Chapter 26
Direct-Current Circuits
Lecture by Dr. Hebin Li
Assignment
Due at 11:59pm on Sunday, March 06
 HW set on Masteringphysics.com
Due before the lecture on Monday, March 07
Read Chapter 27 (p 883 ~ 911)
PHY 2049, Dr. Hebin Li
Goals for Chapter 26
 To analyze circuits having resistors in series and parallel
 To apply Kirchhoff’s rules to multiloop circuits
 To learn how to use various types of meters in a circuit
 To analyze circuits containing capacitors and resistors
 To study power distribution in the home
PHY 2049, Dr. Hebin Li
Introduction
How can we apply series/parallel
combinations of resistors to a
complex circuit board?
In this chapter, we will learn
general methods for analyzing
more complex networks.
We shall look at various
instruments for measuring
electrical quantities in circuits.
PHY 2049, Dr. Hebin Li
Resistors in series
 For resistors in series, the current going through each resistor is
the same.
 For each resistor, we have
 The potential difference across all resistors is
 The equivalent resistance is
PHY 2049, Dr. Hebin Li
Resistors in parallel
 For resistors in parallel, the potential difference across each
resistor is the same.
 For each resistor, we have
 The total current is
 The equivalent resistance 𝑅𝑒𝑞 = 𝑉𝑎𝑏 𝐼, so
PHY 2049, Dr. Hebin Li
Series and parallel combinations
 A resistor network with resistors that are neither all in series nor all
in parallel.
 The strategy is to combine the resistors that are in series or in
parallel, find their equivalent resistance, and replace them with the
equivalent resistor.
 With the equivalent resistor in place, repeat the process until the
network is reduced to a single equivalent resistor.
PHY 2049, Dr. Hebin Li
Example: equivalent resistance
For a circuit shown in the figure, find
(a) The equivalent resistance
(b)The potential difference and current for
each resistor.
PHY 2049, Dr. Hebin Li
Kirchhoff’s Rules I
 There are more complicated
circuits that cannot be simplified
by considering components in
series or parallel.
 Kirchhoff’s rules
 A junction is a point where three
or more conductors meet.
 A loop is any closed conducting
path.
PHY 2049, Dr. Hebin Li
Kirchhoff’s Rules II
 Kirchhoff’s junction rule: The algebraic sum of the currents into
any junction is zero: I = 0.
 Kirchhoff’s loop rule: The algebraic sum of the potential
differences in any loop must equal zero: V = 0.
PHY 2049, Dr. Hebin Li
Sign convention for the loop rule
Figure below shows the sign convention for emfs and
resistors.
Practice:
PHY 2049, Dr. Hebin Li
Example: Charging a battery
Find 𝑟, ℰ and the current I through the power supply.
Apply the junction rule at point a:
Apply the loop rule to loop (1)
Apply the loop rule to loop (2)
We can check our result by evaluating loop (3):
PHY 2049, Dr. Hebin Li
Reducing the number of unknown currents
• Figure below shows how to use the junction rule to reduce the
number of unknown currents.
PHY 2049, Dr. Hebin Li
R-C circuits
 In the steady state, there is no current.
 There is current during the charging
and discharging.
 The current varies over time and lasts
for a limit period of time.
 What happens during charging and
discharging?
PHY 2049, Dr. Hebin Li
Charging a capacitor
 The capacitor is initially uncharged, the potential
difference across the capacitor is zero. Therefore,
𝑉𝑎𝑏 = ℰ, and the initial current 𝐼0 = ℰ/𝑅.
 The instantaneous potential differences are
 Using these in Kirchhoff’s loop rule, we find
 Rearrange the equation:
PHY 2049, Dr. Hebin Li
Charging a capacitor
After a time equal to RC, the current in the R-C circuit has decreased to 1/e of its
initial value. The capacitor charge has reached 1-1/e of its final value 𝑄𝑓 = 𝐶ℰ.
The product RC is a measure of how quickly the capacitor charges, it is called
time constant or relaxation time of the circuit
𝜏 = 𝑅𝐶
PHY 2049, Dr. Hebin Li
Discharging a capacitor
 The capacitor initially has charge 𝑄0 , it
discharges through a resistor and the charge
eventually goes to zero.
 Kirchhoff’s loop rule gives
PHY 2049, Dr. Hebin Li
Discharging a capacitor
PHY 2049, Dr. Hebin Li
Power distribution systems
Household power distribution system:
PHY 2049, Dr. Hebin Li