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ELECTROSTATICS
ELECTROSTATICS
CHARGES, THEIR PROPERTIES, HISTORY AND SOME PEP TALK! ☺
(Mostly adapted from the book “Electricity and Magnetism” by Benjamin Crowell)
We all know that a lot of people have done a lot of experiments and finally proved that there are 2 kinds of charges. In
particular, historically, Benjamin Franklin (1706 – 90) is the one who named these charges as ‘positive’ and ‘negative’. He
could have named them black and white or even as good and bad! But he named it positive and negative surely for some
clear logical reason 1 . However, one thing was clear whatever the nomenclature, like charges repelled and the unlike
charges attracted. A fundamental reason for using positive and negative signs for electrical charge is that experiments show
that charge is conserved according to this definition: in any closed system, the total amount of charge is a constant. This
is why we observe that rubbing initially uncharged substances together always has the result that one gains a certain
amount of one type of charge, while the other acquires an equal amount of the other type. Conservation of charge seems
natural in our model in which matter is made of positive and negative particles. If the charge on each particle is a fixed
property of that type of particle, and if the particles themselves can be neither created nor destroyed, then conservation of
charge is inevitable.
It is also noticed that an electrically charged object can attract objects that are uncharged. The key is that even though a
piece of paper, say, has a total charge of zero, it has at least some charged particles in it that have some freedom to move.
Suppose a positively charged tape is brought near paper which is neutral. The mobile particles in the paper will respond to
the tape’s forces, causing the end near the tape to become negatively charged and the other to become positive. The
attraction between the paper and the tape is now stronger than the repulsion, because the negatively charged end is closer to
the tape.
Exercise 1: What would have happened if the tape was negatively charged?
The second idea about charges is that they are quantized! It simply means that they come in discreet packets of a
fundamental charge which we know equals the charge of one electrone = 1.602 × 10
C. This was experimentally
proved by Robert Millikan in his famous Oil Drop Experiment. The idea of the experiment was to find the charge per mass
ration of tiny oil drops. Millikan explained the observed charges as all being integer multiples of a single number,
1.64×10−19 C (Keep in mid that the experiment was done around 1910 so error is expected) Millikan states in his paper
that these the result of his experiment (that won him the Nobel Prize) were a “ …direct and tangible demonstration . . . of
the correctness of the view advanced many years ago and supported by evidence from many sources that all electrical
charges, however produced, are exact multiples of one definite, elementary electrical charge, or in other words, that an
electrical charge instead of being spread uniformly over the charged surface has a definite granular structure, consisting,
in fact, of . . . specks, or atoms of electricity, all precisely alike, peppered over the surface of the charged body”
In other words, he had provided direct evidence for the charged particle model of electricity and against models in which
electricity was described as some sort of fluid. The basic charge is notated ‘e’, and the modern value is = 1.60 ×
10
C. The word “quantized” is used in physics to describe a quantity that can only have certain numerical values
(whole numbers usually), and cannot have any of the values between those. In this language, we would say that Millikan
discovered that charge is quantized. The charge ‘e’ is referred to as the quantum of charge.
Exercise 2: Is money quantized? What is the quantum of money?
A Historical note on Millikan’s fraud: Very few Physics textbooks mention the well-documented fact that although Millikan’s
conclusions were correct, he was guilty of scientific fraud. His technique was difficult and painstaking to perform, and his
original notebooks, which have been preserved, show that the data were far less perfect than he claimed in his published
scientific papers. In his publications, he stated categorically that every single oil drop observed had had a charge that was a
multiple of e, with no exceptions or omissions. But his notebooks are replete with notations such as “beautiful data, keep,” and
“bad run, throw out.” Millikan, then, appears to have earned his Nobel Prize by advocating a correct position with dishonest
descriptions of his data. Why do textbook authors fail to mention Millikan’s fraud? It may be that they think students are too
unsophisticated to correctly evaluate the implications of the fact that scientific fraud has sometimes existed and even been
rewarded by the scientific establishment. Maybe they are afraid students will reason that fudging data is OK, since Millikan got
the Nobel Prize for it. But falsifying history in the name of encouraging truthfulness is more than a little ironic. English
teachers don’t edit Shakespeare’s tragedies so that the bad characters are always punished and the good ones never suffer!
Another possible explanation is simply a lack of originality; it’s possible that some venerated textbook was uncritical of
Millikan’s fraud, and later authors simply followed suit. Biologist Stephen Jay Gould has written an essay tracing an example of
how authors of biology textbooks tend to follow a certain traditional treatment of a topic, using the giraffe’s neck to discuss the
1
The reason was Mathematical of course
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non-heritability of acquired traits. Yet another interpretation is that scientists derive status from their popular images as
impartial searchers after the truth, and they don’t want the public to realize how human and imperfect they can be. (Millikan
himself was an educational reformer, and wrote a series of textbooks that were of much higher quality than others of his era).
We have begun to encounter complex electrical behavior that we would never have realized was occurring just from the
evidence of our eyes. Unlike the pulleys, blocks, and inclined planes of mechanics, the actors on the stage of electricity and
magnetism are invisible phenomena alien to our everyday experience. For this reason, the flavour of the second half of
your physics education is dramatically different, focusing much more on experiments and techniques. Even though you
will never actually see charge moving through a wire, you can learn to use an ammeter to measure the flow. Students also
tend to get the impression from their first experience of physics that it is a dead science. NOT SO! We are about to pick up
the historical trail that leads directly to the cutting edge physics research you read about in the newspaper. The atom
smashing experiments that began around 1900, were not that different from the ones of year 2008 – just smaller, simpler,
and much cheaper!!!
COULOMB’S LAW AND PRINCIPLE OF SUPERPOSITION
We have talked about the properties of charges being Conservation and Quantization. The unit of charge is Coulomb (C)
named after the French scientist Charles Coulomb who formulated the Coulombs law. 1C of charge equals the amount of
charge that flows though conductor when one Ampere of current passes for one second. It has to be kept in mind that 1C of
charge is a very large amount of charge. The earth for instance can be considered to be having only 10 C of charge in total.
Exercise 3: Given the charge of one electron, = . × find the number of electrons that constitute 1C of
charge.
Charles Coulomb showed using experiments that the force of attraction between any two given charges is directly
proportional to the product of the charge and inversely proportional to the square of the distance between them.
Mathematically, we have,
1 1
. where =
∝ ⇒= ⇒=
4 4
and εo is known as the permittivity of free space. This εo is replaced by ε when we talk about the force in any medium
where
ε = ε% ε& and εr is known as the relative permittivity of the medium or the dielectric constant which is 1 for vacuum and
greater than 1 for all other media (εair = 1.0006 and εwater = 81!). This means that, for a given two charges, the force of
attraction or repulsion between them will be 81 times less in water than in air. It must always be kept in mind that the
above equation of force only gives the magnitude of the force between the two charges and that this force will be attractive
for unlike charges and repulsive for like charges and will act along the line joining the two charges. If we include the signs
for the charge ‘q’ in the above equation, then a negative force will signify attraction and a positive force will signify
repulsion (two negatives will still give you a + force which again signifies repulsion). The value of εo in SI unit is
8.8542 × 10
C /Nm and therefore we have the value of K = 8.98 × 10 . 9 × 10 Nm /C We know that force is a vector. Therefore, it is important that we express coulombs law in the vector form. Imagine two
positive charges q1 and q2 separated by a vector distance ‘r’ with magnitude ‘r’. We know from mathematical convention
that the displacement vector from point 1 to point 2 is written as vector r21. Keeping this in mind, the force on charge +q1
due to +q2 is written as
1 3
0 = 1
2 34 where34 =
/
4 is the unit vector that points from +q2 to +q1 (as the charge q1 goes away from the
charge q2 in the line joining the two charges. As evident from the equation, this is
also the direction of action of F12. On the other hand,
1 3
0 = 1
2 34 where34 =
/
4 is the unit vector that points from q1 to q2 along the line joining the two charges.
Here again, as evident from the equation,34 is also the direction of action of F21. It
must be always known that
0 = 5/
0 andthat34 = 534 /
The above formulae give us the force between two charges that are positive. You will get the same direction of forces if we
use both negative charges instead. On the other hand, as you can see by now, the vector equations given above are sign
sensitive. So if we have one positive and one negative, then the force flips over and we will have attraction. Remember that
0 = 5/
0 is simply Newton’s Third law of action and reaction. Coulomb’s law is valid only for point
the relation /
charges and only when the point charges are at rest.
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When there are more than two charges, to find the resultant force on any one charge due to the other charges we use the
PRINCIPLE OF SUPERPOSITION that states that ‘the net force on any one charge will equal the vector sum of the
forces exerted on it by all other source charges’. The Principle of Super position says that the force on, say, charge q1 due
to another charge, say q2, is unaffected by the presence of the any other charges, q3, q4, q5 etc. However, since this applies
to all pairs of charges, to find the total force on any one particular charge, we have to resort to vector addition. So for a
0 given by/
0 = /
0 + /
0 ; + /
0 < + ⋯ + ⋯ + /
0 > . This has
system of ‘N’ charges, we have the force on any charge say /
to be added NOT using algebra but using the rules of VECTOR ADDTION.
1
A A A B
A C
0 ? + /
0 ? + /
0 ?; + ⋯ + ⋯ + /
0 ?> = 1
2 @ 34? + 34? + 34?; + ⋯ + ⋯ +
/? = /
34 E =
4
A
A
AB
AD ?>
/? = 1
THE IDEA OF ELECTRIC FIELD
H D
P
3S?M
3S?M
A J
1
2 GI 34?M O Qℎ34?M =
=
4 G
T3S?M T AJ
O
JK AJ
F JLA
N
The idea of a FIELD is the product of brilliant imagination by one man – an experimental genius – Michael Faraday. The
field of forces or the force field can act through space, producing an effect even when there is no physical contact between
the objects. Electricity, Magnetism and Gravity are all examples for such field forces (or “force at a distance” unlike the
Newtonian “contact forces” that act only when two bodies are in contact). Luckily for us, the three laws of motion holds
good for these field forces as well. It must be remembered that the concept of a field is a mathematical concept that helps
us to understand how charges and magnets attract each other and how they influence the space around. But this concept
does NOT tell us why this happens2
The gravitational field at a point in space is defined to be equal to the gravitational force acting on a test particle of a given
mass or as the force per unit mass given by the equation U = mV or more specifically,
U
V = WZXY
m
A similar approach to electric forces was developed by Faraday and is of such practical use. However, generally speaking,
since we cannot physically see the force pulling or pushing two charges (just as we don’t see any force pulling an object to
the surface of the earth), we develop this into a mathematical construct that becomes a very convenient methodology in
solving practical problems. In this approach, an electric field is said to exist in the region of space (called the ‘sphere of
influence’) around a charged object. When another charged object enters this electric field an electric force acts on it. By
definition, to find the electric field at any point, keep a positive test charge at that point. Now calculate the Force acting on
that charge. The force divided by the test charge value gives the value of electric field.
U
[ = NZC
However, it is of utmost importance in this definition that the test charge is small enough that it does not change the field
produced by the source charges. Hence we amend the equation as
[ = lim bFZa d NZC
^_ →a
The simplest example would be to find the E due to a point charge q at a point P distance r from the charge q. To do this,
we follow our definition and place a test charge qo at the point P. Now the force on the charge at P due to the charge q is
given by coulomb’s law
1 U
1 N
U=
e4Nandtherefore,
[=
=
e4 ZC
4 4 So, the Electric Field at a point is in the direction of the force exerted by a unit positive charge at that point. This can be
extended to Field due to a collection of charges using the principle of super position.
C
1
B
C
1
A
2 @ e4 + e4 + e4; + ⋯ + ⋯ + e4i E ⇒ [ =
[ = [ + [ + [; + ⋯ + ⋯ + [i ⇒ [ = 1
I e4j
4 B
C
4
A
AK
2
One can always ask the question “How does one electron know that the other electron is there to repel it?” OR “How does one magnet know that
there is another magnet next to it so that it can attract or repel?” These questions are not answered by the concept of the Electric and Magnetic FIELDS.
But it helps us to build theories on what happened and we can use them to our advantage… The fan and light and every other imaginable electrical
appliance that you can think of all works on the basis of these ideas!!!
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REPRESENTING ELECTRIC FIELDS – THE ELECTRIC LINES OF FORCE
One way of representing the E is to draw all the electric field vectors
in space around a given charge distribution. This gives us the
magnitude and direction of the electric field at any given point.
Remember that to draw such a representation all one has to do (in
theory of course) is to put the source charge in place and then bring a
positive test charge (q0) at the position where the E vector needs to be
drawn3. Calculate the Force on the test charge, and then, by definition
we have [ = UZa . The length of each vector will be proportional to
the strength of the field at that point. So, to know the magnitude of E
we have the length of the vector and to know the direction, we have
the direction in which the vector is pointing. This method however, can be quiet tedious and impractical. So we have a
more convenient way of visualizing electric field pattern. Join all these field vectors together continuously to get what we
call as a ‘field line’. These lines, called electric field lines, are related to the electric field in any region of space in the
following manner:
• The electric field vector E is tangent to the electric field line at each point.
• The number of lines per unit area (or the density of field lines) through a surface perpendicular to the lines is
proportional to the magnitude of the electric field in that region.
Thus, E is great when the field lines are close together and small when they are far apart. These properties are illustrated in
the density of lines through surface A is greater than the density of lines through surface B in the first figure above.
Therefore, the E is more intense on surface A than on surface B. Furthermore, the fact that the lines at different locations
point in different directions indicates that the field is non uniform. Representative E lines for the field due to a single
positive point charge are shown above in the second and third diagram. Note that in this two-dimensional drawing we show
only the field lines that lie in the plane containing the point charge. The lines are actually directed radially outward from
the charge in three dimensions. Thus, instead of the “flat wheel” of lines shown, you should picture an entire sphere of
lines (like a ball stuck with pins all around). Since a positive test charge placed in this field would be repelled by the
positive point charge, the lines are directed radially away from the positive point charge. The E lines representing the field
due to a single negative point charge are directed toward the charge. In either case, the lines are along the radial direction
and extend all the way to infinity. Note that the lines become closer together as they approach the charge; this indicates that
the strength of the field increases as we move toward the source charge. The rules for drawing E lines are as follows:
• The lines must begin on a positive charge and terminate on a negative charge.
• The number of lines drawn leaving a +ve charge or approaching a –ve charge is proportional to the magnitude of the
charge.
• No two field lines can cross. (If they cross, then by definition, they will have two tangents and therefore two directions
of E at that point which is absurd)
Exercise 4: Learn the properties of Electric Field Lines from page 10 in our text
Is this visualization of the E in terms of field lines consistent with the equation?
U
1 N
[=
=
e4 ZC ?
4 To answer this question, consider an imaginary spherical surface of radius r concentric with a point charge. From
symmetry, we see that the magnitude of the E is the same everywhere on the surface of the sphere. The number of lines N
that emerge from the charge is equal to the number that penetrates the spherical surface. Hence, the number of lines per
unit area on the sphere is W/4 (where the surface area of the sphere is 4 ). Since E is proportional to the number of
lines per unit area, we see that E varies as 1/ ; this finding is consistent with the above equation.
3
Like how you drew the Magnetic lines of forces keeping a bar magnet in your lower classes.
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ELECTRIC DIPOLE: Two equal and opposite charges (+q and –q separated by a small vector distance L(We will use
|m| = 2n instead of l because in problems we very often come across half the distance between the charges. So taking 2l
S ⇒ |o
0 = . m
0| =
will avoid fractions in our calculation). We also define a vector quantity called the dipole moment as o
p = 2n where vectors L and p are in the same direction which is directed form the negative charge to the positive charge.
ELECTRIC FLUX
The concept of electric field lines is described qualitatively in the previous section. We now use the concept of Electric
Flux Φr to treat electric field lines in a more quantitative way. Consider an E that is uniform in both magnitude and
direction, as shown in the figure below. The field lines penetrate a rectangular surface of area A, which is perpendicular to
the field. Recall that the number of lines per unit area (in other words, the line density) is proportional to the magnitude of
the electric field. Therefore, the total number of lines penetrating the surface is proportional to the product s × t. This
product of the magnitude of the electric field E and surface area A perpendicular to the field is called the Electric flux Φr
Φr = [ ∙ v
From the SI units of E and A, we see that w[ has units of (Nm2/C). Electric flux is proportional to the number of
electric field lines penetrating some surface.
If the surface under consideration is not perpendicular to the field, the flux through it must be less than that given by Φr = E. A. We can understand this by considering the second figure above, in which the normal to the surface of area A is
at an angle to the uniform E. Note that the number of lines that cross this area A is equal to the number that cross the area
A′, which is a projection of area A aligned perpendicular to the field. From second figure we see that the two areas are
related by A′ = A cos θ. Since the flux through A equals the flux through A~ , we conclude that the flux through A′is
Φr = EA~ = EA cos θ = [ ∙ v
From this result, we see that the flux through a surface of fixed area A′ has a maximum value EA when the surface is
perpendicular to the field (in other words, when the normal to the surface is parallel to the field that is  = 0°) and the flux
is zero when the surface is parallel to the field (in other words, when the normal to the surface is perpendicular to the field,
that is,  = 90°.
We assumed a uniform E in the preceding discussion. In more general situations, the E may vary over a surface. Therefore,
our definition of flux given by Φr = [. v has meaning only over a small element of area. Consider a general surface
divided up into a large number of small elements, each of area ∆A (Figure on the left above). The variation in the E over
one element can be neglected if the element is sufficiently small. It is convenient to define a vector ∆Ai whose magnitude
represents the area of the ith element of the surface and whose direction is defined to be perpendicular to the surface
element, as shown in the third figure in the previous page. The electric flux through this element is Φr = E ΔA cos θ =
[j ∙ ƒvj where we have used the definition of the scalar dot product of two vectors v ∙ „ = ABcosθ
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By summing the contributions of all elements, we obtain the total flux through the surface. If we let the area of each
element approach zero, then the number of elements approaches infinity and the sum is replaced by an integral. Therefore,
the general definition of electric flux is
Φr = lim I [j ∙ ƒvj = † [ ∙ ‡v
ƒv j →
This is a surface integral, which means it must be evaluated over the surface in question. In general, the value of flux
depends both on the field pattern and on the surface.
We are often interested in evaluating the flux through a closed surface, which is defined as one that divides space into an
inside and an outside region, so that one cannot move from one region to the other without crossing the surface. The
surface of a sphere, for example, is a closed surface. Consider the closed surface in the figure above on the right. The
vectors ƒvj point in different directions for the various surface elements, but at each point they are normal to the surface
and, by convention, always point outward. This is the Area Vector and in this case of course is a small element. At the
element labeled (1), the field lines are crossing the surface from the inside to the outside and hence, the flux Φr = [ ∙ ƒvj
through this element is positive. For element (2), the field lines ‘graze’ the surface (perpendicular to the vector ƒvj ) thus,
and the flux is zero. For elements such as (3), where the field lines are crossing the surface from outside to inside, 180° >
 > 90° and the flux is negative because ‰Š‹ is negative. The net flux through the surface is proportional to the net
number of lines leaving the surface, where the net number means the number leaving the surface minus the number
entering the surface. If more lines are leaving than entering, the net flux is positive. If more lines are entering than
leaving, the net flux is negative. Using the symbol for integral over a closed surface, we write, the flux through a closed
surface as
Φr = Œ  ∙ Ž = Œ s. Žt
where En represents the component of the E normal to the surface. Evaluating the net flux through a closed surface can be
very cumbersome. However, if the field is normal to the surface at each point and constant in magnitude, the calculation is
pretty straightforward.
GAUSS LAW
We have seen that the flux through any closed surface is given by the expression
Φr = Œ  ∙ Ž =
a
Coulombs Law is the governing law in electrostatics, but it is not cast in a form that particularly simplifies the work in
situations involving symmetry. The Gauss’ Law formulated by Karl Friedrich Gauss, a German mathematician is a new
formulation of the coulomb’s law. It relates the net flux of an electric field through a closed surface (a Gaussian Surface) to
the net charge qenc that is enclosed by that surface. It tells us that
Φ = C‘
Substituting this equation in the above equation, we get
ӐC‘
Φr = Œ  ∙ Ž =
(GaussLawinElectrostatics)
This equation can be used to take advantage of special symmetry situations and to thereby find the E due to certain charge
distribution in 2D and 3D situations. The following examples demonstrate ways of choosing the Gaussian surface over
which the surface integral given by the above equation can be simplified and the E determined. In choosing the surface, we
should always take advantage of the symmetry of the charge distribution so that we can remove E from the integral and
solve for it. The goal in this type of calculation is to determine a surface that satisfies one or more of the following
conditions:
1.
2.
3.
4.
The value of the E can be argued by symmetry to be constant over the surface.
The dot product in the above equation can be expressed as a simple algebraic product E.dA because E and dA are
parallel.
The dot product in the above equation is zero because E and dA are perpendicular.
The E can be argued to be zero over the surface.
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ELECTRIC POTENTIAL ENERGY, ELECTRIC POTENTIAL AND APPLICATIONS
We have dealt with the idea of potential energy in class. I again urge you to read potential energy as the ‘POTENTIAL4’
energy rather than as potential energy! And, also to be kept in mind is, as I always did remind you in class that the potential
energy has to do with a system rather than with any particular single object. So it is the potential energy of a system of
charges that we would talk about here and not that of a single charge alone because a single charge under no other
influence what so ever from any other fields of any other kind will not have any potential energy. When we talk of the
potential energy of a pendulum, it is in fact the potential energy of the pendulum – earth system that we talk about! If the
earth is not there, there is no gravity (g) and there is no force (mg) and there is no work (˜ = / ∙ ™) and therefore no
potential energy (gulp!).
Unlike Kinetic energy that comes into being due to the motion of an object, potential energy is the result of an object’s
position. If you ‘stress’ the object from its natural way of being, you increase its potential energy. Therefore, a compressed
spring, a stretched rubber band, two North poles of two magnets together etc are all examples of systems with potential
energy. We know that this energy is stored in it only if we supply external work to it because the body all by itself will
NOT increase its energy in any way whatsoever! So once the work is done on the system, the energy of the system
increases. This energy can be tapped back because we have to make some arrangement to keep the system in that stressed
state (it will not stay that way on its own) and if we were to let go of this ‘stressed’ state, it would come to its original
relaxed state where the energy is less! Let us now derive the mathematical expression for potential energy and potential.
Imagine a point charge +Q in space. Since there are no other charges
anywhere around, no work has to be done to keep it fixed at a point say
A. Now in order to bring a test charge +q0 to a point P at a distance ‘r’
from +Q, we have to do external work. Why? Because, given the
chance, these two positive charges want to be as far as possible from
each other. This external work that is done will be stored as the Potential
Energy of the system of the two charges. So the system’s energy
increases. Can we prove this? Yes, if these two charges are tied using a string, there will be tension on the string and if we
cut the string, the charges will fly away from each other, in the line joining the two charges. So all that potential energy
was just converted to Kinetic Energy (this is just like compressing a spring!).
If the test charge q0 is infinitely far away, it will experience zero force. But as it gets nearer and nearer, the force will
increase with the square of the distance between the two charges (Coulomb’s law). So we a applying an external Force
(Fext) which must be equal and opposite to the Electrostatic force (Fele). So, the displacement of the charge is in the
direction of the external force and opposite to the direction of the electrostatic force. Since the value of force keeps
changing at every point in space (due to the inverse square relation) to total work done must be the line integral of bringing
the charge from infinity to that point P.
Ž˜š› = /œž ∙ Ž3
However since /œ¢œ = 5/œž we get,
⟹ ˜š› = † /œž ∙ Ž3 = † /œž ∙ Ž3
¡
¡
¡
⟹ ˜š› = † /œž ∙ Ž3 = 5 † /œ¢œ ∙ Ž3 = + † /œ¢œ ∙ Ž3 (Q£¤‰ℎŠ¥¤¦Š¤ℎ§¨¤Y£nn§©§¤‹Q£pp§¨Y)
¡
¡
ª¥¤,
⟹ ˜š›
«a
34
4a ¡
«a
=†
34 ∙ Ž3
4
a
/œ¢œ =
Since e4 and de are in same direction, direction, we get
¡
¡
«a
«a ¡ 1
«a
Ž
«a
1¡
«a 51 51
«a
1
⟹ ˜š› = †
Ž =
† Ž =
¬† ­ =
®5 ¯ ° =
@
5
E=
@+ E
4
4
4
4
4
∞
4
a
a
a
a
a
a
«a
˜š› =
4a This external work done is stored as Potential Energy hence we have the relation
«a
²=
4a When you have more than two charges, then we have to find the individual work done in bringing all the charges at its
corresponding position and hence we will give a relation as follows…
4
Potential meaning – Existing in possibility, Expected to become or be; in prospect, The inherent capacity for coming into being
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In the case of a charge distribution due to charges , , B , … C , the total potential energy is given by the expression…
²=
C
C
A J
B
B
´
C C
1
+
+ ⋯+
+
+ ⋯+
= ² = µI I ¬
­¶
4a 4a B
4a B 4a ´
4a CC
4a AJ
2
AK JK
§ ≠ ¸
Now this is the definition of Potential energy of a system of charges. Now let us revisit some of the equations used to get
the above expression…
¡
¡
«
34
˜š› = † /œ¢œ ∙ Ž3 = † a  ∙ Ž3 ; Qℎ =
4
a
Ok so we have the formula for Work done and hence the Potential Energy. Then how about Electric potential? It turns out
that if you do a certain work on one charge and the system increases its energy by a certain amount. If the same work is
done on 2 charges, then the energy increased doubles! This means that the potential energy is proportional to the charge
under consideration. The idea of defining the potential of a field is so that we can have a quantity independent of the
charges used to assess the ‘quality’ of the system. So we have, the Electric Potential or simply Potential5 of a system is
defined as the Potential at a point in space is defined as the work done per unit charge to bring a positive test charge
from infinity to that point. It’s unit is Volt (V). So we have,
¡
˜ ¼ a  ∙ Ž3
«
=
= †  ∙ Ž3 =
a
a
4
a
For a point A and B, we have potentials VA and VB. given by
º» =
¡
¡
¡
º½ = †  ∙ Ž3 ;º¾ = †  ∙ Ž3
½
¡
¾
¡
¾
∆º½¾ = (º½ 5 º¾ ) = ®†  ∙ Ž3 5 †  ∙ Ž3° = ®†  ∙ Ž3°
½
¾
½
Potential difference between two points in space is defined as the work done per unit charge to bring a positive test
charge one point to the other.
Now, we know that CONSERVATIVE fields are those fields where the change in energy of a body (which is the same as
work done) is only dependant on the initial and final position and has nothing to do with the path that was traversed to get
there. We see that the concept of potential energy is also of great value in the study of electricity since the electrostatic
force given by Coulomb’s law is conservative, electrostatic phenomena can be conveniently described in terms of an
electric potential energy. This idea enables us to define a scalar quantity known as electric potential. Because the electric
potential at any point in an E is a scalar function, we can use it to describe electrostatic phenomena more simply than if we
were to rely only on the concepts of the E and electric forces.
When a test charge qo is placed in an electric field E created by some source charges, the electric force acting on the test
charge is U = [(∵ [ = U⁄ ). Don’t forget however that if the field is produced by more than one charged object, this
force acting on the test charge is the vector sum of the individual forces exerted on it by the various other charged objects.
The force U = [ is conservative because the individual forces described by Coulomb’s law are conservative.
Now that we have seen that º» =
Â
^_
= ¼  ∙ Ž3 and since we know that electrostatic fields are conservative fields, if we
¡
start at a given location in space where the potential is V and then return back to the same position through ANY path we
chose, then the work done will be zero and hence the potential difference is zero. Mathematically, it is written like this…
»
º» 5 º» = †  ∙ Ž3 = Œ  ∙ Ž3 = 0
»
This is said as “CLOSED LOOP integral E dot dr equals zero”. Now since we do not follow a direct displacement from
one point to the other and since taking any path will still not affect the work done (because it is a conservative field and
because the work done depends only on the initial and final position), we often replace the “dr” with a “dl” and so we
write,
»
Here, the ‘dl’ refers to the path taken!
º» 5 º» = †  ∙ Žn = Œ  ∙ Žn = 0
»
Exercise 5: If the path between A and B does not make any difference in Equation˜ = ¼½  ∙ Ž3, why don’t we just
use the expression where d is the straight-line distance between A and B ÃÄ =  ∙ ™ without using Integration
¾
5
Remember “POTENTIAL” of a system
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qed_amk_gesXIphy0211
ELECTROSTATICS
REMEMBER THE FOLLOWING RESULTS…
C
C
A J
1
² = µI I ¬
­¶
2
4a AJ
AK JK
i ≠ j
¡
º½ = º½ 5 º¡ = †  ∙ Ž¢
Ǻ¾½
Æ
½
¾
Dz
= º¾ 5 º½ =
= †  ∙ Ž¢ = 5 †  ∙ Ž¢ 6and
¾
½
Œ  ∙ Ž¢ = 0
Potential difference should not be confused with difference in potential energy. The potential difference is proportional to
the change in potential energy, and we see from the above equation that the two are related by electric potential is a scalar
characteristic of an electric field independent of the charges that may be placed in the field. However, when we speak of
potential energy, we are referring to the charge – field system. Since we are usually interested in knowing the electric
potential at the location of a charge and the potential energy resulting from the interaction of the charge with the field, we
follow the common convention of speaking of the potential energy as if it belonged to the charge.
Because the change in potential energy of a charge is the negative of the work done by the electric field on the charge the
potential difference ∆V between points A and B equals the work per unit charge that an external agent must perform to
move a test charge from A to B without changing the kinetic energy of the test charge. Just as with potential energy, only
differences in electric potential are meaningful. To avoid having to work with potential differences, however, we often take
the value of the electric potential to be zero at some convenient point in an electric field. This is what we do here:
arbitrarily establish the electric potential to be zero at a point that is infinitely remote from the charges producing the field.
Having made this choice, we can state that the electric potential at an arbitrary point in an electric field equals the work
required per unit charge to bring a positive test charge from infinity to that point. Thus, if we take point A in the above
equation to be at infinity, the electric potential at any point P is
É
ΔV = VÉ 5 V¡ = VÉ = 5 † [ ∙ dÊ
¡
In reality, VP represents the potential difference ∆V between the point P and a point at infinity. Since the electric potential
is a measure of potential energy per unit charge, the SI unit of both electric potential and potential difference is joule per
Ë
coulomb, which is defined as a volt (V), 1V = Ì. That is, 1 J of work must be done to move a 1C charge through a
potential difference of 1V. However, the equation ΔV = 5 ¼Î [ ∙ dÊ shows that potential difference also has units of
electric field times distance. From this, it follows that the SI unit of electric field (N/C) can also be expressed in volts per
meter
N
V
1@ E = 1@ E
C
m
EQUIPOTENTIAL SURFACES are surfaces that have the same potential. Any imaginary sphere concentric with a point
charge will be an equipotential surface! They look like contour lines on a map that join places with same height from sea
level.
Í
REFERENCE AND BIBLIOGRAPHY (Kindly get in touch with me if you want access to these books)
1.
2.
3.
4.
6
SERWAY AND JEWETT, Principles of Physics, 3rd Edition, Thomson, Brooks/Cole [2006]
BENJAMIN CROWELL, Electricity and Magnetism, e-copy
HALADAY, RESNICK & WALKER, Fundamentals of Physics, 6th Edition, John Wiley & Sons Inc [2004]
KUMAR, MITTAL, GUPTA, Nootan ISC Physics, Nageen Prakashan Pvt. Ltd [2008]
Notice that the limits changed the value when the minus sign was taken into consideration.
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