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.

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

=||||

()

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)

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.

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.

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.

= ()

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.

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.

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)

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)

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.

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.

.