Resistance
These notes are to supplement the reading of Chapter 19 and 20 in the Holt Text Physics Text Book by Serway and Faughn.
Other useful Resources:
-
The Physics Classroom is useful, click on links for more info. http://www.physicsclassroom.com/Class/circuits/
Mit lecture on resistance and Ohm’s law http://ocw.mit.edu/OcwWeb/Physics/8-02Electricity-andMagnetismSpring2002/VideoAndCaptions/detail/embed09.htm
Khan academy explanation of circuits http://khanexercises.appspot.com/video?v=3o8_EARoMtg
An explanation by a different author http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html#c1
Very basic video on Ohm’s law http://www.youtube.com/watch?v=-mHLvtGjum4
Eeris Fritz explains how to solve a circuit (3 parts) http://www.youtube.com/watch?v=fSuHIQZROmo
 http://www.youtube.com/watch?v=PR1G24E4ls4
 http://www.youtube.com/watch?v=9hH2FNJ9tio
VA SOL’s
PH.13 The student will investigate and understand how to diagram and construct basic electrical circuits and explain the
function of various circuit components. Key concepts include
a) Ohm’s law;
b) series, parallel, and combined circuits; and
c) circuit components including resistors, batteries, generators, fuses, switches, and capacitors.
Let’s start with some definitions.
Voltage (v) is a description of potential difference between two charges. It is often defined as the energy per
charge that is available. However it is perhaps most use to think of it as a force pushing electrons through a circuit. The
SI unit is the volt, not the Newton, because it is not an actual force. However, at one point in history it was thought of as
a force; and even today, occasionally voltage will be referred to as EMF or “electromotive force.”
Current (I) is the rate that electrons move through a circuit. The SI unit is the Ampere often abbreviated
Amp or “A.” The ampere is actually a coulomb/second, you can think of current like this. If you were watching electron go
by in a wire and counting how many pass every second, that number would be the current (just not in SI units.)
DC stands for direct current, with DC electrons will only move one direction through the circuit. Batteries
provide DC current, with a fixed positive and negative terminal.
AC stands for alternating current, with AC electrons change direction constantly. The hot wire will change
from a positive to a negative voltage and then back again. This causes the electrons to move back and forth in the wire
constantly changing direction. House hold outlets in the US use AC and the direction of the current changes 60 times every
second, which is why we say our electricity is 60Hz.
The above photo is an oscilloscope as it graphs AC voltage vs. time. Obtained from
http://www.soundbuggy.com/Eric/Car%20Audio/exile/x800_4/24_08_oscope.JPG Notice how the AC voltage is always
changing.
Dc voltage would look more like the picture to the
left. Image obtained from
http://people.sinclair.edu/nickreeder/EET114/PageArt/Scope
DC.gif
The product of voltage and current is power, the equation is P=vi . The SI unit for power is the watt, and
when multiplied by time is the amount energy an electrical circuit uses.
Resistance is what slows the flow of electrons down . If resistance were zero, then with even the smallest
voltage the current would rise toward infinitely. But, since objects have resistance, this does not happen. Resistance is
caused by the nucleus of atoms in the conducting material blocking the way for electrons to move. As the electrons hit a
nucleus, they are slowed down and the nuclease is shaken, this causes the material to heat up. Remember that heat is
just a measure of how much the atoms are vibrating in a substance.
The SI unit of Resistance is the OHM (Ω), named after Georg Ohm a German physicist.
An electric stove element is a device specially made to convert electrical energy into
thermal energy using the resistance in the element.
The resistance of a material is equal to the resistivity of the material times the length of the material, divided by the
cross sectional area of the material.
R=ρL/A
This means for a given material: longer = more resistance and thicker = less resistance
In the US the thickness of wire is rated in terms of AWG. The bigger AWG number the thinner the wire.
Resistivity is a property of the material, item with a high resistivity are insulators, medium resistivity are semiconductors,
and with a low resistivity they are called conductors. In the cart below you can see some common resistivity. Notice that
silver is the best conductor, however it is not often used for wiring because of cost. Copper is used far more often.
However, if budget is a big concern, aluminum wire can also be used.
Note that resistivity changes with the temperature, as a conductor heats up the resistance increases. However, for
semiconductors the resistance actually decreases as the temperature rises.
Example Problems.
EX. What is the resistance of 50m of copper wire with a cross sectional area of 0.000 005 5m2 at 20 degrees Celsius?
L=50m
A= .000 005 5m2
Ρ= 1.72 x10-8Ωm
EX. What material are you using if it has 2.50 Ω of resistance and is 51.4m long with a radius of 0.0012m?
Caution: This problem involves both a radius and a resistance. Both use the symbol R! You will have to keep them
distinguished.
list
L =51.4m
R=2.50 Ω
r =.0012 m
A resistor is a device specially designed to slow down electrons (reduce current and voltage) in a circuit.
Typically a resistor is labeled with 4 color bands. The first three bands correlate to the amount of resistance of the resistor,
while the fourth band describes how accurate the resistance is.
With the above resistor, the colors are brown, black, red, gold.
So Brown Black Red gold
1
0
x 100 Ω ± 5%
1000 Ω ± 5%
Or 1K ohm
Let’s try one more.
Here is a RED BLUE GREEN GOLD
2
6 x100,000 Ω ± 5%
Or
2,600,000 Ω
or 2.6M ohms
The schematic for a resistor looks like this. The schematic for a variable resistor looks like this.
Resistors in Series and Parallel.
In Series the resistance adds. RT= R1 + R2 + R3 + . . . So that the total resistance will be the sum of all of the resistors.
The total resistance in the circuit above would be RT= R1 + R2 + R3
RT= 150Ω + 200Ω + 160Ω
RT= 410 Ω.
In parallel the resistance adds inversely. 1/RT= 1/R1 + 1/R2 + 1/R3 + . . . So that the total resistance will be less than any of
the individual resistors.
The total resistance in the circuit to the left would be 1/RT=
1/R1 + 1/R2 + 1/R3
1/RT= 1/150Ω + 1/200Ω + 1/160Ω
1/RT=.01791
RT= 55.8 Ω
Ohm’s Law
Ohm’s law, unlike most laws encountered in physics, is not always valid. Although it holds true for most materials, there
are some materials that it does not work for, they are called non-ohmic materials, and Ohm’s law simply does not apply to
them. Furthermore, since resistance changes with temperature, and putting electricity through a circuit heats the circuit
up, Ohm’s law does not hold true when the temperature changes a lot. Having said that, it is still one of the most useful
tools for analyzing circuits.
Ohm’s law states that the ratio of voltage to current will remain constant for a given resistance. We write this
mathematically as.
R=v/I or rearranged v=ir
Example: Use Ohm’s law to determine the current, in the circuit to the left. Then determine the power dissipated by the
resistor.
List
R= 390 Ω
V= 9v
i=?
p=?
v=ir
i=v/r
i=9v/390 Ω
i = .023A
p=vi
p= (9v) (.023A)
p= 0.207w
RVIP Charts
In order to properly evaluate a circuit the voltage, current, and power for each resistor must be known.
Series Circuit.
In a series circuit the current stays the same!
STEP 1: Find total resistance. RT= R1 + R2 + R3
RT= 45Ω+25Ω+50Ω
RT=120Ω
STEP 2: Find total current, because this is a series
the same throughout.
v=i/r
i=v/r
i= 18v/120Ω = 0.15A
circuit the current is
We put this in for the current in each resistor because in series the current stays the same.
R
v
i
p
45 Ω
0.15A
25 Ω
0.15A
50 Ω
0.15A
STEP 4: Find voltage by v=ir. Do this separately for each resistor. Then fill the voltages into the chart.
v=ir=(0.15A)(45 Ω)=6.75v
v=ir=(0.15A)(25 Ω)=3.75v
v=ir=(0.15A)(50 Ω)=7.5v
R
45 Ω
25 Ω
50 Ω
v
6.75 v
3.75 v
7.5 v
I
0.15A
0.15A
0.15A
p
R
45 Ω
25 Ω
50 Ω
v
6.75 v
3.75 v
7.5 v
I
0.15A
0.15A
0.15A
p
1.01 w
0.563 w
1.13 w
STEP 5: Use p=vi to calculate the power.
p=vi=(6.75v)(0.15A)=1.01w
p=vi=(3.75v)(0.15A)=0.563w
p=vi=(7.5v)(0.15A)= 1.13w
STEP 6: Add the individual voltages to see if they add up to the total voltage.
6.75 v + 3.75 v + 7.5 v = 18v √
Parallel Circuit
In a parallel circuit the voltage stays the same.
STEP 1: Find the total Resistance
1/RT= 1/R1 + 1/R2 + 1/R3
1/RT= 1/50Ω + 1/25Ω + 1/45Ω
RT=12.2 Ω
STEP 2: Find the total current, although we will not use it until
i=v/r=(18v)/(12.2 Ω)=1.47A
later.
STEP 3: Fill in the voltage for every resistor, remember in parallel the resistance stays the same.
R
45 Ω
25 Ω
50 Ω
v
18 v
18 v
18 v
I
p
I
0.40 A
0.72 A
0.36 A
p
STEP 4: Calculate the individual currents and put them in the chart.
i=v/r=18v /45 Ω=0.40A
i=v/r=18v /25 Ω= 0.72 A
R
45 Ω
25 Ω
50 Ω
v
18 v
18 v
18 v
i=v/r=18v /50 Ω= 0.36A
STEP 5: Add the individual currents and see if it equals the total current form step 2.
.40A + 0.72A +0.36A = 1.48A √ is close enough to 1.47A that rounding can explain the difference.
STEP 6: Calculate the power and put it in the chart.
P=vi=(18V)(0.40A)= 7.2 watts
P=vi=(18V)(0.72A)= 13 watts
P=vi=(18V)(0.36A)= 6.5 watts
R
45 Ω
25 Ω
50 Ω
v
18 v
18 v
18 v
I
0.40 A
0.72 A
0.36 A
p
7.2 w
13 w
6.5 w
Combination Circuits
Combination circuits can be the most difficult to solve. However, following a few simple steps can make it easier.
STEP 1: Find the circuit with the equivalent resistance that is
either entirely parallel or entirely series.
For this one we will reduce the 50 ohm and 25 ohm resister into
a single 16.7 ohm resistor
STEP 2: Find the total resistance.
RT=R1+R2
RT= 45ohm + 16.7 ohm = 61.6 ohms
STEP 3: Find the total current.
i=v/r=18v/61.6 ohms=0.292A
STEP 4: Since the total current goes through the resistor in series , we can use this to get the individual voltages.
V=ir= (0.292A) (45 ohms) =13.1v
V=ir= (0.292A) (16.7 ohms) =4.9v
R
45 Ω
25 Ω
50 Ω
v
13.1 v
4.9 v
4.9 v
I
p
Remember because the 16.7 ohm represents the other two and voltage stays the same in parallel, the voltage we
get for the 16.7 ohm resistor is the voltage for the other two.
STEP 5: Calculate individual Currents.
I=v/r=13.1v/45ohms=0.291 Amps
I=v/r=4.9v /25 ohms=0.196 Amps
I=v/r=4.9 v/ 50 ohms=0.098 Amps
R
45 Ω
25 Ω
50 Ω
v
13.1 v
4.9 v
4.9 v
I
0.291 A
0.196 A
0.098 A
p
STEP 6: Calculate the individual powers.
P=vi=(13.1v)(0.291A)=3.81 w
P=vi=(4.9v)(0.196A)=.960 w
P=vi=(4.9v)(0.098A)= .480w
R
45 Ω
25 Ω
50 Ω
v
13.1 v
4.9 v
4.9 v
I
0.291 A
0.196 A
0.098 A
p
3.81 w
0.960 w
0.480 w