Chapter 25
description
Transcript of Chapter 25
Chapter 25
Electric Circuits
Direct Current
• When the current in a circuit has a constant direction, the current is called direct current
• Most of the circuits analyzed will be assumed to be in steady state, with constant magnitude and direction
• Because the potential difference between the terminals of a battery is constant, the battery produces direct current
• The battery is known as a source of emf
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 (e.g. batteries, generators, etc.)
• The emf (ε) is the work done per unit charge
• Emf of a source is the maximum possible voltage that the source can provide between its terminals
• SI units: Volts
Electric Circuits• A circuit is a collection of objects usually containing a
source of electrical energy connected to elements that convert electrical energy to other forms
• A circuit diagram – a simplified representation of an actual circuit – is used to show the path of the real circuit
• Circuit symbols are used to represent various elements
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
III 21
21 IRIRV )( 21 RRI eqIR
Resistors in Series
• The equivalent resistance has the effect on the circuit as the original combination of resistors (consequence of conservation of energy)
• For more resistors in series:
• The equivalent resistance of a series combination of resistors is greater than any of the individual resistors
eqIRV 21 RRReq
...321 RRRReq
emf and Internal Resistance
• A real battery has some internal resistance r; therefore, the terminal voltage is not equal to the emf
• The terminal voltage: ΔV = Vb – Va
ΔV = ε – Ir• For the entire circuit (R – load resistance):
ε = ΔV + Ir
= IR + Ir
emf and Internal Resistance
ε = ΔV + Ir = IR + Ir• ε is equal to the terminal voltage when the current is
zero – open-circuit voltage
I = ε / (R + r)• The current depends on both the resistance external to
the battery and the internal resistance
• When R >> r, r can be ignored
• Power relationship: I ε = I2 R + I2 r
• When R >> r, most of the power delivered by the battery is transferred to the load resistor
Resistors in Parallel
VRIRI 2211
21 III 21 R
V
R
V
• 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 (conservation of charge)
• The currents are generally not the same
Resistors in Parallel
VRIRI 2211
21 III 21 R
V
R
V
21
11
RRV
eqR
V
21
111
RRReq
• For more resistors in 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
Resistors in Parallel
321
1111
RRRReq
Problem-Solving Strategy
• Combine all resistors in series
• They carry the same current
• The potential differences across them are not necessarily the same
• The resistors add directly to give the equivalent resistance of the combination:
Req = R1 + R2 + …
Problem-Solving Strategy
• Combine all resistors in parallel
• The potential differences across them are the same
• The currents through them are not necessarily the same
• The equivalent resistance of a parallel combination is found through reciprocal addition:
...111
21
RRReq
Problem-Solving Strategy
• A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistor
• Replace resistors in series or in parallel with a single resistor
• 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
Problem-Solving Strategy
• If the current in or the potential difference across a resistor in the complicated circuit is to be identified, start with the final circuit and gradually work back through the circuits (use formula ΔV = I R and the procedures describe above)
Chapter 25Problem 20
What resistance should you place in parallel with a 56-kΩ resistor to make an equivalent resistance of 45 kΩ?
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 (Laws) can be used instead:
• 1) Junction (Node) Rule
• 2) Loop Rule
Gustav Kirchhoff1824 – 1887
Kirchhoff’s Rules
• Junction (Node) Rule (A statement of Conservation of Charge): The sum of the currents entering any junction must equal the sum of the currents leaving that junction
• Loop Rule (A statement of Conservation of Energy): The sum of the potential differences across all the elements around any closed circuit loop must be zero
Junction Rule
I1 = I2 + I3
• 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
Loop Rule• When applying the loop rule, choose
a direction for transversing the loop
• Record voltage drops and rises as they occur
• If a resistor is transversed in the direction of the current, the potential across the resistor is – IR
• If a resistor is transversed in the direction opposite of the current, the potential across the resistor is +IR
Loop Rule• If a source of emf is transversed in
the direction of the emf (from – to +), the change in the electric potential is +ε
• If a source of emf is transversed in the direction opposite of the emf (from + to -), the change in the electric potential is – ε
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
• The number of times the junction rule can be used is one fewer than the number of junction points in the circuit
• 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
Equations from Kirchhoff’s Rules
Problem-Solving Strategy
• 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
Chapter 25Problem 72
The figure shows a portion of a circuit used to model muscle cells and neurons. All resistors have the same value R = 1.5 MΩ, and the emfs are ε1 = 75 mV, ε2 = 45 mV and ε3 = 20 mV. Find the current through ε3, including its direction.
RC Circuits• If a direct current circuit contains
capacitors and resistors, the current will vary with time
• At the instant the switch is closed, the charge on the capacitor is zero
• When the circuit is completed, the capacitor starts to charge, and the potential difference across the capacitor increases, until the charge reaches its maximum (Q = Cε)
Charging Capacitor in an RC Circuit
• Once the capacitor is fully charged (maximum charge is reached), the current in the circuit is zero, and the potential difference across the capacitor matches that supplied by the battery
0 IRC
q IRC
q
R
RC
q
Rdt
dq
RC
qC
Charging Capacitor in an RC Circuit
RC
qC
dt
dq
RC
dt
dq
RC
dt
dq
'ln ConstRC
tQq ConsteQq RC
t
ConsteQtq RC
t
)(
ConsteQq RC
0
)0(
ConstQ 0 QConst
)1()( RC
t
eQtq
Charging Capacitor in an RC Circuit
• The charge on the capacitor varies with time
q = Q (1 – e -t/RC )
• The time constant, = RC, represents the time required for the charge to increase from zero to 63.2% of its maximum
• In a circuit with a large (small) time constant, the capacitor charges very slowly (quickly)
• After t = 10 , the capacitor is over 99.99% charged
Discharging Capacitor in an RC Circuit
• When a charged capacitor is placed in the circuit, it can be discharged
Constetq RC
t
)(
Constq )0( RC
t
Qetq
)(Q
0 IRC
q
RC
qI
RC
q
dt
dq
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dt
q
dq
RC
t
Qedt
dtI
)( RC
t
eRC
Q
Discharging Capacitor in an RC Circuit
• The charge on the capacitor varies with time
q = Qe -t/RC
• The charge decreases exponentially
• At t = = RC, the charge
decreases to 0.368 Qmax; i.e., in one
time constant, the capacitor loses 63.2% of its initial charge
Chapter 25Problem 34
An uncharged 10-µF capacitor and a 470-kΩ resistor are in series, and 250 V is applied across the combination. How long does it take the capacitor voltage to reach 200 V?
Meters in a Circuit – Ammeter, Voltmeter
• An ammeter is used to measure current in line with the bulb – all the charge passing through the bulb also must pass through the meter
• A voltmeter is used to measure voltage (potential difference) – connects to the two ends of the bulb
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
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
Answers to Even Numbered Problems
Chapter 25:
Problem 18
200 kJ
Answers to Even Numbered Problems
Chapter 25:
Problem 26
2 A
Answers to Even Numbered Problems
Chapter 25:
Problem 30
1 kW
Answers to Even Numbered Problems
Chapter 25:
Problem 58
7.5 mJ