DC Electrical Circuits

1
DC Electrical Circuits

• Chapter 28
• Electromotive Force
• Potential Differences
• Resistors in Parallel and Series
• Circuits with Capacitors

2
Resistors in Series
The pair of resistors, R1 and R2, can be replaced
by a single equivalent resistor R one which,
given I, has the same total voltage drop as the
original pair.
Note the current I is the same, anywhere between
a and b, but there is a voltage drop V1 across
R1, and a voltage drop V2 across R2.
3
Resistors in Series
I
R1
R2
a
b
V1
V2
The pair of resistors can be replaced by a single
equivalent resistor Req one which, given I, has
the same total voltage drop as the original pair.

• V V1 V2 I R1 I R2

4
Resistors in Series
I
R1
R2
a
b
V1
V2
The pair of resistors can be replaced by a single
equivalent resistor Req one which, given I, has
the same total voltage drop as the original pair.

• V V1 V2 I R1 I R2
• We want to write this as V I Req

5
Resistors in Series
I
R1
R2
a
b
V1
V2
The pair of resistors can be replaced by a single
equivalent resistor Req one which, given I, has
the same total voltage drop as the original pair.

• V V1 V2 I R1 I R2
• We want to write this as V I Req
• hence Req R1 R2

6
Resistors in Series
I
R1
R2
a
b
V1
V2
7
Resistors in Parallel
R1
I1
I
a
b
R2
I2
V

• Again find the equivalent single resistor which
  has the same V if I is given.

8
Resistors in Parallel
R1
I1
I
a
b
R2
I2
V

• Again find the equivalent single resistor which
  has the same V if I is given. Here the total I
  splits
• I I1I2 V / R1 V / R2 V(1/ R1 1/ R2)

9
Resistors in Parallel
R1
I1
I
a
b
R2
I2
V

• Again find the equivalent single resistor which
  has the same V if I is given. Here the total I
  splits
• I I1I2 V / R1 V / R2 V(1/ R1 1/ R2)
• We want to write this as I V / Req

10
Resistors in Parallel
R1
I1
I
a
b
R2
I2
V

• Again find the equivalent single resistor which
  has the same V if I is given. Here the total I
  splits
• I I1I2 V / R1 V / R2 V(1/ R1 1/ R2)
• We want to write this as I V / Req
• Hence 1 / Req 1/ R1 1/ R2

11
Parallel and Series
Resistors Capacitors
Parallel 1/R1/R11/R2 CC1C2
Series RR1R2 1/C1/C11/C2
12
Batteries and Generators

• Current is produced by applying a potential
  difference across a conductor (IV/R) This is
  not equilibrium so there is an electric field
  inside the conductor.
• This potential difference is set up by some
  source, such as a battery or generator that
  generates charges, from some other type of
  energy, i.e. chemical, solar, mechanical.
• Conventionally an applied voltage is given the
  symbol E (units volts).
• For historical reasons, this applied voltage is
  often called the electromotive force (emf)
  even though its not a force.

13
The Voltaic Pile
Voltas original battery
Carbon
Ag

wet cloth
-
Zn
Mixture of Ammonium Chloride Manganese Dioxide
Zinc case
electrical converter... .....converts chemical
energy to electrical energy
14
Electrical Description of a Battery
I

symbol for resistance
symbol for battery
E
R
-

• A battery does work on positive charges in moving
  them to higher potential (inside the battery).
• The EMF E, most precisely, is the work per unit
  charge exerted to move the charges uphill (to
  the terminal, inside)
• ... but you can just think of it as an applied
  voltage.
• Current will flow, in the external circuit, from
  the terminal,
• to the terminal, of the battery.

15
Resistors in Series
R1
R2

-
I
E

• Given Req R1 R2, the current is IE/(R1
  R2)
• One can then work backwards to get the voltage
• across each resistor

16
Resistors in Series
R1
R2

-
I
E

• Given Req R1 R2, the current is IE/(R1
  R2)
• One can then work backwards to get the voltage
• across each resistor

17
The Loop Method
Go around the circuit in one direction. If you
pass a voltage source from to , ? the voltage
increases by E (or V). As you pass a resistor
the voltage decreases by V I R. The total
change in voltage after a complete loop is zero.
18
Analyzing Resistor Networks
1. Replace resistors step by step.
2. The loop method
R2
R1

-
I
E

3 W
E
-
E IR1 IR2 0
2 W
E I (R1 R2)

I E / (R1 R2)
5W
E
-
I
I E / R
19
Analyzing Resistor Networks
Often you can replace sets of resistors step by
step.
6 W

E
6 W
-
2 W
20
Analyzing Resistor Networks
Often you can replace sets of resistors step by
step.
6 W

E
6 W
-
2 W
step 1
21
Analyzing Resistor Networks
Often you can replace sets of resistors step by
step.
6 W

E
6 W
-
2 W
1/61/61/(3)
2 W
step 1
step 2
22
Internal Resistance of a Battery
r
R
battery
E

• One important point batteries actually have an
• internal resistance r
• Often we neglect this, but sometimes it is
  significant.

23
Effect of Internal ResistanceAnalyzed by the
Loop Method

• Start at any point in the circuit. Go around the
  circuit in a loop.
• Add up (subtract) the potential differences
  across each element
• (keep the signs straight!).
• E - Ir - IR 0 (using VIR) ? I E /
  (R r)
• VR I R E R / (R r) E / 1
  (r/R)
• if r ltlt R ? VR E

24
Kirchhoffs Laws
Kirchoff devised two laws that are universally
applicable in circuit analysis

• 1. At any circuit junction,
• currents entering must
• equal currents leaving.

I2
I1
I3 I1 I2
I
2. Sum of all DVs across all circuit
elements in a loop must be zero.
r
R

E
-
E - Ir - IR 0