1. Series and Parallel Circuits Direct Current Circuits

2. Series circuits A series circuit is formed in a circuit like this one, where current flows through one path that travels through one device after another (the two light bulbs). If you measure the current at any point in this circuit, it will be the same.

3. Resistors in Series When two or more resistors are connected end-to-end, they are said to be 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 resistors in series because any charge that flows through one resistor flows through the other. The current is the same in resistors in series because any charge that flows through one resistor flows through the other. The sum of the potential differences (voltage drops) across the resistors is equal to the total potential difference across the combination (the battery or power supply voltage). The sum of the potential differences (voltage drops) across the resistors is equal to the total potential difference across the combination (the battery or power supply voltage).

4. Resistors in Series Potentials add Potentials add ΔV = IR 1 + IR 2 ΔV = IR 1 + IR 2 = I (R 1 +R 2 ) = I (R 1 +R 2 ) Consequence of Conservation of Energy Consequence of Conservation of Energy The equivalent resistance R 1 +R 2 has the same effect on the circuit as the original combination of resistors The equivalent resistance R 1 +R 2 has the same effect on the circuit as the original combination of resistors

5. Equivalent Resistance – Series R eq = R 1 + R 2 + R 3 + … R eq = R 1 + R 2 + R 3 + … 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 resistances. 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 resistances. Note that the equivalent resistance is always LARGER than any of the individual resistances when in series. Note that the equivalent resistance is always LARGER than any of the individual resistances when in series.

6. Equivalent Resistance – Series An Example

7. QUICK QUIZ 1 When a piece of wire is used to connect points b and c in this figure, the brightness of bulb R 1 (a) increases, (b) decreases, or (c) stays the same. The brightness of bulb R 2 (a) increases, (b) decreases, or (c) stays the same.

8. QUICK QUIZ 1 ANSWER R 1 becomes brighter. Connecting a wire from b to c provides a nearly zero resistance path from b to c and decreases the total resistance of the circuit from R 1 + R 2 to just R 1. Ignoring internal resistance, the potential difference maintained by the battery is unchanged while the resistance of the circuit has decreased. The current passing through bulb R 1 increases, causing this bulb to glow brighter. Bulb R 2 goes out because essentially all of the current now passes through the wire connecting b and c and bypasses the filament of Bulb R 2.

9. QUICK QUIZ 2 With the switch in this circuit (figure a) closed, no current exists in R 2 because the current has an alternate zero-resistance path through the switch. Current does exist in R 1 and this current is measured with the ammeter at the right side of the circuit. If the switch is opened (figure b), current exists in R 2. After the switch is opened, the reading on the ammeter (a) increases, (b) decreases, (c) does not change.

10. QUICK QUIZ 2 ANSWER (b). When the switch is opened, resistors R 1 and R 2 are in series, so that the total circuit resistance is larger than when the switch was closed. As a result, the current decreases.

11. Voltage Divider Circuit

12. Resistors in Parallel For resistors in parallel, the potential difference across each resistor (from a to b) is the same because each is connected directly across the battery terminals. For resistors in parallel, the potential difference across each resistor (from a to b) 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. The current, I, that enters a point must be equal to the total current leaving that point. I = I 1 + I 2 I = I 1 + I 2 The currents I 1 and I 2 are generally not the same. The currents I 1 and I 2 are generally not the same. This is a consequence of conservation of charge. This is a consequence of conservation of charge.

13. Equivalent Resistance – Parallel, Examples 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

14. Equivalent Resistance – Parallel Circuits Equivalent Resistance 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 inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance. For parallel R’s, the equivalent R eq is always less than the smallest resistor in the group.

15. Problem-Solving Strategy, 1 When two or more unequal resistors are connected in series, they carry the same current, but the potential differences across them are not the same. When two or more unequal resistors are connected in series, they carry the same current, but the potential differences across them are not the same. The resistors add directly to give the equivalent resistance of the series combination R eq = R 1 + R 2 The resistors add directly to give the equivalent resistance of the series combination R eq = R 1 + R 2 The equivalent resistance is always larger than any of the individual resistances. The equivalent resistance is always larger than any of the individual resistances.

16. Problem-Solving Strategy, 2 When two or more unequal resistors are connected in parallel, the potential differences (voltage drop) across them are the same. The currents through them are not the same. When two or more unequal resistors are connected in parallel, the potential differences (voltage drop) 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 The equivalent resistance of a parallel combination is found through reciprocal addition The equivalent resistance is always less than the smallest individual resistor in the combination The equivalent resistance is always less than the smallest individual resistor in the combination

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

18. Problem-Solving Strategy, 4 If the current 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 If the current 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 Use ΔV = I R and the procedures in steps 1 and 2

19. QUICK QUIZ 3 With the switch in this circuit (figure a) open, there is no current in R 2. There is current in R 1 and this current is measured with the ammeter at the right side of the circuit. If the switch is closed (figure b), there is current in R 2. When the switch is closed, the reading on the ammeter (a) increases, (b) decreases, or (c) remains the same.

20. QUICK QUIZ 3 ANSWER (a). When the switch is closed, resistors R 1 and R 2 are in parallel, so that the total circuit resistance is smaller than when the switch was open. As a result, the total current increases.

21. Equivalent Resistance – Complex Circuit

22. Kirchhoff’s Rules There are ways in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor There are ways in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor Two rules, called Kirchhoff’s Rules can be used instead Two rules, called Kirchhoff’s Rules can be used instead

23. Statement of Kirchhoff’s Rules Junction Rule Junction Rule The sum of the currents entering any junction must equal the sum of the currents leaving that junction The sum of the currents entering any junction must equal the sum of the currents leaving that junction A statement of Conservation of Charge A statement of Conservation of Charge Loop Rule Loop Rule The sum of the potential differences across all the elements around any closed circuit loop must be zero The sum of the potential differences across all the elements around any closed circuit loop must be zero A statement of Conservation of Energy A statement of Conservation of Energy

24. More About the Junction Rule I 1 = I 2 + I 3 I 1 = I 2 + I 3 From Conservation of Charge From Conservation of Charge Diagram b shows a mechanical analog Diagram b shows a mechanical analog

25. Setting Up Kirchhoff’s Rules Assign symbols and directions to the currents in all branches of the circuit Assign symbols and directions to the currents in all branches of the circuit If a direction is chosen incorrectly, the resulting answer will be negative, but the magnitude will be correct If a direction is chosen incorrectly, the resulting answer will be negative, but the magnitude will be correct When applying the loop rule, choose a direction for traversing the loop When applying the loop rule, choose a direction for traversing the loop Record voltage drops and rises as they occur Record voltage drops and rises as they occur

26. More About the Loop Rule Traveling around the loop from a to b Traveling around the loop from a to b In a, the resistor is traversed in the direction of the current, the potential across the resistor is –IR In a, the resistor is traversed in the direction of the current, the potential across the resistor is –IR In b, the resistor is traversed in the direction opposite of the current, the potential across the resistor is +IR In b, the resistor is traversed in the direction opposite of the current, the potential across the resistor is +IR

27. Loop Rule, final In c, the source of emf is traversed in the direction of the emf (from – to +), the change in the electric potential is +ε In c, the source of emf is traversed in the direction of the emf (from – to +), the change in the electric potential is +ε In d, the source of emf is traversed in the direction opposite of the emf (from + to -), the change in the electric potential is -ε In d, the source of emf is traversed in the direction opposite of the emf (from + to -), the change in the electric potential is -ε

28. Junction Equations from Kirchhoff’s Rules Use the junction rule as often as needed so long as, each time you write an equation, you include in it a current that has not been used in a previous junction rule equation Use the junction rule as often as needed so long as, each time you write an equation, you include in it a current that has not been used in a previous junction rule equation In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit

29. Loop Equations from Kirchhoff’s Rules The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation You need as many independent equations as you have unknowns You need as many independent equations as you have unknowns

30. Problem-Solving Strategy – Kirchhoff’s Rules Draw the circuit diagram and assign labels and symbols to all known and unknown quantities. Assign directions to the currents. Draw the circuit diagram and assign labels and symbols to all known and unknown quantities. Assign directions to the currents. Apply the junction rule to any junction in the circuit Apply the junction rule to any junction in the circuit Apply the loop rule to as many loops as are needed to solve for the unknowns Apply the loop rule to as many loops as are needed to solve for the unknowns Solve the equations simultaneously for the unknown quantities. Solve the equations simultaneously for the unknown quantities. For example 2 equations and 2 unknowns For example 2 equations and 2 unknowns or 3 equations and 3 unknowns or 3 equations and 3 unknowns

31. Example 1 by standard method I1I1 I2I2 Left loop: 18 - 5I 1 - 5I 2 - 1.5 I 1 = 0 Whole loop: 18 - 5I 1 - 5 I 3 - 5I 3 -1.5I 1 = 0 Junction: I 1 = I 2 + I 3 Three equations, three unknowns, solve. You should get I 1 = 1.83A, I 2 = 1.22A, I 3 = 0.61A I3I3 I3I3 I1I1

32. Example 1- mesh method I1I1 I2I2 Loop 1: 18 - 5I 1 - 5(I 1 -I 2 ) - 1.5 I 1 = 0 Loop 2: 0 - 5I 2 - 5 I 2 - 5(I 2 -I 1 ) = 0 By using the mesh method, you have one less equation to solve. Two equations, two unknowns, solve. You should get I 1 = 1.83A, I 2 = 0.61

33. Example 2 – stand.method I1I1 I2I2 Full loop: 12 – 1(I 1 ) – 3(I 1 ) – 8(I 3 ) = 0 Left loop: 4 – 1(I 2 ) – 5(I 2 ) – 8(I 3 ) = 0 Junction: I 1 + I 2 = I 3 3 equations, 3 unknowns, solve for I 1, I 2, I 3 I 1 = 1.31A, I 2 = -.46A, I 3 = 0.85A Try this one I1I1 I3I3

34. Example 2 mesh method I1I1 I2I2 Loop 1: 4 – 1(I 1 -I 2 ) – 5(I 1 -I 2 ) – 8(I 1 ) = 0 Loop 2: 12 – 1(I 2 ) – 3(I 2 ) – 5(I 2 -I 1 ) – 1(I 2 -I 1 ) – 4 = 0 2 equations, 2 unknowns, solve for I 1, I 2 I 1 = 0.85, I 2 = 1.31, I 2 – I 1 = 0.46A Try this one

35. Example 3 – mesh method I2I2 I1I1 Loop 1: 20 – 30I 1 – 5 (I 1 -I 2 ) – 10 = 0 Loop 2: 10 - 5(I 2 -I 1 ) – 20(I 2 ) = 0 2 equations, 2 unknowns, solve for I 1, I 2 I 1 = 0.353A, I 2 = 0.47A Try this one

36. Capacitors A parallel plate capacitor consists of two metal plates, one carrying charge +q and the other carrying charge –q. It is the capacity to store charge that resulted in the name capacitor. It is common to fill the region between the plates with an electrically insulating substance called a dielectric.

37. The relation between charge and voltage for a capacitor The magnitude of the charge on each plate of the capacitor is directly proportional to the magnitude of the potential difference between the plates. q = CV The capacitance C is the proportionality constant. SI Unit of Capacitance: coulomb/volt = farad (F) Typical measurements are in microfarads ( μF) or picofarads (pF).

38. Capacitors THE DIELECTRIC CONSTANT The insulating material between the plates is called a dielectric. If a dielectric is inserted between the plates of a capacitor, the capacitance can increase markedly. E o and E are, respectively, the magnitudes of the E field between the plates without and with a dielectric. Dielectric constant

39. Capacitors

40. Capacitance for parallel plate capacitors Parallel plate capacitor filled with a dielectric o = permittivity of free space = 8.85E-12 C 2 /N m 2 = 8.85E-12 C 2 /N m 2

41. Energy storage in a capacitor V =Ed

42. Capacitors in parallel Parallel capacitors

43. Capacitors in series Series capacitors

44. Capacitor charging time constant

45. Capacitor discharging time constant

46. Electrical Safety Electric shock can result in fatal burns Electric shock can result in fatal burns Electric shock can cause the muscles of vital organs (such as the heart) to malfunction Electric shock can cause the muscles of vital organs (such as the heart) to malfunction The degree of damage depends on The degree of damage depends on the magnitude of the current the magnitude of the current the length of time it acts the length of time it acts the part of the body through which it passes the part of the body through which it passes

47. Effects of Various Currents 5 mA or less 5 mA or less can cause a sensation of shock can cause a sensation of shock generally little or no damage generally little or no damage 10 mA 10 mA hand muscles contract hand muscles contract may be unable to let go a of live wire may be unable to let go a of live wire 100 mA 100 mA if passes through the body for 1 second or less, can be fatal if passes through the body for 1 second or less, can be fatal

48. Ground Wire Electrical equipment manufacturers use electrical cords that have a third wire, called a ground Electrical equipment manufacturers use electrical cords that have a third wire, called a ground Prevents shocks Prevents shocks

49. Ground Fault Interrupts (GFI) Special power outlets Special power outlets Used in hazardous areas Used in hazardous areas Designed to protect people from electrical shock Designed to protect people from electrical shock Senses currents (of about 5 mA or greater) leaking to ground Senses currents (of about 5 mA or greater) leaking to ground Shuts off the current when above this level Shuts off the current when above this level

50. Three types of safety devices There are three types of safety devices that are used in household electric circuits to prevent electric overcurrent conditions: There are three types of safety devices that are used in household electric circuits to prevent electric overcurrent conditions: Fuses Fuses Circuit Breakers Circuit Breakers Ground Fault Interrupters Ground Fault Interrupters