Transcript Chapter 28

Chapter 28
Direct Current 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
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 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
I1  I 2  I
V  IR1  IR2  I ( R1  R2 )  IReq
Resistors in Series
V  IReq
Req  R1  R2
• 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:
Req  R1  R2  R3  ...
• The equivalent resistance of a series combination of
resistors is greater than any of the individual resistors
Resistors in Parallel
I1R1  I 2 R2  V
V V
I  I1  I 2 

R1
R2
• 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
I1R1  I 2 R2  V
V V
I  I1  I 2 

R1
R2
1
1  V
 V    
 R1 R2  Req
1
1 1
 
Req R1 R2
Resistors in Parallel
• For more resistors in parallel:
1
1 1
1
   
Req R1 R2 R3
• 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
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:
1
1 1
   ...
Req R1 R2
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 28
Problem 15
Calculate the power delivered to each resistor in the circuit shown in
the figure.
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:
• 1) Junction Rule
• 2) Loop Rule
Gustav Kirchhoff
1824 – 1887
Kirchhoff’s Rules
• Junction 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 28
Problem 17
Determine the current in each branch of the circuit shown
in the Figure.
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, the current in
the circuit is zero
• Once the maximum charge is reached, the current in
the circuit is zero, and the potential difference
across the capacitor matches that supplied by the
battery
q
   IR  0
C

q

I
R RC
dq 
q
C  q
 

dt R RC
RC
Charging Capacitor in an RC Circuit
dq C  q

dt
RC
dq
dt

Q  q RC
dq
dt
 Q  q   RC
t
 ln q  Q  
 Const ' q  Q  e
RC t

q(t )  Q  e RC  Const
0

RC
q(0)  Q  e
 Q  Const  0
q(t )  Q(1  e

 Const
Const  Q
t
RC
)
t

RC
 Const
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
q
q dq
q
 IR  0 I  

C
RC dt
RC
t

dq
dt
RC


q
(
t
)

e
 Const
 q  RC
q (0)  Const  Q
q(t )  Qe

t
RC
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 28
Problem 29
A 2.00-nF capacitor with an initial charge of 5.10 μC is discharged through a
1.30-kΩ resistor. (a) Calculate the current in the resistor 9.00 μs after the
resistor is connected across the terminals of the capacitor. (b) What charge
remains on the capacitor after 8.00 μs? (c) What is the maximum current in
the resistor?
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 28:
Problem 2
(a) 4.59 Ω
(b) 8.16%
Answers to Even Numbered Problems
Chapter 28:
Problem 24
starter 171 A downward in the diagram;
battery 0.283 A downward
Answers to Even Numbered Problems
Chapter 28:
Problem 28
587 kΩ