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Electric charges and Fields.
Charge is one of the fundamental property of all matter, due to
which it puts electrical force on
other charged particles. (Just like how mass is responsible for
gravity, or spin is responsible for
magnetism)
Some of the properties of charges are
a) Charges are conserved. They cannot be created or destroyed
but only transferred
b) Charges are quantized. Any charge must be an integral
multiple of
Or
= . Where = 1.6 1019 and is an integer ( = 1,2 )
c) Charges are additive in nature.
(I advise you to first write these above 3 properties in the
exam)
d) Charges will always tend to redistribute (delocalize)
themselves on the outer surface of a
conductor. (This is proved using Gausss Law). But in insulators,
they tend to remain localized.
e) Charges tend to crowd more towards pointed regions of a
conductor (Proved in class)
f) Charges can also leak out from pointed regions. This is
called as action of points or corona
discharge. (Will be discussed in class)
Electrification:
This is the process of charging up any body. There are three
ways to do it.
a) Friction: When two objects are rubbed together, one gets
positive charge and the other equal
and opposite negative charge.
Eg: Glass when rubbed with silk, ends up with +ve charge, and
therefore, silk gets equal amount
of ve charge.
Rubber rubbed with fur, or wool, ends up with ve charge.
b) Conduction: This method only works for conductors (obviously
:-P). When a charged body is
brought in contact with an uncharged conductor, charges tend
flow to the conductor.
(Conduction process happens very very very fast)
c) Induction: This method is also usually for conductors. When a
charged body is brought CLOSE
to an uncharged conductor, charges in the conductor get
separated (we call this as electrostatic
polarization creating two poles), and we can then charge up the
conductor as shown in the
diagram.
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Coulombs Law:
French scientist Charles Coulomb, was able to figure out the
strength of the force between two
point charges. (We treat charged body as points, if the distance
of separation is way bigger than
their sizes, like two dust particles meters apart). His law
states
The force between two point charges is always directly
proportional to the magnitude of charges
and inversely proportional to the square of the distance between
them.
Note: So the strength of force doesnt depend upon whether it is
positive
or negative. Positive or negative only decides the direction of
the force.
Remember this guys!
Mathematically Coulombs law can be stated as follows.
If 1 and 2 are two charges separated by distance and is the
unit
vector from 2 1 then force on 1 would be
=||||
()
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And force on 2 would be
2 = |1||2|
2 ()
Clearly both forces are equal in magnitude and opposite in
direction making Mr. Newton very
happy. (Consistent with his third law).
Superposition principle:
This principle is used to calculate force on one charge due to
many other charges. The principle
states that the total force on charge 1 due to charge 2, 3, 4
etc. is the vector sum of the
individual forces on it.
1 = 21 + 31 + 41
Where 1 is the total force on charge 1, and 21 , 31 , 41 etc.
are the individual forces on charge 1
due to 2, 3, 4, etc.
For example from the given figure,
Force on 1 due to 3 alone is 31 = 31
Force on 1 due to 2 alone is 21 = 21
Therefore from super position principle the total force
on 1 is
1 = 21 + 31
1 = 31 + 21
Electric field:
How can charges separated by some distance attract or repel each
other? This action at a
distance can be explained by the concept of Electric field.
Consider two positive charges
(dominant one) and (small negligible test charge). Charge
produces an influence around
itself. This influence produced by a charge around itself is
called as the electric field. Since
charge is in contact with the electric field it experiences a
force. (Thus electric forces are due to
these electric fields)
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We define the strength (or intensity) of this electric field at
any
point as force per unit positive charge. (Remember that our
test
charge will always be positive)
=
(/)
Note:
The E field has the same direction as the force experienced by
the positive charge.
At point B, the force acts because there is a charge . If the
charge is removed there is no force
at B, but the field exists.
The electric force on a charge in a field is = , just like
gravitational force on a mass
in a field is = (Easy right?)
Field due to a point charge:
From the above figure, we calculate the electric field at B due
to the charge +.
First we put a test charge + at B, and find the force using
Coulombs law.
=
2
But the field is
=
We get
=
You can visualize this field, the same way Faraday did, we
call
these visualizations as Faraday lines or field lines or flux
lines. All we have to do is keep the test charge all around
the + charge and figure out the direction of the field and
draw continuous line. The field due this point charge would
look like this. This is called as radial field. A negative
point
charge creates a similar inward radial field.
This gives us our first property of the Faraday lines.
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1) Field lines always originate from positive charge (Source)
and terminate into negative charge
(Sink).
2) From the first property we can conclude, that field lines
can
never form closed loops.
3) Since field lines indicate direction of force on a
positive
charge, they can never intersect.
Thus if you had two similar charges (say both positive) then
the
field lines would bend as shown.
4) Since field lines can bend, the field at any point is
always
tangential. In fact we define the electric field lines as
curves
tangent to which gives the direction of the electric field.
5) If field lines are more crowded, the field is stronger. Thus
parallel
equidistant lines represent uniform field.
= =
Electric dipole: Two charges equal in magnitude opposite in
direction constitute a dipole. Dipoles
are very common in nature, dipole fields are quite unique.
Field due to an electric dipole:
a) Case 1: Along the dipole axis
Consider a dipole having charge magnitude and seperated by a
distance of 2 as shown in the
figure. Let be a point at a distance of from the centre of the
dipole on the axial line as shown.
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To evaluate the net electric field at that point, we first find
out the indivisual electric fields due to
each charges and then sum them up.
The field due to + (it is along chosen positive)
+ =
( )
The field due to (It is along negative)
=
( + )
Thus the total field is
= + +
=
( )
( + )
=
[
( )
( + )]
=
( + ) ( )
( )( + )
=
( )
Usually we are interested in field far away, which is when (a
point dipole)
Then 2 2 2
Thus the above equation reduces to
=
=
Note that, this field is very different from field due to any
other charge configurations
a) The field dies out much faster.
b) The field depends not on the charge, but the product of
charge and the distance between
them. Thus we assign this product a name called the dipole
moment.
= ()
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Dipole moment is a vector quantity, and its direction is from
negative charge to positive. Clearly
along the axis, the electric field is along the dipole moment.
Thus in the vector form, our field
equation becomes
=
b) Case 2: Along the Equatorial axis:
Consider a dipole with charge and distance 2. To
find field at x, we first calculate the electric field due
to
individual charges. Since the distance to x is same (let
it be b) the field is same in magnitude, which is
=
In vector form,
+ = () + ()
And
= () ()
Hence the total field is
= + +
= ()
= [
] [
]
=
=
( + )
Again when we consider field far away 2 2, then 2 + 2 2
thus our equation now becomes
=
(Notice, the minus sign indicates, that the electric field is in
the
opposite direction of the dipole moment direction) The
dipole
field is shown to the right.
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Dipole in uniform electric field:
Consider a dipole with charge and length 2 kept in a uniform E
field with the axis making an
angle with it. The field puts an equal and opposite force on
each charge of magnitude
= Thus the net force is zero, but the two forces produce a
couple,
and the torque is given by
= = [2 sin()] = sin ()
This can be written as vector product, thus the torque is
given
by
= ()
Note:
a) Torque is zero, when = 0. Thus the E field always tries to
align the dipole along the field. In
this position, the dipole is in stable equilibrium.
Torque is also zero when = 1800. In this position it is in
unstable equilibrium.
b) Torque is max, when = 900.
c) If the field was non uniform, the dipole would also
experience a net force towards the stronger
region of the field.
Electric flux
Electric flux is the amount of electric field flowing through
any surface. Mathematically the flux
through a small elemental surface is defined as
= . = () (
)
The total flux through an entire surface is given by
= .
This is a surface integral. Is always along the normal to the
elemental surface. The dot product
indicates, that the flux is a scalar. Usually the flux is
evaluated for a closed surface, in which case,
the normal is always towards the outside of the surface.
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Gauss Law
Consider an imaginary spherical surface of radius at the centre
of which sits a positive charge .
To calculate the total flux through the entire surface, we chose
small elemental surface as
shown. The flux through that surface would be
= .
Thus the total flux through the entire surface would be
= .
Since and are in the same direction everywhere,
. =
Also since is a constant through the surface, it can be pulled
out of the integral. Thus
The total flux is
= = (42) =
4(42) =
This result is true in general regardless of the position of the
charge inside the surface, and the
shape of the surface AS LONG AS THE SURFACE IS CLOSED.
Thus we can now state the Gauss law,
The total outward flux through a closed surface, is equal to
times the
total charge inside the surface. And this is the first of the
four deadly, awesome, epic, Maxwells Equations!
(P.S. If you ever forget this, I WILL kill you)
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This one equation can derive any equation you want in
electrostatics including Coulombs law.
Some important points about flux.
a) If the flux is positive (negative), there is net outward
(inward) flow, and total charge inside is
positive (negative).
b) If flux is zero, then either the electric field over the
entire surface is zero OR there is an inward
flux in some region of the surface and an equal outward flux in
some other region giving net flux
zero (example if it encloses a dipole). Thus if
Field at every point on the surface is zero, then flux through
it MUST be zero, but the converse
NEED NOT be true.
Also if flux through a closed surface is zero, then TOTAL CHARGE
enclosed sums up to be zero
(doesnt necessarily mean there arent any charges inside)
Applications
1) Spherical symmetry: Consider a conducting shell
(negligible thickness) of radius uniformly charged to
(positive). We wish to evaluate electric field
everywhere.
First consider at some point outside at distance
from the spheres centre. From spherical symmetry, the
field must be radially outwards
So we choose a Gaussian surface (a closed surface) to be
a sphere itself, concentric with the conductor, you
notice that the radial field makes the and in the
same direction, and also is constant through the
entire spherical surface. Thus the flux now becomes
= . = = = (42)(1)
Applying Gausss Law we get
=0
=
0(2)
From (1) and (2) we have
(42) =
0 =
420
Or
=
being the unit vector along the radius of the sphere.
On the surface, the field becomes maximum =
-
= =
Inside the shell, as the Gaussian surface encloses no
charge,
the flux must be zero. And from the symmetry, the field on
every point of the surface must be zero. Thus
=
A plot of vs is given below.
2) Cylindrical symmetry
Consider an infinitely long line of charge, with linear
charge
density (charge per unit length - / ). We will calculate
the electric field at a point at a distance from the line
using Gauss law.
Since the field is cylindrical, the Gaussian surface is going
to
be a cylinder of radius and length .
Again, the E field vector and the vector are always in
the same direction, and the field strength is the same
everywhere over the curved surface. Thus the flux becomes
= . = = = (2)(1)
Using Gauss law
=0
=
0(2)
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From (1) and (2) we have
(2) =
0
Or
=
20
In vector form
=
Where is the unit vector along the radius of the cylinder.
3) Planar symmetry
Consider an infinitely long plane uniformly charged with
surface
charge density ( /2). The field
lines are going to be parallel and straight outwards. So we
choose
a cuboidal surface of area A, which has length of 2, as
shown.
There is no flux through the lateral surface, only through
front
and behind. Since the Area is flat and field is uniform we
really
dont need any integrals here.
Flux through front surface (Area and E same direction)
= . = (0) =
Flux through back surface (Area and E again same direction)
= . = (0) = Thus total flux through the surface is
= + = 2 . (1)
Using Gausss law we have
=0
=
0 . (2)
From (1) and (2) we have
2 =
0
=
Clearly this is a uniform field, so uniformly charged infinite
long sheets produce uniform E field.
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4) Faraday Caging ( Awesome application Electrostatic
Shield)
Any conductor (hollow or solid) when charged up, the
charges must always reside on the OUTER SURFACE. This
can be easily proved using Gausss law. Consider an
irregular conductor as shown. We choose a Gaussian
surface, such that each point of it lies within the
thickness
of the conductor.
Within the conductor, E field must be zero (else the field
would put a force on electrons causing a current). Thus flux
through the surface must be zero.
From Gausss law, the total charge enclosed must be zero, thus
there cannot be any charges
inside or on the inner surface.
Even if this conductor is exposed to an external electric field
as shown
The electric field induces charges on its surface, (left
gets negative, the right side gets equal positive). This
creates a field inside, which is equal and opposite and
makes total field inside zero. Thus the field lines
cannot penetrate into a closed conductor (in
electrostatic conditions of course).
Note that the field gets a little modified. The
field lines near the conductor are always
perpendicular to the surface. (Try to think
about this why, this is always true for any
conductor)
Hence a closed conductor is always electrostatically shielded
from outside. A closed conductor is
called as a Faraday Cage. Aero planes are protected from
lightning due to faraday caging. Also
people who work on high voltage power lines, use this concept to
protect themselves from
electrocution.
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