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**Direct Current Circuits**

Chapter 18 Direct Current Circuits

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DC Circuits

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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

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**EMF and Terminal Voltage**

This resistance behaves as though it were in series with the emf.

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**emf and Internal Resistance**

A real battery has some internal resistance Therefore, the terminal voltage is not equal to the emf

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**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

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**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

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**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

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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 potential differences across the resistors is equal to the total potential difference across the combination

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

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**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

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**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

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**Equivalent Resistance – Series: An Example**

Four resistors are replaced with their equivalent resistance

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**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.

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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? 9 V

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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? 1) 12 V 2) zero 3) 3 V 4) 4 V 5) you need to know the actual value of R 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. 9 V

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**In the circuit below, what is the voltage across R1?**

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

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**In the circuit below, what is the voltage across R1?**

2) zero 3) 6 V 4) 8 V 5) 4 V 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. 12 V R1= 4 W R2= 2 W

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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

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

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**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

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**Equivalent Resistance – Parallel**

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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**In the circuit to the right, what is the current through R1?**

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

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**In the circuit below, what is the current through R1?**

2) zero 3) 5 A 4) 2 A 5) 7 A 10 V R1= 5 W R2= 2 W 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.

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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?

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**Points P and Q are connected to a battery of fixed voltage**

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? 1) increases 2) remains the same 3) decreases 4) drops to zero 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.

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**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?

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**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.

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**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:

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**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.

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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)

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

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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 of 1 W . By how much is the brightness of bulb B greater or smaller than the brightness of bulb A? (brightness power) 10 V A B C

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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 of 1 W . By how much is the brightness of bulb B greater or smaller than the brightness of bulb A? (brightness power) 10 V A B C 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

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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: V R1 R3 R2 S

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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: V R1 R3 R2 S With the switch closed, the addition of R2 to R3 decreases the equivalent resistance, so the current from the battery increases. This will cause an increase in the voltage across R1 .

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1) increases 2) decreases 3) stays the same What happens to the voltage across the resistor R4 when the switch is closed? V R1 R3 R4 R2 S [CORRECT 5 ANSWER]

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**What happens to the voltage across the resistor R4 when the switch is closed?**

1) increases 2) decreases 3) stays the same V R1 R3 R4 R2 S A B C 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!

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

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**Which resistor has the greatest current going through it**

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

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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: Req = R1 + R2 + …

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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 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

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

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**Equivalent Resistance – Complex Circuit**

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

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

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Example 2 A 9.0-Ω resistor and a 6.0-Ω resistor are connected in series with a power supply. (a) The voltage drop across the 6.0-Ω resistor is measured to be 12 V. Find the voltage output of the power supply. (b) The two resistors are connected in parallel across a power supply, and the current through the 9.0-Ω resistor is found to be 0.25 A. Find the voltage setting of the power supply.

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Example 3 (a) Find the equivalent resistance of the circuit in Figure P18.8. (b) If the total power supplied to the circuit is 4.00 W, find the emf of the battery.

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

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Practice 1 What is the equivalent resistance of the combination of resistors between points a and b in Figure P18.7? Note that one end of the vertical resistor is left free.

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Practice 2 (a) You need a 45-Ω resistor, but the stockroom has only 20-Ω and 50-Ω resistors. How can the desired resistance be achieved under these circumstances? (b) What can you do if you need a 35-Ω resistor?

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**Some circuits cannot be broken down into series and parallel connections.**

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**Gustav Kirchhoff 1824 – 1887 Invented spectroscopy with Robert Bunsen**

Formulated rules about radiation

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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 Two rules, called Kirchhoff’s Rules can be used instead

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**Statement of Kirchhoff’s Rules**

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

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**More About the Junction Rule**

I1 = I2 + I3 From Conservation of Charge Diagram b shows a mechanical analog

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**What is the current in branch P?**

[CORRECT 5 ANSWER]

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**What is the current in branch P?**

junction 6 A The current entering the junction in red is 8 A, so the current leaving must also be 8 A. One exiting branch has 2 A, so the other branch (at P) must have 6 A. S

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**Setting Up Kirchhoff’s Rules**

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 When applying the loop rule, choose a direction for transversing the loop Record voltage drops and rises as they occur

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**More About the Loop Rule**

Traveling around the loop from a to b In a, the resistor is transversed in the direction of the current, the potential across the resistor is –IR In b, the resistor is transversed in the direction opposite of the current, the potential across the resistor is +IR

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

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**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 In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit

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**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 You need as many independent equations as you have unknowns

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**EMFs in series in the same direction: total voltage is the sum of the separate voltages**

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EMFs in series, opposite direction: total voltage is the difference, but the lower-voltage battery is charged.

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EMFs in parallel only make sense if the voltages are the same; this arrangement can produce more current than a single emf.

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**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. 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 Solve the equations simultaneously for the unknown quantities Check your answers

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1) both bulbs go out 2) intensity of both bulbs increases 3) intensity of both bulbs decreases 4) A gets brighter and B gets dimmer 5) nothing changes The lightbulbs in the circuit are identical. When the switch is closed, what happens? [CORRECT 5 ANSWER]

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1) both bulbs go out 2) intensity of both bulbs increases 3) intensity of both bulbs decreases 4) A gets brighter and B gets dimmer 5) nothing changes The lightbulbs in the circuit are identical. When the switch is closed, what happens? When the switch is open, the point between the bulbs is at 12 V. But so is the point between the batteries. If there is no potential difference, then no current will flow once the switch is closed!! Thus, nothing changes.

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**Which of the equations is valid for the circuit below?**

1) 2 – I1 – 2I2 = 0 2) 2 – 2I1 – 2I2 – 4I3 = 0 3) 2 – I1 – 4 – 2I2 = 0 4) I3 – 4 – 2I = 0 5) 2 – I1 – 3I3 – 6 = 0 Which of the equations is valid for the circuit below? 2 V 2 6 V 4 V 3 1 I1 I3 I2

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**Which of the equations is valid for the circuit below?**

1) 2 – I1 – 2I2 = 0 2) 2 – 2I1 – 2I2 – 4I3 = 0 3) 2 – I1 – 4 – 2I2 = 0 4) I3 – 4 – 2I = 0 5) 2 – I1 – 3I3 – 6 = 0 Eqn. 3 is valid for the left loop: The left battery gives +2V, then there is a drop through a 1W resistor with current I1 flowing. Then we go through the middle battery (but from + to – !), which gives –4V. Finally, there is a drop through a 2W resistor with current I2. 2 V 2 6 V 4 V 3 1 I1 I3 I2

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Example 5 The ammeter shown in Figure P18.16 reads 2.00 A. Find I1, I2, and ε.

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Example 6 Determine the potential difference Є1 for the circuit in Figure P18.18.

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Example 7 In the circuit of Figure P18.20, the current I1 is 3.0 A while the values of ε and R are unknown. What are the currents I2 and I3?

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Example 8 Four resistors are connected to a battery with a terminal voltage of 12 V, as shown in Figure P Determine the power delivered to the 50-Ω resistor.

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Practice 3 Calculate each of the unknown currents I1, I2, and I3 for the circuit of Figure P18.25.

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RC Circuits A direct current circuit may contain capacitors and resistors, the current will vary with time When the circuit is completed, the capacitor starts to charge The capacitor continues to charge until it reaches its maximum charge (Q = Cε) Once the capacitor is fully charged, the current in the circuit is zero

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**When the switch is closed, the capacitor will begin to charge.**

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**The voltage across the capacitor increases with time:**

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**The charge follows a similar curve:**

This curve has a characteristic time constant:

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**If an isolated charged capacitor is connected across a resistor, it discharges:**

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Notes on Time Constant In a circuit with a large time constant, the capacitor charges very slowly The capacitor charges very quickly if there is a small time constant After t = 10 t, the capacitor is over 99.99% charged

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Example 9 An uncharged capacitor and a resistor are connected in series to a source of emf. If ε = 9.00 V, C = 20.0 μF, and R = 100 Ω, find (a) the time constant of the circuit, (b) the maximum charge on the capacitor, and (c) the charge on the capacitor after one time constant.

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Example 10 Consider a series RC circuit for which R = 1.0 MΩ, C = 5.0 μF, and ε = 30 V. Find the charge on the capacitor 10 s after the switch is closed.

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Practice 4 A series RC circuit has a time constant of s. The battery has an emf of 48.0 V, and the maximum current in the circuit is mA. What are (a) the value of the capacitance and (b) the charge stored in the capacitor 1.92 s after the switch is closed?

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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, cont.**

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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**A person receiving a shock has become part of a complete circuit.**

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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

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**Ground Fault Interrupts (GFI)**

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

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Example 11 An electric heater is rated at W, a toaster at W, and an electric grill at 1 500 W. The three appliances are connected in parallel to a common 120-V circuit. (a) How much current does each appliance draw? (b) Is a 30.0-A circuit breaker sufficient in this situation? Explain.

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Example 12 A lamp (R = 150 Ω), an electric heater (R = 25 Ω), and a fan (R = 50 Ω) are connected in parallel across a 120-V line. (a) What total current is supplied to the circuit? (b) What is the voltage across the fan? (c) What is the current in the lamp? (d) What power is expended in the heater?

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Practice 5

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