  38. R1 B A S R3 R4 V R2 C 1) increases 2) decreases 3) stays the same What happens to the voltage across the resistor R4 when the switch is closed? We just saw that closing the switch causes an increase in the voltage across R1 (which is VAB). The voltage of the battery is constant, so if VAB increases, then VBC must decrease!

  39. R5 R2 R3 R1 R4 V 1) R1 2) both R1and R2 equally 3) R3and R4 4) R5 5) all the same Which resistor has the greatest current going through it? Assume that all the resistors are equal.

  40. R5 R2 R3 R1 R4 V Which resistor has the greatest current going through it? Assume that all the resistors are equal. 1) R1 2) both R1and R2 equally 3) R3and R4 4) R5 5) all the same The same current must flow through left and right combinations of resistors. On the LEFT, the current splits equally, so I1 = I2. On the RIGHT, more current will go through R5 than R3+ R4 since the branch containing R5 has less resistance.

  41. Problem-Solving Strategy, 1 • Combine all resistors in series • They carry the same current • The potential differences across them are not the same • The resistors add directly to give the equivalent resistance of the series combination: Req = R1 + R2 + …

  42. Problem-Solving Strategy, 2 • Combine all resistors in parallel • The potential differences across them are the same • The currents through them are not the same • The equivalent resistance of a parallel combination is found through reciprocal addition:

  43. Problem-Solving Strategy, 3 • A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistor • Replace any resistors in series or in parallel using steps 1 or 2. • Sketch the new circuit after these changes have been made • Continue to replace any series or parallel combinations • Continue until one equivalent resistance is found

  44. Problem-Solving Strategy, 4 • If the current in or the potential difference across a resistor in the complicated circuit is to be identified, start with the final circuit found in step 3 and gradually work back through the circuits • Use ΔV = I R and the procedures in steps 1 and 2

  45. Equivalent Resistance – Complex Circuit

  46. An analogy using water may be helpful in visualizing parallel circuits:

  47. Example 1 A 4.0-Ω resistor, an 8.0-Ω resistor, and a 12-Ω resistor areconnected in series with a 24-V battery. What are (a) theequivalent resistance and (b) the current in each resistor?(c) Repeat for the case in which all three resistors are connectedin parallel across the battery.

  48. Example 2 A 9.0-Ω resistor and a 6.0-Ω resistor are connected inseries with a power supply. (a) The voltage drop acrossthe 6.0-Ω resistor is measured to be 12 V. Find the voltageoutput of the power supply. (b) The two resistors areconnected in parallel across a power supply, and the currentthrough the 9.0-Ω resistor is found to be 0.25 A. Find thevoltage setting of the power supply.

  49. Example 3 (a) Find the equivalent resistance of the circuit in FigureP18.8. (b) If the total power supplied to the circuit is4.00 W, find the emf of the battery.

  50. Example 4 Three 100-Ω resistors are connected as shown in FigureP18.12. The maximum power that can safely be deliveredto any one resistor is 25.0 W. (a) What is the maximumvoltage that can be applied to the terminals a and b?(b) For the voltage determined in part (a), what is thepower delivered to each resistor? What is the total powerdelivered?

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