Download Unit 7 Part 2---Electric Field Notes

Unit 7, Part 2: The Electric Field (a vector quantity) and (a lot of) other stuff
You learned in the first part of this unit that charges (either positive or negative charges) can exert forces on each other over
some distance (Coulomb’s Law). They are able to do this due to an invisible force field called the electric field (E, a vector
quantity). We can’t see this field, but we know it’s there. This is very similar to us not being able to see the gravitational force
field, but we know it exists or we wouldn’t be able to experience the force due to it (such as being able to stand on the earth).
A little fact before we start: Charges produce electric force fields. The strength of the electric field around a charge decreases
as you move further away from the charge creating the field. Therefore, other charged objects in the field will experience
smaller forces acting on them (due to the decreased strength of the electric field) the further away they are from the charge
creating the field and vice-versa. This is similar to the strength of the gravitational field decreasing as you get further away
from the center of the earth (so the gravitational force that you would experience, i.e. your weight, gets smaller the further
away from the earth).
You can calculate the strength of the electric field (E) at any point in space around the charge that is creating the field by first
calculating the size of the electric force (Fe—this is the same force that Coulomb’s Law describes) that another (different)
charge (q, in Coulomb’s) is experiencing at that point in the original charge’s field.
The equation for the above is:
E = F (acting on q)
q
where
E = the strength of the electric “force” field at some point around the charge creating it; the units of E are N/C
(Newtons per Coulomb)
q = the size of the charge in the other charge’s electric field; this is NOT the size of the charge creating the field.
Once again, the above equation says that the strength of the electric field (E) generated by some random charge is equal to the
force another charge (q) experiences at that point in the field divided by the size of the charge, q (in Coulomb’s), in the other
charge’s field. The further away q is from the charge creating the field, the smaller the force it will experience (from
Coulomb’s Law---you know this already). And…the smaller the force it experiences, the weaker the electric field must be.
Therefore, the electric field intensity (E) decreases as you move away from the charge creating it. Kind of common sense.
The size (strength) of the electric field (E) produced by a charge is directly proportional to the size of the force (F) acting on
another charge (q) in its field. As E increases, the force on q (the little charge in the force field) will increase, too.
The Direction of the Electric Field
KNOW THIS:
The electric field (E) is a vector quantity so it has both magnitude (size) and direction. The direction of the electric field is
(by definition) the same direction that a force would act on a small POSITIVE test charge (q) if placed in the Electric
Field at some point. This is the definition of the direction of the electric field. Know this. It is that important.
For Example:
If the charge creating the field is positive, then its electric field would radiate away from it in all directions (think about the
direction the force would act on a positive test charge (q) in its field…it would push it away).
+
Positive Charge Creating the field
Unit 7, Part 2---The Electric Field
q+
Positive TEST Charge
1
If the charge creating the field is negative, its electric field would radiate IN toward it. Think about the direction a positive test
charge (q) would move if it was near a negative charge. It would move toward it, so the force must be acting in the same
direction to push it (the positive test charge) toward the negative charge that is creating the field in the first place.
q+
Negative Charge Creating the field
Positive TEST Charge
We use electric field lines to pictorially represent the strength and direction of the electric field that exists around a charge.
These lines don’t really exist, they’re simply how we portray the electric field around a charge (so we have a visual). The
more line’s drawn around a charge (and the closer they are together), the stronger the field. Notice in the diagrams below that
the field lines are the closest together the nearer they are to the charge creating the field. This shows that the strength of the
electric field (E) is the strongest the closer it is to the charge creating it. And… the further away from the charge creating the
electric field, the weaker (smaller E) the electric field. The electric field exists in 3 dimensions but I can’t represent that here (I
can only do 2-D on a flat piece of paper). Any way, the direction of the electric field extends radially AWAY from positive
charges and radially INTO negative charges as represented in the following diagrams. (Once again, think how a positive test
charge would move if placed in the field of a charge… that’s the direction of the electric field).
+
In the case where there is more than one charge near each other, the electric fields interact with each other (causing the charges
to either attract or repel) and will look something like the next diagrams.
In the diagram below in the middle, the electric field comes OUT of the positive charge. But since the electric field extends
INTO negative charges, it begins to curve around so it ends up entering the negative charge. The electric field will extend into
infinity if it doesn’t get pulled into another charge, but it’s impossible to show that here (on an 8.5 x 11 piece of paper). The
electric field that comes out of the right side of the positive charge will eventually curve around and enter the negative charge
(or another negative charge that’s close by). This is the reason that opposite charges attract---their fields pull them toward each
other.
If the two charges are both positive (as in the diagram above to the right), the two electric fields coming out of them will repel
each other (causing the two positive charges to move away from each other) as shown in the diagram above. Somewhere in
between the two positive charges the electric field will be zero.
Unit 7, Part 2---The Electric Field
2
In the case where you have
multiple random charges
interacting with each other, their
field line patterns become
complicated. However, you can
still tell by the spacing of the field
lines where the field is strong and
where it is weak.
Some practice problems.
A negative charge of 2 E –8 C experiences a force of 0.06 N to the right in an electric field. What is the electric field
magnitude and direction?
1.
q = 2 x 10-8 C
F (on q) = 0.060N
E=F
q
E=?
*The direction of the electric field would be to the left. If the negative charge experiences a force to the right, a positive charge
would experience a force to the left and the direction of the force on a positive charge is the direction of the electric field.
A positive test charge of 5 E –4 C is in an electric field that exerts a force of 2.5 E –4 N on it. What is the magnitude of the
electric field at the location of the test charge?
2.
q = +5 x 10--4 C
F = 2.5 x 10 -4 N
E=?
Make sure you know that the strength of the electric field will change with the proximity to the charge creating the
field. If the test charge (q) is close to the charge creating the field, it will experience a greater force than if it were
further away from the charge creating the field. This is because the strength of the field decreases as you get further
away from the charge creating the field.
Unit 7, Part 2---The Electric Field
3
Why do we care about the electric field?
The electric field is a storehouse of energy that can be transported over long distances and the energy contained in it can be
used to do work for us (for example, it can power our toaster or light a light bulb). The electric field is responsible for pushing
charge and energy through electrical circuits to give electrical devices power.
Background: If a ball is some height off the ground in the gravitational field, is has gravitational potential energy. If the ball
is then released, the gravitational field will do conservative work on the ball and move it down in its gravitational field. Since
the work done on the ball by the gravitational force is conservative, the total energy of the ball won’t change as it moves
naturally in the field. However, its energy will get transferred from gravitational PE to KE. The ball loses gravitational PE as
is falls but it speeds up in doing so and its KE increases (remember from a while back that anytime work is done on an object,
energy is transferred).
Remember that Wconservative forces = KE = -PE
Likewise, if a charge moves the way it will naturally want to move in an electric field, the electric field does conservative work
on the charge and it will lose electrical potential energy and gain KE in the process. The larger the amount of charge (in
coulomb’s) moved in the field, the greater the conservative work done by the field and the greater the loss in the total electrical
potential energy and the greater the gain in KE. Think of two negative charges close to each other. They naturally want to
move away from each other (since like charges repel). If you hold them in place close to each other, they have electrical
potential energy simply due to their positions in each other’s electric fields. Once you release them, they are going to move
away from each other and convert their electrical potential energy into KE and we can harness that to do stuff for us. We call
the amount of potential energy per coulomb of charge the “electric potential”. The electric potential (V) is simply the amount
of electrical potential energy (PEe) each coulomb of charge has based on where it is in an electric field compared to some
reference point (that has “zero electric potential”). The electric potential (V) is a scalar quantity and the units of electric
potential are volts. The electric field possesses electric potential (voltage), and if a charge is in the electric field it will have a
certain amount of electrical potential energy based on the charge’s size (in coulomb’s) and how much electrical potential
(voltage) the field has at that point. For example, a 12 Volt battery has the potential to give 12 J of electrical potential energy
to a ONE coulomb of charge (between the positive terminal and the negative terminal). As the one coulomb of charge exiting
the battery flows around a circuit, it will gradually lose the 12 J of potential energy and convert it to either KE or useful work
for us (to whatever the charge is powering in the circuit).
Electric potential (AKA voltage) refers to the change in electrical potential energy per ONE coulomb of charge (6.25 x 1018
electrons) from a start place (with maximum electrical PE) to an end place (with zero electrical PE). The equation for that is
V = - PE
q
the units of voltage are Joules
Coulomb
/
1 Volt (1V) = 1 joule 1 Coulomb
= J/C
= Volt (V)
(one Joule of energy per every coulomb of charge)
(ex: 12 V = 12 J/C)
(And. remember from a while back in the course that work (by conservative forces) =KE= - PE. So, if one coulomb of
charge loses 12 J of PEe as it naturally moves from X to Y in the electric field, then the charge gains 12 J of KE and that KE
can eventually be transferred into useful work for us).
Electric potential (AKA Voltage) simply refers to how much electrical potential energy ONE COULOMB of charge has at
some point in an electric field---i.e., how much electrical potential energy 6.25 x 10 18 electrons have the ability to lose as they
move across an electrical circuit to deliver power to a toaster or another electrical device. The more electrical potential energy
ONE coulomb of charge (6.25 x 1018 electrons) has, the greater the electric potential (voltage) difference between a start place
and an end place in the circuit. Your outlets at home are set for ~110 V. That means that each Coulomb of electrons (6.25 x
1018 electrons) that come out of the outlets will collectively have ~110 Joules of PE e to use to power all the electrical devices in
your home. The electrical potential energy (voltage) is like an electrical pressure that can produce a flow of charge called a
current (more on that in the next unit). The greater the electrical pressure (electric potential , AKA Voltage), the greater the
current of charge that can be moved. Think about the water pressure in your house, it’s the same thing. The greater the
pressure in your water pipes, the more water will flow out. The greater the electrical pressure [voltage], the more electrons will
flow out. The electric field is the invisible force field that gives these charges their electrical potential energy and causes the
energy to flow (out of the electrical sockets) in the first place.
So, to sum it up: Like charges repel, opposite charges attract. The “electric field” around charged objects causes the
attraction or repulsion of the charges. If two like charges (like two protons) are close together they have electrical PE due to
Unit 7, Part 2---The Electric Field
4
their positions in the electric fields (the closer they together, the more electric potential energy they have). Their natural
tendency is to move away from each other and move toward regions on lower electrical PE. Once released and free to move
naturally, as the charges move away from each other, they will lose their electric potential energy and it will be converted to
KE and eventually used to do work on something else. We can use the transfer of energy to power our electrical devices like
our toaster and our computer. In the real world, the energy that is needed to create the electric field to give the charges their
electric potential energy in the first place comes from burning of fossil fuels, solar power, water power, etc. The electric power
plants create HUGE electric fields and convert other types of energy into electrical potential energy. Neat, huh?
Charges will always tend to move in the direction of lowest PEe for that type of charge (just like things in the gravitational
force field will always tend to move toward the lowest PE g). If a proton was in an electric field that had a higher potential
(AKA voltage) to the left and a lower potential (AKA voltage) to the right, the proton will want to move to the right. Charges
will continue moving in any conductor until the electrical potential (the voltage) is the same everywhere in the conductor.
3.
4.
If 120J of work is done by the electric field (a conservative force) as it pushes one Coulomb of charge from a
negatively charged plate to a positively charged plate, what is the potential difference (voltage) between the plates?
(In this problem the charge that is moving is negative….)
W = 120J
q = 1C
(remember that W(conservative forces) =KE= -PE)
V=?
V =  PE
q
ANS: 120 J/C or 120 V
How much work is done by the electric field as it pushes 0.15 C of charge through a potential difference (voltage) of
9.0 volts?
W=?
(remember that W(conservative forces) = KE = -PE)
q = 0.15C
V=9V
V =  PE
q
Both of the above problems deal with how much work was done by the electric field on charges as they move the way they
naturally want to move in the electric field. This amount of work that was done on the charges by the conservative forces of
electric field is the same numerical value as how much electrical potential energy the charges lose as they move the way they
naturally want to move in an electric field (remember that W = -PE) . The “voltage” the problems are referring to is the
electrical potential difference that exists between where the charge is and where it is going to end up. Picture two parallel
plates (below) that have an electrical potential (voltage) difference between them. One plate has an electrical potential
(voltage) of 0V and the other one has an electrical potential (voltage) of 10V (I’m making these numbers up). The proton in
the electric field between the two plates will naturally want to move in a direction away from the 10V (higher electrical
potential) toward the 0V (lower electrical potential) plate. The work that is done on the charge as it moves is conservative as
the force doing the work is the electric field force (which is conservative, just like the gravitational force and the elastic force).
So, the work done will not increase or decrease the total energy of the system…it will simply change it from one form to
another. If work is done on the charge (by the electric field) to move it from the 10V to the 0V plate, the charge will lose
potential energy but it will speed up as it is moving so it will gain Kinetic Energy (like when you drop a book ---it loses
gravitational potential energy as it’s falling in the gravitational field but it gains the same amount of KE that it loses as it falls.
This is due to the work that is done on the book by the gravitational field being conservative, so the total energy of the book
will never change W = KE = -PE). If a charge is released and free to move how it wants, it will move toward the lower
potential energy for that type of charge and lose its electrical potential energy. As it moves freely in the electric field, the
electrical potential energy it loses is then converted to KE and we can then use the motion of the charges to produce thermal
energy, light energy, etc for us, i.e., we can harness the energy and use it to do work for us---like power a toaster or a light
bulb.
p+
0V
Unit 7, Part 2---The Electric Field
10V
5
Sometimes you’ll be asked to calculate the strength of the electric field (E) that exists between two oppositely charged parallel
plates (like the ones above) that are d (in meters) apart with a potential difference of V between the positive and the negative
plates. The important thing to understand in this situation is that the electric field strength between oppositely charged parallel
plates is uniform (constant, the same) everywhere between the plates (unlike the electric field around a single point charge).
This constant electric field will produce constant forces and constant accelerations (so you can use kinematics to solve
problems). The equation to do this is
Eavg = -V/d (disregard the negative sign in this formula)
The above equation will also tell you the potential difference (voltage, V) that exists between the two oppositely charged
parallel plates that are “d” meters apart with an electric field “E” between them.
5.
The electric field intensity between two parallel plates is 8000 N/C. The plates are 0.05m apart. What is the potential
difference between them?
ANS: 4 E2 volts
6.
A voltmeter (a device that measures voltage) reads 500 volts when placed across two oppositely charged parallel
plates that are 0.02m apart. What is the electric field strength at any point between them?
ANS: 2.5 E4 N/C
7.
What potential difference exists between two parallel metal plates 0.5m apart if the field between them is 2.5 E3
N/C ?
ANS: 1.25 E3 V
Capacitors
Sometimes it is necessary to store electrical energy (the static kind) until enough charge is built up so a useful amount of work
can be done. We are able to store electrical charge on devices called capacitors. An example of a capacitor is in a camera that
has a “flash” capability. Think about this: When you want to take a picture that needs a flash and you try to press the button to
take the picture, it usually takes a moment of pressing before the flash goes off and the picture is actually taken. The reason
behind this is there needs to a relatively huge burst of electrical energy released by the battery in order to power the flash.
However, the little batteries (usually 1.2 volts) in a camera don’t
have the ability to generate a huge burst of energy at one time.
They are designed to put out small amounts of steady energy over a
longer time. So, to generate the huge burst of energy that is needed,
a capacitor is used in the camera. A capacitor is made from two
parallel plates of a certain area each that are a tiny distance apart
(like the width of a hair). Each of the plates are connected to the
battery via a wire. When you press down on the camera button and
want to take a picture using the flash, charge (electrons) begins to
be released from the battery and is then stored on one of the plates
in the capacitor (this plate becomes negatively charged very fast).
As the one plate becomes negatively charged, the other becomes
positively charged (due to the negative charges that were originally
on the plate being repelled by the strong negative plate a short
distance, d, away) and an electric field is generated between the
plates. The more charge that is crammed together on the negative
plate, the greater the electric potential difference (voltage) across
the two plates. The charge will continue to accumulate on the
negative plate of the capacitor until the capacitor senses there is
enough charge (with enough potential energy) to power the flash.
When this point is reached, all the charge is released at once and as
the charge moves to a region of lower electric potential energy, the
burst of energy lost is harnessed and used to power the flash. Neat,
huh? But, of course, with every neat thing comes math…..
Unit 7, Part 2---The Electric Field
6
Honors/AP only (but AP kids will get a LOT more info on capacitors than is represented below)….. Capacitors are made
of (usually) two parallel conducting sheets (such as metal) with an insulating space between them. Capacitance (C, don’t
confuse this symbol with the symbol for the unit coulomb---they refer to completely different things) basically tells how much
charge can be stored on a capacitor. Capacitance is defined as the “ratio of charge to potential difference between the parallel
plates”. The capacitance of any given capacitor is independent of the amount of charge stored on it… every
manufactured capacitor can only have a certain capacitance, and it will always have that capacitance. If there is a small
amount of charge on it, there will also be a small potential difference between the two plates. If there is a large amount of
charge on the capacitor, there will also be a large potential difference between the plates. As the charge on the capacitor
increases, so will the voltage between the plates and vice versa. The ratio of the two will always be whatever the capacitance
of the particular capacitor is. However, if the charge and the voltage become too great, you can fry a capacitor. The equation
to measure capacitance is
C=
Q
V
The amount of charge when discussing capacitors is symbolized with a large Q rather than a small q; a small q represents a
point charge whereas large Q represents a large amount of charge relating to capacitors.
The units of capacitance are C/V (Coulombs /volts) which smooshes down to a new unit called a Farad (F).
One Coulomb is a lot of charge (remember how many electrons are in a coulomb???), so one Farad (F) is a LOT of
capacitance. One Farad is one Coulomb of charge stored across one volt of potential difference (between the parallel plates of
the capacitor). Capacitors that are used in real-life are typically measured in pico- to microfarads (10-12 to 10-6 Farads).
8.
A sphere has a potential difference of 60 volts between it and the earth when it’s charged with 3 E –6 C. What is its
capacitance?
C=?
q = 3 E –6 C
V = 60 volts
9.
C = q/V
ANS : 5 E –8 F
A 27μF capacitor has a potential difference of 25V across it. What is the charge on the capacitor?
ANS: 6.8 E –4 C
10.
Both a 3.3 μF and a 6.8 μF capacitor are connected across a 15V potential difference. Which capacitor has the greater
charge?
ANS: the larger capacitor (1 E –4 C)
11.
A 2.2 μF capacitor is first charge so that the potential difference is 6V. How much additional charge is needed to
increase the potential difference between the plates to 15V?
ANS: 2 E –5 C
Unit 7, Part 2---The Electric Field
7