Chapter 11 Electric Resistance and Circuits
We have developed expressions for voltage and current. The most
basic relation between these two concepts is called Ohm’s Law
which states:
∆V = I R
The voltage difference across some passive component in a circuit,
is equal to the current passing through that component multiplied
by the resistance of that component, measured in ohms with a
symbol of Ω. It’s important to realize that we are now talking
about circuits; wires connected together with various components
(resistors, capacitors, inductors, transformers, etc. etc.). These will
include some voltage source which should be thought of as a
source of energy for the circuit; such as a battery or the power
outlet in your house. Ohm’s law defines resistance, so one ohm is
one volt/amp.
Does not follow
Ohm’s Law
I
Slope = 1/R
∆V
Before looking at circuits in more detail, consider just wires. They
have a low resistance but when long distances are relevant this
resistance can be important. The resistance is greater with the
length of the wire, and lower with the cross-sectional area of the
wire. For wires we may write:
L
R=ρ
A
where ρ (rho) is the resistivity, L is the length, and A is the crosssectional area. Table 11.2 lists value of ρ for common materials.
Note that a metal has a value a billion billion times smaller than an
insulator.
Resistivity changes with temperature, and over a small range we
may write:
ρ = ρ0 [1 + α (T - T0 )]
where ρ0 is the resistivity at temperature T0 and ρ the resistivity at
temperature T. α is called the temperature coefficient of
resistivity.
If one multiplies the above equation by L/A on both sides one
obtains:
R = R 0 [1 + α (T - T0 )]
The temperature dependence of resistance is the same as that for
resistivity.
Electric Circuits
A battery is an energy source that has a given voltage between two
terminals. Think of it as something capable of pushing charges
through a wire or producing a current in a wire.
Figure 11.6 in your text lists the symbols that we will use in our
circuit diagrams. Note that for circuits we will consider wires to
have zero resistance.
V
I
V=IR
R
V
V = I (R1+R2)
R1
R2
Parallel circuits
V
I
R1
I1
R2
I2
V = I1 R1 = I2 R2
I = I1 + I2
For resistors in series:
R = R1 + R2 + R3 + ...
For resistors is parallel we have:
1 1
1
1
=
+
+
+ ...
R R1 R 2 R 3
Electric Power
Remember that
and
q/t = I
∆U = q ∆V
But power is energy divided by time or ∆U/t, so
∆U q
P=
= ∆V = I ∆V
t
t
or power is current times voltage. Now using Ohm’s law we note
P = I ∆V = I ( I R) = I2 R
∆V
∆V 2
P=
∆V =
R
R
which give two expressions for the power.
AC Circuits
For many kinds of items, like light bulbs, space heaters and other
simple resistive devices, it makes no difference which way the
current flows. It is easier to build generators which produce
alternating currents (AC), and alternating voltages than it is to
build ones that produce direct currents (DC). A graph of voltage
versus time is produced on the following page. It is a simple sine
wave.
One can still apply Ohm’s law but the relevant voltage is the
effective voltage which is the peak voltage divided by √2. For
house voltage, Veff is 120 volts while Vp is 170 volts.
Vp
voltage
Veff
time