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Chapter 20

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20.1

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Series Circuits Have only ONE LOOP or circuit for the current to travel through.

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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 voltages across the resistors is equal to the total voltages across the combination Kirchoffs Voltage Law, the Conservation of Voltage

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

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Equivalent Resistance Series R e = 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 resistors Batteries and even wires contribute small amounts of resistance but we can ignore this for now

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Equivalent Resistance Series Four resistors are replaced with their equivalent resistance An Example

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Resistors in Parallel The voltage 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 = I 1 + I 2 The currents are generally not the same Consequence of Kirchoffs Second Law, the Conservation of Charge

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Equivalent Resistance Parallel 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 An Example

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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

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Example 1 In the depicted circuit, the voltage supplied by the battery is 12V, and the resistors have values of R 1 = 10, R 2 = 5 and R 3 = 15. What is the current flowing through each branch?

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Example 1 The current through each resistor can be found with Ohms Law I 1 = 1.2A;I 2 = 2.4A;I 3 = 0.8A To check this, find the current through the whole circuit by finding the total resistance, then using Ohms Law again. R tot = 2.7I tot = 4.4A This matches the sum of the individual currents

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Example 2 If, V = 24V; R 1 = 2; R 2 = 3.3; R 3 = 7 and R 4 = 12.2 Find the current through the circuit and the current through each resistor.

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Example 2 Find the total resistance: R tot = o.973 Use this to find the current through the circuit I = 24.7A The current through each resistor is just the voltage divided by each individual resistance: I 1 = 12A;I 2 = 7.3A; I 3 = 3.4AI 4 = 2A

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Series Circuit A series of sources separated by resistors is equivalent to a single source having the net voltage and a single resistor having the combined resistance. R1R1 A B R2R2 R3R3 R4R4 V1V1 V2V2 V3V3 R = R 1 + R 2 + R 3 + R 4 V = V 1 - V 2 + V 3 The same current passes through every resistor in a given branch, regardless of the presence of sources in that branch, and the resistors are in series even though they are not directly connected to one another..

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Parallel Circuit The current in each branch of a parallel circuit depends inversely on the total resistance: the larger the resistance, the less current flows through the branch If we know I but not V R1R1 R2R2 V

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20.2

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Review of Circuit Rules For SERIES Circuits: There is ONE path for the current Current is CONSTANT Voltage DROPS across each resistance Resistors add simply Additional resistances DECREASE current

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Review of Circuit Rules For PARALLEL Circuits: There are MULTIPLE paths for the current Current may NOT be CONSTANT Voltage is CONSTANT to each branch Resistors add RECIPROCALLY Additional resistance INCREASES current

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Review of Circuit Rules Sometimes you will have BOTH series AND parallel resistors in the SAME circuit!! You then need to SIMPLIFY the circuit in your analysis.

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Example 3 Consider this circuit. (a) If possible, simplify it and determine an equivalent resistance between C and G. (b) What current is provided by the source? (c) What is the voltage across points G and E? Given: Nine resistors, R = 1.0 k each, and V = 12V Find: R e, I, and V between G and E

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Example 3 Solution To solve this one we'll need the equivalent resistance of the circuit Procedure Redraw this circuit to make it look more manageable Lift up the inside square, with the resistor and source attached, and place it outside E-F-G-H see diagram on right Branches A-B-C and A-D-C are in parallel, as are E-F-G and E-H-G, and each has a resistance of 1.0 k + 1.0 k = 2.0 k (resistors add in series)

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Example 3 The equivalent circuit is shown to the right The resistance of each square E-F-G-H and A-B- C-D reduces to and R = 1.0 k .

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Example 3 The three 1.0-k resistors then are in series with the source, (a) The equivalent resistance is 3.0 k . (b) Since V = IR e,

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Example 3 (c) A current of 4.0 mA leaves the battery and splits at C Because the two branches C-D-A and C-B-A have the same resistance, the current divides into two equal streams of 2.0 mA each The voltage drop in going from C to A is given by V AC = IR = (2.0 mA)(1.0 k + 1.0 k ) = 4.0 V In going from A to E, there is another drop of V EA = IR = (4.0 mA)(1.0 k ) = 4.0 V. C is 12V above G, A is 8.0 V above G, and E is 4.0V above G.

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Example 4 Consider the following circuit: A battery supplying 12 volts leads to a resistor (R 1 = 1.3), then splits into three branches. The first branch has R 2 = 4.5, the second branch has R 3 = , and the third branch contains R 4 = 5 AND R 5 = 2.2 in series. Finally, the three branches reunite, and lead to R 6 = 7 before reconnecting to the battery. Draw a diagram of this circuit Find the total resistance Find the overall current in the circuit.

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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: R e = R 1 + R 2 +

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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:

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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 1. Replace any resistors in series or in parallel using steps 1 or 2. 2. Sketch the new circuit after these changes have been made 3. Continue to replace any series or parallel combinations 4. Continue until one equivalent resistance is found

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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 V = I R and the procedures in steps 1 and 2

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Equivalent Resistance Complex Circuit

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Capacitors in Circuits A circuit is a collection of objects usually containing a source of electrical energy (such as a battery) connected to elements that convert electrical energy to other forms A circuit diagram can be used to show the path of the real circuit

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Capacitors in Parallel When capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged The flow of charges ceases when the voltage across the capacitors equals that of the battery The capacitors reach their maximum charge when the flow of charge ceases

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Capacitors in Parallel The total charge is equal to the sum of the charges on the capacitors Q total = Q 1 + Q 2 The potential difference across the capacitors is the same And each is equal to the voltage of the battery

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More About Capacitors in Parallel The capacitors can be replaced with one capacitor with a capacitance of C eq The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors Capacitors in parallel all have the same voltage differences as does the equivalent capacitance

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Capacitors in Parallel The equivalent capacitance of several capacitors in parallel is the sum of all the individual capacitors. C = C 1 + C 2 + The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors

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Example 5 The figure below shows two capacitors attached to a 12-V battery. Determine the equivalent capacitance and the charge it would carry. What is the charge on each of the capacitors in the figure? Given: C 1 = 20 F, C 2 = 30 F, and V = 12 V Find: C, Q, Q 1, and Q 2 +++ -- - 20 F30 F 12 V

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Example 5 Solution: Capacitors are in parallel and the potential across each capacitor is 12 V Q = CV = (50 x 10 -6 F)(12 V) = 6.0 x 10 -4 C

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Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C 1 to the right plate of C 2 through the battery As this negative charge accumulates on the right plate of C 2, an equivalent amount of negative charge is removed from the left plate of C 2, leaving it with an excess positive charge All of the right plates gain charges of Q and all the left plates have charges of +Q

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More About Capacitors in Series An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage Capacitors in series all have the same charge, Q, as does their equivalent capacitance

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Capacitors in Series The equivalent capacitance of several capacitors in series The equivalent capacitance of a series combination is always less than any individual capacitor in the combination

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Example 6 The circuit shown in the figure consists of a 12-V battery and three capacitors. It is redrawn from Fig. 12.27a in the book. Determine both the voltage across and charge on each capacitor after the switch S is closed and electrostatic equilibrium is established. Find the equivalent capacitance of the network. Given: C 1 = 2.0 F, C 2 = 2.0 F, C 3 = 5.0 F, and V = 12 V Find: C, V 1, V 2, V 3, Q 1, Q 2, and Q 3 + 2.0 F 12 V 2.0 F 5.0 F C3C3 C1C1 C2C2

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Example 6 Combining C 1 and C 2 which are in series + 12 V 1.0 F 5.0 F C3C3 C 1 + C 2

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Example 6 Combining C 3 and (C 1 + C 2 ) which are in parallel + 12 V 6.0 F C The equivalent capacitance of the network

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Example 6 There is 12 V across C 3 so Q 3 = C 3 V 3 = (5.0 F)(12 V) = 60 C + 2.0 F 12 V 2.0 F 5.0 F C3C3 C1C1 C2C2 There is 12 V across the combination of the two 2.0 mF capacitors so there is a potential difference of 6.0 V across each Q 1 = Q 2 = (2.0 F)(6.0 V) = 12 C

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Problem-Solving Strategy Be careful with the choice of units Combine capacitors following the formulas When two or more unequal capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor

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Problem-Solving Strategy, cont Combining capacitors When two or more capacitors are connected in parallel, the potential differences across them are the same The charge on each capacitor is proportional to its capacitance The capacitors add directly to give the equivalent capacitance

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Problem-Solving Strategy, final Repeat the process until there is only one single equivalent capacitor A complicated circuit can often be reduced to one equivalent capacitor Replace capacitors in series or parallel with their equivalent Redraw the circuit and continue To find the charge on, or the potential difference across, one of the capacitors, start with your final equivalent capacitor and work back through the circuit reductions

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20.3

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Household Circuits The utility company distributes electric power to individual houses with a pair of wires Electrical devices in the house are connected in parallel with those wires The potential difference between the wires is about 120V

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Household Circuits A meter and a circuit breaker are connected in series with the wire entering the house Wires and circuit breakers are selected to meet the demands of the circuit If the current exceeds the rating of the circuit breaker, the breaker acts as a switch and opens the circuit Household circuits actually use alternating current and voltage

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Household Usage Electricity usage is measured in kilowatt-hours (kWh) Watts are units of Power P = VI (for DC) Usually measured in Kilowatts or Horsepower 1hp = 746W 1kWh = 3.6 x 10 6 J

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Types of Current Direct Current (DC) charge flows uniformly in ONE direction Alternating Current (AC) charge flows in opposite directions alternating in a regular, periodic way with a given frequency. Peak vs. Average Voltage in AC the difference is the Average Voltage coming out of the wall is a percentage of the Peak Voltage supplied. **Alternating Current is both easier to generate AND to transmit long distances**

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Power in AC Power in AC circuits is calculated in the same way as in DC circuits but using average voltage values instead of peak. Peak values for current (I) can be found using Peak Voltage when resistance (R) is known. I freakin HATE Circuit analysis!!

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Reactance in AC In AC circuits, where the current is constantly (and ver rapidly) reversing, if you have anything other than simple resistance (Capacitors or Inductors), the response time or Reactance of those components will create a lag time in voltage response to the current shift. Inductive Reactance Capacitive Reactance

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

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Effects of Various Currents 5 mA or less Can cause a sensation of shock Generally little or no damage 10 mA Hand muscles contract May be unable to let go a of live wire 100 mA If passes through the body for just a few seconds, can be fatal

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