Ch. 25 Current,
Resistance, and EMF
AP Physics C
1
Electric Current
• There needs to be an E-field
present.
2
Motion of ball analogous to an
electron moving in a metallic
conductor with the presence of an
electric field.
3
Conventional current
• Direction of the net flow of positive
charges
• The rate that the charge is moving
is given by:
I 
dq
dt
4
Concentration of charges
• Suppose there
are n charged
particles per
unit volume.
This is called
the
concentration
of charges.
5
Drift velocity
• If all the
particles are
moving with
the same drift
velocity, vd,
how many
particles flow
at the end of
the conductor
in a time
interval of dt?
6
Current density
• Current density
is defined to
the ratio of
current to the
cross-sectional
area; that is,
J 
I
A
7
Sample Problem #1
• An 18-gauge copper wire (the size usually
used for lamp cords) has a nominal
diameter of 1.02 mm. This wire carries a
constant current of 1.67 A to a 200-watt
lamp. The density of free electrons is 8.5 x
1028 electrons per cubic meter. Find the
magnitudes of
• The current density and
• The drift velocity.
8
Sample Problem #2
• Suppose that we replaced the wire
in the previous sample problem
with a 12-gauge copper wire,
which has twice the diameter of
the 18-gauge wire. If the current
remains the same, what is the new
magnitude of the drift velocity?
9
Sample Problem #3
• The 12-gauge copper wire in a typical
residential building has a cross-sectional
area of 3.31 x 10-6 m2. If it carries a current
of 10.0 A, what is the drift speed of the
electrons? Assume that each copper atom
contributes one free electron to the
current. The density of copper is 8.95 g/cm3
and its molar mass is 63.54 g/mol.
10
Sample Problem #4
• If a copper wire carries a current of
80.0 mA, how many electrons flow
past a cross-section of the wire in
10.0 min?
11
Non-Electrostatic Electric Field
• An electric field
exists in the
conductor because
the charges are in
motion due to the
difference in
potential. In some
materials, the
current density is
proportional to the
electric field.
12
Ohm’s Law
• σ is the conductivity
of the conductor,
J E
• The reciprocal of
the conductivity is
known as the
resistivity of the
conductor, ρ.
• Pg. 948, Table 25.1.
13
Ohmic Materials
• Materials that obey
Ohm’s law
V  EL
J   E  
V
L
J 
I
 
A
R 
L
A
V
L

V
I
V  IR
14
Different Materials
• Ohmic
• Vacuum-tube
diode
• Semiconductor
diode
15
Resistance
• R = resistance
• 1 volt/ampere=1 ohm (Ω)
• Factors affecting resistance
• Temperature
• R(T) = Ro [1 + a (T - To)]
• Type of material
• Length of material
• Cross-sectional area of material
16
Sample Problem #5
• Calculate the resistance of an
aluminum cylinder that is 10.0 cm
long and has a cross-sectional area
of 2.00 x 10-4 m2. The resistivity for
aluminum is 2.75 x 10-8 Ω-m.
17
Sample Problem #6
• Calculate the resistance per unit length of a 22-gauge
Nichrome wire, which has a radius of 0.321 mm. The
resistivity of Nichrome is 100 x 10-8 Ω-m.
• If a potential difference of 10 V is maintained across a 1.0
m length of the Nichrome wire, what is the current in the
wire?
• What is the resistance of a 6.0 m length of 22-gauge
Nichrome wire?
• How much current does the wire carry when connected
to a 120 V source of potential?
• Calculate the current density and electric field in the 22gauge Nichrome wire when it carries a current of 2.2 A.
18
Sample Problem #7
• Coaxial cables are used extensively for cable
television and other electronic applications. A
coaxial cable consists of two cylindrical conductors.
The gap between the conductors is completely filled
with silicon and current leakage through the silicon is
unwanted. The cable is designed to conduct
current along its length. The radius of the inner
conductor is 0.500 cm, the radius of the outer one is
1.75 cm, and the length of the cable is 15.0 cm.
Calculate the resistance of the silicon between the
conductors. The resistivity of silicon is 2300 Ω-m.
19
Sample Problem #8
• A resistance thermometer, which
measures temperature by measuring the
change in resistance of a conductor, is
made from platinum and has a resistance
of 50.0 Ω at 20.0o C. When immersed in a
vessel containing melting indium, its
resistance increases to 76.8 Ω. Calculate
the melting point of the indium. The
temperature coefficient of resistivity for
platinum is 3.92 x 10-3 (oC)-1.
20
Sample Problem #9
• An 18-gauge copper wire has a diameter
of 1.02 mm and it carries a current of 1.67
A. Find:
• The cross-sectional area of the wire
• The electric field magnitude in the wire
• The potential difference between two
points in the wire 50.0 m apart
• The resistance of a 50.0 m length of the
wire
• Suppose that the temperature is at 20o C,
what is its resistance at 100o C?
21
Internal Resistance and EMF
• Sources of emf, ε
• Batteries
• Generators
• Internal resistance,
r
• Resistance that
the charge
encounters as it
moves through
the source
22
Voltage rises and drops in a circuit:
• Ideal source
•r=0
  V ab  IR
• Real source
V ab    Ir  IR
23
Sample Problem #10
• Draw a schematic diagram showing a
source (a battery) with an emf of 12 V
and an internal resistance of 2 Ω. Also,
show a voltmeter connected across the
battery and ammeter connected to the
battery. There is no external resistance
connected. This is an example of an
open circuit. What are the readings of
the idealized voltmeter and idealized
ammeter?
24
Sample Problem #11
• A 4 Ω resistor is connected to the
battery in the circuit described in
sample problem #10. What are the
readings of the voltmeter and
ammeter now?
25
Sample Problem #12
• The external resistance in sample
problem #11 is replaced with a
zero resistance. This is called a
short circuit. What are the meter
readings now?
26
Sample Problem #13
• A battery has an emf of 1.5 V and an
internal resistance of 0.10 Ω. When the
battery is connected to a resistor, the
terminal voltage is 1.3 V. What is the
resistance of the resistor?
27
Power
• V = U / Q = dU / dQ
• I = dQ / dt
• V = dU / dQ = dW / dQ
• dW = V dQ = V I dt
• P = dW / dt = V I
• the rate of delivering energy to a
circuit element having a potential
difference across it of V
28
Power Output
• P = dW / dt = V I
• P = VI = (ε – Ir)I = (IR)I
• P = VI =  I  I 2 r  I 2 R
• Rate of conversion of
chemical energy to electric
energy,  I
• Rate of electrical energy
2
dissipated by the internal resistance, I r
• Rate of electrical energy dissipated by the external
resistance, I 2 R
29
Sample Problem #14
• A battery has an emf of 12.0 V and
internal resistance of 0.05 Ω. Its
terminals are connected to a load
resistance of 3.00 Ω.
• Find the current in the circuit and
the terminal voltage of the battery.
• Calculate the power delivered to
the load resistor, the power
delivered to the internal resistance
of the battery, and the power
delivered by the battery.
30
Sample Problem #15
• Show that the maximum power
delivered to the load resistance R
in a circuit occurs when the load
resistance matches the internal
resistance; that is, when R = r.
31
Power Input
• How is this equation different when
you are charging a battery?
• P = VI =  I  I 2 r  I 2 R
32