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Chapter 18
LECTURE NOTES
The battery provides a continuous source of voltage (also called potential difference or
electromotive force), charging chemical into electrical energy. If a continuous conducting path
connects the + and – terminals of a battery, we have a circuit.
The battery is like a pump, pushing electric charge through the circuit. A flow of electric charge
Q
is called a current: I =
where  Q is the amount of charge (in coulombs) that pass any point
t
in the circuit per unit time. The units for electric current are thus C/s. We write 1 ampere (A) =
1C/s. Physicists and engineers still picture the direction of current flow as the early electricians
(following Ben Franklin) did: A flow of positive charge from the + to the – terminals of a battery
through the circuit. This is called conventional current. We have known for the past century that
in metals it is really the negative electrons that are flowing (in the opposite direction of
conventional current), but conventional current flow is still the accepted model.
Ohm’s Law
In many materials, I  V; the current that flows is proportional to the potential difference across
the material. Note we are now writing just V for voltage or potential difference, but the
difference in potential is required for current conventional flow from high to low potential; we
simply omit the  from the  V.
Today we write V = IR where resistance R is the reciprocal of the above proportionality current.
This is a very important relation. Voltage or potential difference is measured in volts, electric
current in amps, and resistance in ohms (  ): 1  = 1V/A.
The flow of current through a wire is analogous to the flow of water through a pipe. Voltage
plays the role of water pressure (recall the battery was like a pump). Electric current (how many
coulombs of charge pass a point per second) corresponds to rate of flow (current of water in cfm
(cubic feet per minute). Electrical resistance is the opposition to current flow like the resistance
offered by narrow, clogged pipes.
Resistors are electrical devices made to offer a certain amount of resistance, or opposition to
current flow. The wires in a circuit usually offer a very small amount of resistance by
comparison.
Example: A resistor with bands colored red, green, orange, silver is in a circuit connected to a
9.00 V battery. What current will flow through the circuit? Using the resistor color code we find
R = 25,000   (10%). Then Ohm’s Law gives I = V/R = 9V/25,000  = 0.36mA.
Resistivity
l
where  is the resistivity in  ۰ m, a property of the
A
wire’s material, l is the length of the wire (in meters), A is the cross-sectional area of the wire (in
m2), and the resistance R is in ohms (  ).
The resistance of a wire is given by R =
Chapter 18
LECTURE NOTES
Resistivity, hence resistance, is temperature dependent; at lower temperatures the slower moving
wire atoms have fewer collisions with the conduction electrons resulting in lower resistance. In
general,  =  0 [1+  (T-T0)], where  0 is a tabulated resistivity at temperature T0 and  is the
temperature coefficient of resistivity in units of /C°. Similarly R = R0 [1+  (T-T0)].
We expect from these relations that R will decrease as T decreases but for some materials R
abruptly vanishes (R = 0) at some temperature. This is called superconductivity. Find a material
with a transition temperature Tc near 300K and collect lots of money – to date, we still must cool
materials to 100K or so (most materials must be much cooler) before they superconduct.
Electric Power
Recall we must do work q  V to move a charge q through potential difference  V. Power is
the rate at which work is done: P = W/t = q  V/t = I  V where we have used the definition of
current I as the rate of flow of charge. We write more simply P = IV where power is in watts
(W), current I is an amperes or amps (A) and voltage V is in volts (V).
This relation can be written in other ways for power consumed in a resistance. Using Ohm’s
Law we find P = IV = I2R = V2/R. Again note the last two forms are only valid for resistors.
The electric company charges us for energy, not power. Power is energy/time, so we must
multiply the power of an appliance (in watts) by the time (in seconds) it is used to find the
energy consumed (in joules). Utilities prefer to use more convenient units. Multiply the power
used by the appliance in kW by the time in hours it is used to find energy used (in units of kWh).
My last electric bill showed my family used 1545 kWh of electric energy during the month of
July, for which I paid $89.34 (excluding tax) or a bit less than 6¢/kWh. This is a good rate due,
in part, to Indiana’s reliance on low cost (but highly polluting) coal burning power plants.
Try reading your own electric meter and estimate your next electric bill.
Example
Mrs. Watson has an electric toaster with a label on the bottom stating the unit consumes 480 W
of power. When plugged into a 120V outlet, the toaster draws how much current? If used for 1
minute, how much does the energy cost at 6¢/kWh?
480 W
Use P = IV to fine I = P/V =
= 4A
120V
 1 
Now energy = (.480kW)  h  = .008kWH.
 60 
Multiply this by 6¢/kWh to find a cost of .048¢; Mrs. Watson could toast a loaf of bread for
about a penny on our electric bill.
Note: Toasters are simply wires with high resistivity and convert the power P = I2R they
consume into heat.
Chapter 18
LECTURE NOTES
Example:
Mr. Watson plugs 4 toasters like the one above into two outlets on his kitchen counter and turns
them all on at once. What happens?
If each toaster draws 4A of current and the two outlets are on the same circuit, the toasters would
try to draw 16A of current from the service panel. This would heat the household wiring beyond
design limits and perhaps cause a fire, so a circuit breaker (or fuse in older homes) limits the
current to 15A. Thus my toast-making attempt would “blow a fuse” or “trip the breaker” and no
current would flow to the toasters or any other device on the circuit. I would be wise to unplug
one of the toasters, reset the breaker and try again.
Do you know where the breaker box or fuse box is in your house? Are the circuits labeled? Can
you recover from a toast faux-pas on your own? Knowing where the water cut-off valve for the
house is located is also a good idea.
So far, we have considered circuits connected to a battery. In such a circuit, current flows in one
direction and is called direct current (DC). When toasters are plugged into a household circuit,
the current reverses direction 60 times per second. This is alternating current (AC).
Alternating Current
The voltage pushing charges through the household wiring varies from
+V0 = 170V to –V0 = -170V at a frequency of 60Hz. The average voltage is 0V, but it is more
useful to consider the root mean square or effective voltage Veff = Vrms= 120V. The current
flowing back and forth still encounters resistance and produces I2R heat in resistors. Averaging
over the square of the sinusoidal varying current shows the average power dissipated in a
1
1
resistance is P  I 02 R or P  V02 / R . We define a Veff pr Vrms that deposits the same power as
2
2
V
1
DC voltage by writing Vrms2 = V02 so Vrms = 0 = .707 V0 and similarly
2
2
I0
Irms =
= .707 I0.
2
In Europe, Vrms = 240V at a frequency of 50Hz. These different values date from the earliest
power generating stations of Edison (US) and Swan (Great Britain).
Finally, we offer a note about the speed of electricity. The individual electrons drift through a
wire connected to a battery, suffering repeated collisions, at roughly 0.05mm/s. However, the
effect of turning on a switch is communicated at the speed of light (3۰108m/s). This is similar to
water in a full hose coming out at once though the spigot is far away; the water that exits is not
the water just entering the hose (that water pushes the water already in the hose out).