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Chapter 1 Charge & Coulomb's Law
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Calculus-Based Physics II
by Jeffrey W. Schnick Copyright 2006, Jeffrey W. Schnick, Creative Commons Attribution Share-Alike License 2.5. You can copy, modify, and re-release this work under the same license provided you give attribution to the author. See http://creativecommons.org/
- 1 Charge & Coulomb's Law ....................................................................................................... This book is dedicated to Marie, Sara, and Natalie.
- 2 The Electric Field: Description and Effect ...............................................................................
- 3 The Electric Field Due to one or more Point Charges ............................................................
- 4 Conductors and the Electric Field ..........................................................................................
- 5 Work Done by the Electric Field, and, the Electric Potential..................................................
- 6 The Electric Potential Due to One or More Point Charges .....................................................
- 7 Equipotential Surfaces, Conductors, and Voltage ..................................................................
- 8 Capacitors, Dielectrics, and Energy in Capacitors..................................................................
- 9 Electric Current, EMF, Ohm's Law........................................................................................
- 10 Resistors in Series and Parallel; Measuring I & V ................................................................
- 11 Resistivity, Power ...............................................................................................................
- 12 Kirchhoff’s Rules, Terminal Voltage...................................................................................
- 13 RC Circuits .........................................................................................................................
- 14 Capacitors in Series & Parallel ..........................................................................................
- 15 Magnetic Field Intro: Effects .............................................................................................
- 16 Magnetic Field: More Effects ............................................................................................
- 17 Magnetic Field: Causes .....................................................................................................
- 18 Faraday's Law, Lenz's Law................................................................................................
- 19 Induction, Transformers, and Generators ...........................................................................
- 20 Faraday’s Law and Maxwell’s Extension to Ampere’s Law...............................................
- 21 The Nature of Electromagnetic Waves...............................................................................
- 22 Huygens’s Principle and 2-Slit Interference.......................................................................
- 23 Single-Slit Diffraction .......................................................................................................
- 24 Thin Film Interference.......................................................................................................
- 25 Polarization .......................................................................................................................
- 26 Geometric Optics, Reflection.............................................................................................
- 27 Refraction, Dispersion, Internal Reflection ........................................................................
- 28 Thin Lenses: Ray Tracing..................................................................................................
- 29 Thin Lenses: Lens Equation, Optical Power ......................................................................
- 30 The Electric Field Due to a Continuous Distribution of Charge on a Line ..........................
- 31 The Electric Potential due to a Continuous Charge Distribution.........................................
- 32 Calculating the Electric Field from the Electric Potential ...................................................
- 33 Gauss’s Law......................................................................................................................
- 34 Gauss’s Law Example .......................................................................................................
- 35 Gauss’s Law for the Magnetic Field, and, Ampere’s Law Revisited ..................................
- 36 The Biot-Savart Law .........................................................................................................
- 37 Maxwell’s Equations .........................................................................................................
Chapter 1 Charge & Coulomb's Law
1 Charge & Coulomb's Law
Charge is a property of matter. There are two kinds of charge, positive “+” and negative “−”.^1 An object can have positive charge, negative charge, or no charge at all. A particle which has charge causes a force-per-charge-of-would-be-victim vector to exist at each point in the region of space around itself. The infinite set of force-per-charge-of-would-be-victim vectors is called a vector field. Any charged particle that finds itself in the region of space where the force-per- charge-of-would-be-victim vector field exists will have a force exerted upon it by the force-per- charge-of-would-be-victim field. The force-per-charge-of-would-be-victim field is called the electric field. The charged particle causing the electric field to exist is called the source charge. (Regarding jargon: A charged particle is a particle that has charge. A charged particle is often referred to simply as “a charge.”)
The source charge causes an electric field which exerts a force on the victim charge. The net effect is that the source charge causes a force to be exerted on the victim. While we have much to discuss about the electric field, for now, we focus on the net effect, which we state simply (neglecting the “middle man”, the electric field) as, “A charged particle exerts a force on another charged particle.” This statement is Coulomb’s Law in its conceptual form. The force is called the Coulomb force, a.k.a. the electrostatic force.
Note that either charge can be viewed as the source charge and either can be viewed as the victim charge. Identifying one charge as the victim charge is equivalent to establishing a point of view, similar to identifying an object whose motion or equilibrium is under study for purposes of
applying Newton’s 2nd^ Law of motion, m
a= ∑F
. In Coulomb’s Law, the force exerted on one
charged particle by another is directed along the line connecting the two particles, and, away from the other particle if both particles have the same kind of charge (both positive, or, both negative) but, toward the other particle if the kind of charge differs (one positive and the other negative). This fact is probably familiar to you as, “like charges repel and unlike attract.”
The SI unit of charge is the coulomb, abbreviated C. One coulomb of charge is a lot of charge, so much that, two particles, each having a charge of +1 C and separated by a distance of 1 meter
exert a force of 9 × 109 N, that is, 9 billion newtons on each other.
This brings us to the equation form of Coulomb’s Law which can be written to give the magnitude of the force exerted by one charged particle on another as:
(^1) It can be argued that, since the net charge on an object consisting of a bunch of particles, each of which has a
positive amount of charge, and a bunch of particles, each of which has a negative amount of charge, is simply the algebraic sum of all the individual values of charge, there is really only one kind of charge and that it is the value of the charge of an object that can be either positive or negative. (See One Kind of Charge by John Denker, http://www.av8n.com/physics/one-kind-of-charge.htm .) However, it is conventional to refer to a negative amount of charge as an amount of negative charge and a positive amount of charge as an amount of positive charge. We use language consistent with that convention.
Chapter 1 Charge & Coulomb's Law
Note the absence of the absolute value signs around q 1 and q 2. A particle which has a certain
amount, say, 5 coulombs of the negative kind of charge is said to have a charge of − 5 coulombs and one with 5 coulombs of the positive kind of charge is said to have a charge of +5 coulombs) and indeed the plus and minus signs designating the kind of charge have the usual arithmetic meaning when the charges enter into equations. For instance, if you create a composite object by combining an object that has a charge of q 1 = +3 C with an object that has a charge of q 2 = −5 C,
then the composite object has a charge of q = q 1 + q 2
q = +3 C + (−5 C)
q = −2 C
Note that the arithmetic interpretation of the kind of charge in the vector form of Coulomb’s Law causes that equation to give the correct direction of the force for any combination of kinds of charge. For instance, if one of the particles has positive charge and the other negative, then the value of the product q 1 q 2 in equation 1-
2 12
1 2
12 r ˆrrrr
q q F =k
has a negative sign which we can associate with the unit vector. Now − rr^ ˆrr 12 is in the direction
opposite “from 1 to 2” meaning it is in the direction “from 2 to 1.” This means that F 12
, the
force of 1 on 2, is directed toward particle 1. This is consistent with our understanding that
opposites attract. Similarly, if q 1 and q 2 are both positive, or both negative in 12 122 ˆrrrr 12
r
q q F =k
then the value of the product q 1 q 2 is positive meaning that the direction of the force of 1 on 2 is
rrrr^ ˆ 12 (from 1 to 2), that is, away from 1, consistent with the fact that like charges repel.
We’ve been talking about the force of 1 on 2. Particle 2 exerts a force on particle 1 as well. It is
given by 21 122 ˆrrrr 21
r
q q F =k
. The unit vector ˆrrrr 21 , pointing from 2 to 1, is just the negative of the unit
vector pointing from 1 to 2:
ˆrrrr 21 =−ˆrrrr 12
If we make this substitution into our expression for the force exerted by particle 2 on particle 1, we obtain:
21 =^122 (^ −rrˆrr 12 )
r
q q F k
2 12
1 2
21 ˆrrrr
r
q q F =−k
Chapter 1 Charge & Coulomb's Law
Comparing the right side with our expression for the force of 1 on 2 (namely,
2 12
1 2
12 r rrrrˆ
q q F =k
), we see that
F 21 F 12
So, according to Coulomb’s Law, if particle 1 is exerting a force F 12
on particle 2, then particle 2 is, at the same time, exerting an equal but opposite force F 12
− back on particle 2, which, as we know, by Newton’s 3rd^ Law, it must.
In our macroscopic^2 world we find that charge is not an inherent fixed property of an object but, rather, something that we can change. Rub a neutral rubber rod with animal fur, for instance, and you’ll find that afterwards, the rod has some charge and the fur has the opposite kind of charge. Ben Franklin defined the kind of charge that appears on the rubber rod to be negative charge and the other kind to be positive charge. To provide some understanding of how the rod comes to have negative charge, we delve briefly into the atomic world and even the subatomic world.
The stable matter with which we are familiar consists of protons, neutrons, and electrons. Neutrons are neutral, protons have a fixed amount of positive charge, and electrons have the same fixed amount of negative charge. Unlike the rubber rod of our macroscopic world, you cannot give charge to the neutron and you can neither add charge to, nor remove charge from, either the proton or the electron. Every proton has the same fixed amount of charge, namely
- 60 × 10 −^19 C. Scientists have never been able to isolate any smaller amount of charge. That amount of charge is given a name. It is called the e, abbreviated e and pronounced “ee”. The e is a non-SI unit of charge. As stated 1 e= 1. 60 × 10 −^19 C. In units of e, the charge of a proton is 1 e (exactly) and the charge of an electron is − 1 e. For some reason, there is a tendency among humans to interpret the fact that the unit the e is equivalent to 1. 60 × 10 −^19 C to mean that 1 e equals − 1. 60 × 10 −^19 C. This is wrong! Rather,
1 e= 1. 60 × 10 −^19 C. A typical neutral atom consists of a nucleus made up of neutrons and protons surrounded by orbiting electrons such that the number of electrons in orbit about the nucleus is equal to the number of protons in the nucleus. Let’s see what this means in terms of an everyday object such as a polystyrene cup. A typical polystyrene cup has a mass of about 2 grams. It consists of roughly: 6 × 10 23 neutrons, 6 × 1023 protons, and, when neutral, 6 × 10 23 electrons. Thus, when neutral it has about 1 × 10 5 C of positive charge and 1 × 105 C of negative charge, for a total of 0 charge. Now if you rub a polystyrene cup with animal fur you can give it a noticeable charge. If you rub it all over with the fur on a dry day and then experimentally determine the charge on the cup, you will find it to be about − 5 × 10 −^8 C. This represents an increase of about 0.00000000005 % in the number of electrons on the cup. They were
(^2) Macroscopic means “of a size that we can see with the naked eye.” It is to be contrasted with microscopic (you
need a light microscope to see it), atomic (of or about the size of an atom), and subatomic (smaller than an atom, e.g. about the size of a nucleus of an atom).
Chapter 1 Charge & Coulomb's Law
of charging by rubbing is called triboelectrification. The following ordered list of the tendency of (a limited number of) materials to give up or accept electrons is called a triboelectric sequence:
Increasing tendency to take on electrons Air Rabbit Fur Glass Wool Silk Steel Rubber Polyester Styrofoam Vinyl Teflon Increasing tendency to give up electrons
The presence and position of air on the list suggests that it is easier to maintain a negative charge on objects in air than it is to maintain a positive charge on them.
Conductors and Insulators
Suppose you charge a rubber rod and then touch it to a neutral object. Some charge, repelled by the negative charge on the rod, will be transferred to the originally-neutral object. What happens to that charge then depends on the material of which the originally-neutral object consists. In the case of some materials, the charge will stay on the spot where the originally neutral object is touched by the charged rod. Such materials are referred to as insulators, materials through which charge cannot move, or, through which the movement of charge is very limited. Examples of good insulators are quartz, glass, and air. In the case of other materials, the charge, almost instantly spreads out all over the material in question, in response to the force of repulsion (recalling that force causes acceleration which leads to the movement) that each elementary particle of the charge exerts on every other elementary particle of charge. Materials in which the charge is free to move about are referred to as conductors. Examples of good conductors are metals and saltwater.
When you put some charge on a conductor, it immediately spreads out all over the conductor. The larger the conductor, the more it spreads out. In the case of a very large object, the charge can spread out so much that any chunk of the object has a negligible amount of charge and hence, behaves as if were neutral. Near the surface of the earth, the earth itself is large enough to play such a role. If we bury a good conductor such as a long copper rod or pipe, in the earth, and connect to it another good conductor such as a copper wire, which we might connect to another metal object, such as a cover plate for an electrical socket, above but near the surface of the earth, we can take advantage of the earth’s nature as a huge object made largely of conducting material. If we touch a charged rubber rod to the metal cover plate just mentioned, and then withdraw the rod, the charge that is transferred to the metal plate spreads out over the earth to the extent that the cover plate is neutral. We use the expression “the charge that was transferred to the cover plate has flowed into the earth.” A conductor that is connected to the earth in the manner that the cover plate just discussed is connected is called “ground.” The act of touching a charged object to ground is referred to as grounding the object. If the object itself is a conductor, grounding it (in the absence of other charged objects) causes it to become neutral.
Charging by Induction
Chapter 1 Charge & Coulomb's Law
If you hold one side of a conductor in contact with ground and bring a charged object very near the other side of the conductor, and then, keeping the charged object close to the conductor without touching it, break the contact of the conductor with ground, you will find that the conductor is charged with the opposite kind of the charge that was originally on the charged object. Here’s why. When you bring the charged object near the conductor, it repels charge in the conductor right out of the conductor and into the earth. Then, with those charges gone, if you break the path to ground, the conductor is stuck with the absence of those charged particles that were repelled into the ground. Since the original charged object repels the same kind of charge that it has, the conductor is left with the opposite kind of charge.
Polarization
Let’s rub that rubber rod with fur again and bring the rubber rod near one end of a small strip of neutral aluminum foil. We find that the foil is attracted to the rubber rod, even though the foil remains neutral. Here’s why:
The negatively charged rubber rod repels the free-to-move negative charge in the strip to the other end of the strip. As a result, the near end of the aluminum strip is positively charged and the far end is negatively charged. So, the rubber rod attracts the near end of the rod and repels the far end. But, because the near end is nearer, the force of attraction is greater than the force of repulsion and the net force is toward the rod. The separation of charge that occurs in the neutral strip of aluminum is called polarization, and, when the neutral aluminum strip is positive on one end and negative on the other, we say that it is polarized.
Polarization takes place in the case of insulators as well, despite the fact that charge is not free to move about within an insulator. Let’s bring a negatively-charged rod near one end of a piece of paper. Every molecule in the paper has a positive part and a negative part. The positive part is attracted to the rod and the negative part is repelled. The effect is that each molecule in the paper is polarized and stretched. Now, if every bit of positive charge gets pulled just a little bit closer to the rod and every bit of negative charge gets pushed a little farther away, the net effect in the bulk of the paper is to leave it neutral, but, at the ends there is a net charge. On the near end, the repelled negative charge leaves the attracted positive charge all by itself, and, on the far end, the attracted positive charge leaves the repelled negative charge all by itself.
As in the case of the aluminum strip, the negative rubber rod attracts the near, positive, end and repels the far, negative, end, but, the near end is closer so the attractive force is greater, meaning that the net force on the strip of paper is attractive. Again, the separation of the charge in the paper is called polarization and the fact that one end of the neutral strip of paper is negative and the other is positive means that the strip of paper is polarized.
Rod
Strip of Paper
Chapter 2 The Electric Field: Description and Effect
and the electric field itself. In this chapter, we focus our attention on the relation between an existing electric field (with no concern for how it came to exist) and the effect of that electric field on any charged particle in the electric field. To do so, it is important for you to be able to accept a given electric field as specified, without worrying about how the electric field is caused to exist in a region of space. (The latter is an important topic which we deal with at length in the next chapter.)
Suppose for instance that at a particular point in an empty region in space, let’s call it point P, there is an eastward-directed electric field of magnitude 0.32 N/C. Remember, initially, we are talking about the electric field at an empty point in space. Now, let’s imagine that we put a particle that has +2.0 coulombs of charge at point P. The electric field at point P will exert a force on our 2.0 C victim:
F E
=q
eastward) C
N
F = 2. 0 C( 0. 32
Note that we are dealing with vectors so we did include both magnitude and direction when we substituted for E
. Calculating the product on the right side of the equation, and including the direction in our final answer yields:
F = 0. 64 N eastward
We see that the force is in the same direction as the electric field. Indeed, the point I want to make here is about the direction of the electric field: The electric field at any location is defined to be in the direction of the force that the electric field would exert on a positively charged victim if there was a positively charged victim at that location.
Told that there is an electric field in a given empty region in space and asked to determine its direction at the various points in space at which the electric exists, what you should do is to put a single positively-charged particle at each of the various points in the region in turn, and find out which way the force that the particle experiences at each location is directed. Such a positively- charged particle is called a positive test charge. At each location you place it, the direction of the force experienced by the positive test charge is the direction of the electric field at that location.
Having defined the electric field to be in the direction of the force that it would exert on a positive test charge, what does this mean for the case of a negative test charge? Suppose that, in the example of the empty point in space at which there was a 0.32 N/C eastward electric field, we place a particle with charge − 2 .0 coulombs (instead of +2.0 coulombs as we did before). This particle would experience a force:
F E
=q
eastward) C
N
F =− 2. 0 C( 0. 32
F =− 0. 64 N eastward
A negative eastward force is a positive westward force of the same magnitude:
F = 0. 64 N westward
Chapter 2 The Electric Field: Description and Effect
In fact, any time the victim particle has negative charge, the effect of the minus sign in the value
of the charge q in the equation F E
= q is to make the force vector have the direction opposite that of the electric field vector. So the force exerted by an electric field on a negatively charged particle that is at any location in that field, is always in the exact opposite direction to the direction of the electric field itself at that location.
Let’s investigate this direction business for cases in which the direction is specified in terms of unit vectors. Suppose that a Cartesian reference frame^2 has been established in an empty region of space in which there is an electric field. Further assume that the electric field at a particular point, call it point P, is:
kkkk C
kN E = 5. 0
Now suppose that a proton ( q = 1. 60 × 10 −^19 C) is placed at point P. What force would the
electric field exert on the proton?
F E
=q
kkkk C
N
F= ( 1. 60 × 10 −^19 C) 5. 0 × 103
F= 8. 0 × 10 −^16 N kkkk
The force on the proton is in the same direction as that of the electric field at the location at which the proton was placed (the electric field is in the +z direction and so is the force on the proton), as it must be for the case of a positive victim.
If, in the preceding example, instead of a proton, an electron ( q = − 1. 60 × 10 −^19 C) is placed at
point P, recalling that in the example kkkk C
kN E = 5. 0
, we have
F E
=q
kk kk C
N
F= ( − 1. 60 × 10 −^19 C) 5. 0 × 103
F= − 8. 0 × 10 −^16 N kkkk
The negative sign is to be associated with the unit vector. This means that the force has a
magnitude of 8. 0 × 10 −^16 Nand a direction of − kkkk. The latter means that the force is in the –z
direction which is the opposite direction to that of the electric field. Again, this is as expected.
(^2) A reference frame is a coordinate system.
Chapter 2 The Electric Field: Description and Effect
the same diagram. The closer the lines are, the stronger the electric field they represent. In the case of the uniform electric field in question, because the magnitude of the electric field is the same everywhere (which is what we mean by “uniform”), the line spacing must be the same everywhere. Furthermore, because the electric field in this example has a single direction, namely eastward, the electric field lines will be straight lines, with arrowheads:
NORTH
EAST
SOUTH
WEST
E
Chapter 3 The Electric Field Due to one or more Point Charges
3 The Electric Field Due to one or more Point Charges
A charged particle (a.k.a. a point charge, a.k.a. a source charge) causes an electric field to exist in the region of space around itself. This is Coulomb’s Law for the Electric Field in conceptual form. The region of space around a charged particle is actually the rest of the universe. In practice, the electric field at points in space that are far from the source charge is negligible because the electric field due to a point charge “dies off like one over r-squared.” In other words, the electric field due to a point charge obeys an inverse square law, which means, that the electric field due to a point charge is proportional to the reciprocal of the square of the distance that the point in space, at which we wish to know the electric field, is from the point charge that is causing the electric field to exist. In equation form, Coulomb’s Law for the magnitude of the electric field due to a point charge reads
r^2
k q E = (3-1)
where E is the magnitude of the electric field at a point in space,
k is the universal Coulomb constant (^2)
2 9 C
N m 8 99 10
k =. × ,
q is the charge of the particle that we have been calling the point charge, and
r is the distance that the point in space, at which we want to know E, is from the point
charge that is causing E.
Again, Coulomb’s Law is referred to as an inverse square law because of the way the magnitude of the electric field depends on the distance that the point of interest^1 is from the source charge.
Now let’s talk about direction. Remember, the electric field at any point in space is a force-per- charge-of-would-be-victim vector and as a vector, it always has direction. We have already discussed the defining statement for the direction of the electric field: The electric field at a point in space is in the direction of the force that the electric field would exert on a positive victim if there were a positive victim at that point in space. This defining statement for the direction of the electric field is about the effect of the electric field. We need to relate this to the cause of the electric field. Let’s use some grade-school knowledge and common sense to find the direction of the electric field due to a positive source charge. First, we just have to obtain an imaginary positive test charge. I recommend that you keep one in your pocket at all times (when not in use) for just this kind of situation. Place your positive test charge in the vicinity of the source charge, at the location at which you wish to know the direction of the electric field. We know that like charges repel, so, the positive source charge repels our test charge. This means that the source charge, the point charge that is causing the electric field under investigation to exist, exerts a force on the test charge that is directly away from the source charge. Again, the electric field at any point is in the direction of the force that would be exerted on a positive test charge if that charge was at that point, so, the direction of the electric field is “directly away from the positive source charge.” You get the same result no matter where, in the region of space
(^1) The point of interest is the point at which we wish to calculate the electric field due to the point charge.
Chapter 3 The Electric Field Due to one or more Point Charges
existence there is implied by the lines that are drawn—we simply can’t draw lines everywhere that the electric field does exist without completely blackening every square inch of the diagram. Thus, a charged victim that finds itself at a position in between the lines will experience a force as depicted below for each of two different positively-charged victims.
The next point is a reminder that a negatively-charged particle that finds itself at a position at which an electric field exists, experiences a force in the direction exactly opposite that of the electric field at that position.
E
F 2
F 1
E
F
Chapter 3 The Electric Field Due to one or more Point Charges
The third and final point that should be made here is a reminder that the direction of the force experienced by a particle, is not, in general, the direction in which the particle moves. To be sure, the expression “in general” implies that there are special circumstances in which the particle would move in the same direction as that of the electric field but these are indeed special. For a particle on which the force of the electric field is the only force acting, there is no way it will stay on one and the same electric field line (drawn or implied) unless that electric field line is straight (as in the case of the electric field due to a single particle). Even in the case of straight field lines, the only way a particle will stay on one and the same electric field line is if the particle’s initial velocity is zero, or if the particle’s initial velocity is in the exact same direction as that of the straight electric field line. The following diagram depicts a positively-charged particle, with an initial velocity directed in the +y direction. The dashed line depicts the trajectory for the particle (for one set of initial velocity, charge, and mass values). The source charge at the origin is fixed in position by forces not specified.
vo
x
y
Chapter 3 The Electric Field Due to one or more Point Charges
Some General Statements that can be made about Electric Field Lines
The following useful facts about electric field lines can be deduced from the definitions you have already been provided:
Every electric field line begins either at infinity or at a positive source charge.
Every electric field line ends either at infinity or at a negative source charge.
Electric field lines never cross each other or themselves.
Superposition
If there is more than one source charge, each source charge contributes to the electric field at every point in the vicinity of the source charges. The electric field at a point in space in the vicinity of the source charges is the vector sum of the electric field at that point due to each source charge. For instance, suppose the set of source charges consists of two charged particles. The electric field at some point P will be the electric field vector at point P due to the first charged particle plus the electric field vector at point P due to the second particle. The determination of the total electric field at point P is a vector addition problem because the two electric field vectors contributing to it are, as the name implies, vectors.
E
Chapter 3 The Electric Field Due to one or more Point Charges
Suppose, for instance, that you were asked to find the magnitude and direction of the electric field vector^2 at point P due to the two charges depicted in the diagram below:
given that charge q 1 is at (0,0), q 2 is at (11 cm, 0) and point P is at (11 cm, 6. 0 cm). The first
thing that you would have to do is to find the direction and magnitude of E 1
(the electric field
vector due to q 1 ) and the direction and magnitude of E 2
(the electric field vector due to q 2 ).
Referring to the diagram above, the direction of E 2
is “the –y direction” by inspection.
(^2) We use the expression “the electric field vector at point P” for added clarity in distinguishing between the electric
field as the infinite set of all electric field vectors and the electric field as the electric field vector at a particular point in space. The reader is warned that it is common practice to use the expression “the electric field at point P” and the reader is expected to tell from the context, that it means “the electric field vector at point P”.
q 2 = − 1 .2 mC
q 1 = − 1 .2 mC^ q 2 =^ −^1 .2 mC
P
x
y
q 1 = − 1 .2 mC
P
x
y
E 2
E 1
