Electromotive Force and Circuits
For a conductor to have a steady current, it must be a closed loop path
If charge goes around a complete circuit and returns to a starting point –
potential energy does not change
As charges move through the circuit they loose their potential energy
due to resistance
“Electromotive force” (emf, ε) is produced by
a battery or a generator and acts as a “charge
pump”. It moves charges uphill and is equal to
the potential difference across such a device
under open-circuit conditions (no current). In
reality, batteries have some internal resistance.
Emf is measured in Volts (so it is not a “force” per say, but potential difference)
Sources of emf – batteries, electric generators, solar cells, fuel cells
Internal Resistance
In ideal situation,   Vab  IR
As the charge flows through the circuit, the potential
rise as it passes through the ideal source is equal to
potential drop via the resistance, Vab  IR

Internal resistance r
Load resistance
R
  IR  Ir
I
Evolution of the
electric potential
in the circuit
with a load

Rr
Voltage between
terminals V    Ir
We measure currents
We measure voltages
with voltmeters
with ammeters
An ideal voltmeter
would have an infinite
resistance
An ideal ammeter
would have a zero
resistance
Example: What are voltmeter
and ammeter readings?
Examples
Bulb B is taken away,
will the bulb A glow differently?
Which bulb glows brighter?
Which bulb glows brighter?
Potential changes around the circuit
Potential gain in the battery
Potential drop at all resistances
In an old, “used-up” battery emf is
nearly the same, but internal resistance
increases enormously
Electrical energy and power
Chemical energy → Electric potential energy
→ Kinetic energy of charge carriers →
Dissipation/Joule heat (heating the resistor
through collisions with its atoms)
As the charge goes through the resistance the
potential energy qV is expended (if both q and V
are positive), but charge does not acquire kinetic
energy (current is constant). Instead, it converted
to heat. The opposite can also happen – if change
in potential energy is positive, the charge acquires
it - battery
For charge Q :
U  QV  heat in a resistor
(or other type s of energy
in devices/ap pliances)
U Q
Power P   V  I  V
t
t
Unit : 1 W  1A 1V
In a resistor : V  I  R
V2
PI R
R
2
Power Output of a Source
Vab    Ir;
P  Vab I   I  I 2r
Maximum power
delivered to load
(load matching) :
PI R
2
dP
0
dR

2
R
(R  r)2
Rr
Power Input to a Source
Current flows “backwards”
Vab    Ir
P  Vab I   I  I 2 R
Rate of conversion of electric energy
into non-electrical energy
Work is being done on, rather than by
the top battery (source of non-electrostatic
force)
Circuits in Series
•Resistance (light bulbs) on same path
•Current has one pathway - same in every part of the circuit
•Total resistance is sum of individual resistances along path
•Current in circuit equal to voltage supplied divided by total resistance
•Sum of voltages across each lamp equal to total voltage
•One bulb burns out - circuit broken - other lamps will not light (think of
string of old Christmas lights)
Water Analogy for Series Circuits
ISNS 3371 - Phenomena of
Nature
Resistors in series
Current is the same in both resistors
V  V1  V2  IR1  IR2  IReq
Equivalent Req  R1  R2
Req  R1  R2    max( Ri )
Parallel Circuits
•Bulbs connected to same two points of electrical circuit
•Voltage same across each bulb
•Total current divides among the parallel branches - equals
sum of current in each branch - current in each branch
inversely proportional to resistance of branch
•Overall resistance of circuit lowered with each additional
branch
•Household wiring (and new Christmas light strings)
designed in parallel - too many electrical devices on - too
much current - trip fuse/breaker
Water Analogy for Parallel Circuits
ISNS 3371 - Phenomena of
Nature
Resistors in parallel
Voltage is the same across both resistors, current splits at a junction :
V V
V
I  I1  I 2  

R1 R2 Req
1
1
1
Equivalent
 
Req R1 R2
1
Req 
 min( Ri )
1
1


R1 R2
Calculating resistance
A variable cross-section resistor treated as a serial
combination of small straight-wire resistors:
a b
r ( x)  b 
x;
h
h
dx 
dr
a b
h
dx
R   dR    2
r ( x)
0
 h
h
 2
dr 
r a  b
ab
b
a
Example: Equivalent resistances
Series versus parallel connection
What about power delivered to each bulb?
P  I 2 R or
P  I 2 R or
Vab2 Vbc2
P

R
R
Vde2
P
R
What if one bulb burns out?
Symmetry considerations to calculate equivalent
resistances
No current through the resistor
All resistors r
Currents : I1  I / 3; I 2  I1 / 2
I2
I1
I1
I1
I2
I2
I2
I2
I1
I1
I1
I2
Total voltage drop between a and b :
1 1 1
5
V  I (   )r  I r
3 6 3
6
5
R r
6
Kirchhoff’s rules
To analyze more complex (steady-state) circuits:
1. For any junction: Sum of incoming currents
equals to sum of outgoing currents
(conservation of charge)
I  0
Valid for any junction
2. For any closed circuit loop: Sum of the voltages
across all elements of the loop is zero
(conservation of energy)
V  0
-
Valid for any close loop
The number of independent equations will be
equal to the number of unknown currents
Loop rule – statement that the electrostatic force is conservative.