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Chapter 28
Direct Current Circuits

Circuit Analysis
Simple electric circuits may contain batteries, resistors, and capacitors in various
combinations.
For some circuits, analysis may consist of combining resistors.
In more complex complicated circuits, Kirchhoff’s Rules may be used for
analysis.
 These Rules are based on conservation of energy and conservation of
electric charge for isolated systems.
Circuits may involve direct current or alternating current.
Introduction

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.
Section 28.1

Electromotive Force
The electromotive force (emf), , of a battery is the maximum possible voltage
that the battery can provide between its terminals.
 The emf supplies energy, it does not apply a force.
The battery will normally be the source of energy in the circuit.
The positive terminal of the battery is at a higher potential than the negative
terminal.
We consider the wires to have no resistance.
Section 28.1

Internal Battery Resistance
If the internal resistance is zero, the
terminal voltage equals the emf.
In a real battery, there is internal
resistance, r.
The terminal voltage, V =  – Ir
The emf is equivalent to the open-circuit
voltage.
 This is the terminal voltage when no
current is in the circuit.
 This is the voltage labeled on the
battery.
The actual potential difference between
the terminals of the battery depends on
the current in the circuit.
Section 28.1

Load Resistance
The terminal voltage also equals the voltage across the external resistance.
 This external resistor is called the load resistance.
 In the previous circuit, the load resistance is just the external resistor.
 In general, the load resistance could be any electrical device.
 These resistances represent loads on the battery since it supplies the energy to
operate the device containing the resistance.
Section 28.1

Power
The total power output of the battery is
P = I ΔV = I ε
This power is delivered to the external resistor (I 2
R) and to the internal resistor
(I2
r).
P = I2
R + I2
r
The battery is a supply of constant emf.
 The battery does not supply a constant current since the current in the circuit
depends on the resistance connected to the battery.
 The battery does not supply a constant terminal voltage.
Section 28.1

Resistors in Series
When two or more resistors are connected end-to-end, they are said to be in
series.
For a series combination of resistors, the currents are the same in all the
resistors because the amount of charge that passes through one resistor must
also pass through the other resistors in the same time interval.
The potential difference will divide among the resistors such that the sum of the
potential differences across the resistors is equal to the total potential difference
across the combination.
Section 28.2

Resistors in Series, cont
Currents are the same
 I = I1 = I2
Potentials add
 ΔV = V1 + V2 = IR1 + IR2
= I (R1+R2)
 Consequence of Conservation of
Energy
The equivalent resistance has the same
effect on the circuit as the original
combination of resistors.
Section 28.2

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
individual resistance.
If one device in the series circuit creates an open circuit, all devices are
inoperative.
Section 28.2

Equivalent Resistance – Series – An Example
All three representations are equivalent.
Two resistors are replaced with their equivalent resistance.
Section 28.2

Some Circuit Notes
A local change in one part of a circuit may result in a global change throughout
the circuit.
 For example, changing one resistor will affect the currents and voltages in all
the other resistors and the terminal voltage of the battery.
In a series circuit, there is one path for the current to take.
In a parallel circuit, there are multiple paths for the current to take.
Section 28.2

Resistors in Parallel
The potential difference across each resistor is the same because each is
connected directly across the battery terminals.
ΔV = ΔV1 = ΔV2
A junction is a point where the current can split.
The current, I, that enters junction must be equal to the total current leaving that
junction.
 I = I 1 + I 2 = (ΔV1 / R1) + (ΔV2 / R2)
 The currents are generally not the same.
 Consequence of conservation of electric charge
Section 28.2

Equivalent Resistance – Parallel, Examples
All three diagrams are equivalent.
Equivalent resistance replaces the two original resistances.
Section 28.2

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.
1 2 3
1 1 1 1
eq
R R R R
   
Section 28.2

Resistors in Parallel, Final
In parallel, each device operates independently of the others so that if one is
switched off, the others remain on.
In parallel, all of the devices operate on the same voltage.
The current takes all the paths.
 The lower resistance will have higher currents.
 Even very high resistances will have some currents.
Household circuits are wired so that electrical devices are connected in parallel.
Section 28.2

Combinations of Resistors
The 8.0- and 4.0- resistors are in
series and can be replaced with their
equivalent, 12.0 
The 6.0- and 3.0- resistors are in
parallel and can be replaced with their
equivalent, 2.0 
These equivalent resistances are in
series and can be replaced with their
equivalent resistance, 14.0 
Section 28.2

Gustav Kirchhoff
1824 – 1887
German physicist
Worked with Robert Bunsen
Kirchhoff and Bunsen
 Invented the spectroscope and
founded the science of
spectroscopy
 Discovered the elements cesium
and rubidium
 Invented astronomical
spectroscopy
Section 28.3

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.
Section 28.3

Kirchhoff’s Junction Rule
Junction Rule
 The sum of the currents at any junction must equal zero.
 Currents directed into the junction are entered into the equation as +I and
those leaving as -I.
 A statement of Conservation of Charge
 Mathematically,
0
junction
I 

Section 28.3

More about the Junction Rule
I1 - I2 - I3 = 0
Required by Conservation of Charge
Diagram (b) shows a mechanical
analog
Section 28.3

Kirchhoff’s Loop Rule
Loop Rule
 The sum of the potential differences across all elements around any closed
circuit loop must be zero.
 A statement of Conservation of Energy
Mathematically,
closed
loop
0
V
 

Section 28.3

More about the Loop Rule
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 (b), the resistor is traversed in the
direction opposite of the current, the
potential across the resistor is is + IR.
Section 28.3

Loop Rule, final
In (c), the source of emf is traversed in
the direction of the emf (from – to +),
and the change in the potential
difference is +ε.
In (d), the source of emf is traversed in
the direction opposite of the emf (from
+ to -), and the change in the potential
difference is -ε.
Section 28.3

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.
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.
In order to solve a particular circuit problem, the number of independent
equations you need to obtain from the two rules equals the number of unknown
currents.
Any capacitor acts as an open branch in a circuit.
 The current in the branch containing the capacitor is zero under steady-state
conditions.
Section 28.3

Problem-Solving Strategy – Kirchhoff’s Rules
Conceptualize
 Study the circuit diagram and identify all the elements.
 Identify the polarity of each battery.
 Imagine the directions of the currents in each battery.
Categorize
 Determine if the circuit can be reduced by combining series and parallel
resistors.
 If so, proceed with those techniques
 If not, apply Kirchhoff’s Rules
Section 28.3

Problem-Solving Strategy, cont.
Analyze
 Assign labels and symbols to all known and unknown quantities.
 Assign directions to the currents.
 The direction is arbitrary, but you must adhere to the assigned directions when
applying Kirchhoff’s rules.
 Apply the junction rule to any junction in the circuit that provides new
relationships among the various currents.
 Apply the loop rule to as many loops as are needed to solve for the
unknowns.
 To apply the loop rule, you must choose a direction in which to travel
around the loop.
 You must also correctly identify the potential difference as you cross
various elements.
 Solve the equations simultaneously for the unknown quantities.
Section 28.3

Problem-Solving Strategy, final
Finalize
 Check your numerical answers for consistency.
 If any current value is negative, it means you guessed the direction of that
current incorrectly.
 The magnitude will still be correct.
Section 28.3

RC Circuits
In direct current circuits containing capacitors, the current may vary with time.
 The current is still in the same direction.
An RC circuit will contain a series combination of a resistor and a capacitor.
Section 28.4

RC Circuit, Example
Section 28.4

Charging a Capacitor
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.
As the plates are being charged, the potential difference across the capacitor
increases.
At the instant the switch is closed, the charge on the capacitor is zero.
Once the maximum charge is reached, the current in the circuit is zero.
 The potential difference across the capacitor matches that supplied by the
battery.
Section 28.4

Charging a Capacitor in an RC Circuit
The charge on the capacitor varies with
time.
 q(t) = C(1 – e-t/RC
)
= Q(1 – e-t/RC
)
The current can be found
  is the time constant
  = RC
I( ) t RC
ε
t e
R


Section 28.4

Time Constant, Charging
The time constant represents the time required for the charge to increase from
zero to 63.2% of its maximum.
 has units of time
The energy stored in the charged capacitor is ½ Q = ½ C2
.
Section 28.4

Discharging a Capacitor in an RC Circuit
When a charged capacitor is placed in
the circuit, it can be discharged.
 q(t) = Qe-t/RC
The charge decreases exponentially.
Section 28.4

Discharging Capacitor
At t =  = RC, the charge decreases to 0.368 Qmax
 In other words, in one time constant, the capacitor loses 63.2% of its initial
charge.
The current can be found
Both charge and current decay exponentially at a rate characterized by  = RC.
 
I t RC
dq Q
t e
dt RC

 
Section 28.4

Household Wiring
The utility company distributes electric power to individual homes by a pair of
wires.
Each house is connected in parallel with these wires.
One wire is the “live wire” and the other wire is the neutral wire connected to
ground.
Section 28.5

Household Wiring, cont
The potential of the neutral wire is
taken to be zero.
 Actually, the current and voltage
are alternating
The potential difference between the
live and neutral wires is about 120 V.
Section 28.5

Household Wiring, final
A meter is connected in series with the live wire entering the house.
 This records the household’s consumption of electricity.
After the meter, the wire splits so that multiple parallel circuits can be distributed
throughout the house.
Each circuit has its own circuit breaker.
For those applications requiring 240 V, there is a third wire maintained at 120 V
below the neutral wire.
Section 28.5

Short Circuit
A short circuit occurs when almost zero resistance exists between two points at
different potentials.
This results in a very large current
In a household circuit, a circuit breaker will open the circuit in the case of an
accidental short circuit.
 This prevents any damage
A person in contact with ground can be electrocuted by touching the live wire.
Section 28.5

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 touched by the live wire
 The part of the body in which the current exists
Section 28.5

Effects of Various Currents
5 mA or less
 Can cause a sensation of shock
 Generally little or no damage
10 mA
 Muscles contract
 May be unable to let go of a live wire
100 mA
 If passing through the body for a few seconds, can be fatal
 Paralyzes the respiratory muscles and prevents breathing
Section 28.5

More Effects
In some cases, currents of 1 A can produce serious burns.
 Sometimes these can be fatal burns
No contact with live wires is considered safe whenever the voltage is greater than
24 V.
Section 28.5

Ground Wire
Electrical equipment manufacturers use
electrical cords that have a third wire,
called a ground.
This safety ground normally carries no
current and is both grounded and
connected to the appliance.
If the live wire is accidentally shorted to
the casing, most of the current takes
the low-resistance path through the
appliance to the ground.
If it was not properly grounded, anyone
in contact with the appliance could be
shocked because the body produces a
low-resistance path to ground.
Section 28.5

Ground-Fault Interrupters (GFI)
Special power outlets
Used in hazardous areas
Designed to protect people from electrical shock
Senses currents (< 5 mA) leaking to ground
Quickly shuts off the current when above this level
Section 28.5

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