**MODULE – 01, ELECTROSTATIC,**

** CHAPTER – 02, ELECTRIC FIELD**

* Electric field :* It is defined as the space around a charge where another charge can experience force due to former charge.

From coulomb’s law of electrostatic, we have

**F = K q _{1}q_{2} / r^{2}**

Where K is proportionality constant which is given by

K = 1 / ( 4πε_{0} ) = 9 * 10^{9} Nm^{2}C^{-2}

where ε_{0} is the absolute permitivity of free space.

If q_{1} and q_{2} are non zero and F = 0 then

r = ∞

that is the mathematically, the electric field is extended up to ∞ but practically its extension is limited.

**Electric Field Intensity : **

From Coulomb’s law we have

**F = K q _{1}q_{2} / r^{2}**

Where K is proportionality constant which is given by

K = 1 / ( 4πε_{0} ) = 9 * 10^{9} Nm^{2}C^{-2}

where ε_{0} is the absolute permitivity of free space.

If q_{1} = q and q_{2} = + 1 unit then

E = q / r^{2}

The electric field intensity at a point in the electric field is defined as the eletrostatic force experienced by unit positive charge placed at point . It is vector quantity.

**Relation Between The Electrostatic Force And Electric Field Intensity : **

From Coulomb’s law of electrostatic we have

**F = K q _{1}q_{2} / r^{2}**

Where K is proportionality constant which is given by

K = 1 / ( 4πε_{0} ) = 9 * 10^{9} Nm^{2}C^{-2}

where ε_{0} is the absolute permitivity of free space.

The electric field intensity is given by

**E = K q _{2} / r^{2}**

From above two equations we can write

F = q_{1} E

The required relation between the force and electric field intensity is given by

**F = qE**

Here, F and E are in the same direction for positive charge while F and E are in opposite direction for negative charge.

**Electric Dipole :**

An electric dipole is said to be formed if equal and opposite charges are separated by a small distance.

**Electric Dipole Moment : **

The product of length of the dipole and the either of the two charges is called as the electric dipole moment.

p = q * 2l

where q is the magnitude of the charge of the electric dipole and 2l is the length of the dipole.

Unit :

1 ) Its SI unit is C- m

2 ) Its CGS unit is stat coulomb – cm

*Electric Lines Of Force :*

It is a path along which a free positive charge moves and tangent drawn at any point on the path gives the direction of the electric field.

**Properties Of Electric Lines Of Force : **

1 ) These are the imaginary lines originating from the positive charge and end on negative charge.

2 ) The direction of the electric field at a point on the electric lines of force is given by the tangent drawn at that point.

3 ) Two electric lines of force can not intersect each other because at the point of intersection two tangents can be drawn and this will indicate the two different directions of the electric field at the same point simultaneously. This is an impossible situation. So, the two electric lines of force can never intersect each other.

4 ) The electric lines of force having tendency to contract along their length. This explains the attraction between two opposite charges.

5 ) The electric lines of force having tendency to separate in a direction perpendicular to their length. This shows the repulsion between two same charges.

6 ) Closer the distance between the lines of force higher is the electric field intensity and vice versa.

7 ) No electric lines of force exist inside the conductor.

8 ) This electric lines of force are imaginary lines but they, represent the electric field is real.

**Effect of Electric Field on point charge :**

If a point charge q is kept in a electric field, it experiences the electrostatic force F due to electric field intensity E. Due to this force, it accelerates.

As we know that

F = qE

ma = q E

a = ( q / m ) E

This is the acceleration through which the point charge of mass ‘ m ‘ accelerates.

( q / m ) is the specific charge of the particle.

**Expression for the Electric Field Intensity due to an Electric Dipole :**

Let us discuss the electric intensity due to presence of electric dipole at three different position.

**1 ) Expression for the electric field intensity due to an electric dipole moment on the axial position / tanA position / end on position : **

Let a unit positive charge is placed at a point P at a distance distance of ‘r’ from an electric dipole of length ‘2l’ and dipole moment ‘ p ‘

p = 2l * q

where q is the either of the charges

Let E_{+q} = electric intensity due to +q = kq / ( r – l )^{2}

E_{-q} = electric intensity due to -q = kq / ( r + l )^{2}

Where

K = 1 / ( 4πε_{0} ) = 9 * 10^{9} Nm^{2}C^{-2}

where ε_{0} is the absolute permitivity of free space.

Resultant electric intensity at point P is given by

E = E_{+q} – E_{-q}

E = { kq / ( r – l )^{2} } – { kq / ( r + l )^{2} }

E = kq { ( r + l )^{2} – ( r – l )^{2} } / { ( r + l )^{2} ( r – l )^{2} }

E = Kq { 4rl / ( r^{2} – l^{2} )^{2} }

E = { k / ( r^{2} – l^{2} )^{2} } * ( 2l * q ) * 2r

**E = 2kpr / ( r ^{2} – l^{2} )^{2}**

This is the expression for the electric intensity due to an electric dipole on axial position.

**Conclusion : **

1 ) It is directed along the axis of the electric dipole.

2 ) It is proportional to the electric dipole moment of the electric dipole.

3 ) For a small dipole r >>>l, then in the above expression ‘ l ‘ may be neglected.

E = 2kpr / ( r^{2} – l^{2} )^{2}

E ≅ 2kpr / ( r^{2} )^{2}

E = 2kp / r ^{3}

**2 ) Expression for the electric field intensity due to an electric dipole moment on the perpendicular bisector position / tanB position / broadside on position : **

Let a unit positive charge is placed at a point P at a a perpendicular distance from an electric dipole of length ‘2l’ and dipole moment ‘ p ‘

p = 2l * q

where q is the either of the charges

sine components of the electric field intensity at point P are acting in opposite direction so, they cancel each other while its cosine components are in the same direction, so they will give the resultant electric intensity.

Let E_{+q} = electric intensity due to +q = E_{-q} = electric intensity due to -q = kq / √( r^{2} + l^{2} )^{2}

Where

K = 1 / ( 4πε_{0} ) = 9 * 10^{9} Nm^{2}C^{-2}

where ε_{0} is the absolute permitivity of free space.

cosθ = l / √( r^{2} + l^{2} )

Resultant electric intensity at point P is given by

E = E_{+q }cosθ + E_{+q }cosθ

E = 2E_{+q} cosθ

E = 2 * { kq / √( r^{2} + l^{2} )^{2} } * { l / √( r^{2} + l^{2} ) }

After calculating, we will get

**E = kp / ( r ^{2} + l^{2} ) ^{3/2}**

**Conclusion : **

1 ) The electric field intensity acts parallel to the axis of the dipole ( from negative to positive )

2 ) For very short dipole r >>>>l, then in the above expression, l can be neglected,

therefore, E = kp / r^{3}

3 ) The electric field intensity is proportional to the electric dipole moment.

**Expression Of The Torque In The Electric Field :**

Torque experienced by an electric dipole placed in an electric dipole is given by

τ = Force * perpendicular distance

τ = q E * x

τ= q E * 2 l sinθ

τ = ( q * 2 l ) * E sinθ

τ = p E sinθ

**τ** = **p** × **E**

where θ is the angle of inclination of the electric dipole, q is the amount of the charges on the ends of the electric dipole and 2 l is the length of the dipole and E is the electric field intensity.

Hence, the torque experienced by an electric dipole placed in an electric field is the cross product of the electric dipole moment and the electric field intensity.

If the electric field is uniform then the dipole will experience only torque. If the electric field is non uniform, then it will experience both the force as well as the torque.

**Conclusion : **

1 ) τ = p E sinθ, if θ = 0 degree i.e the dipole is placed parallel to the electric field.

**Work done in rotating an electric dipole : **

Let us consider that the electric field is uniform and the torque experienced by the dipole is given by

τ = p E sinθ

work done in rotating the electric dipole from θ_{1} to θ_{2} is given by

W = ∫ τ .dθ

W = ∫ p E sinθ dθ

W = pE ∫^{θ2}_{θ1} sinθdθ

W = pE [ -cosθ ] ^{θ2}_{θ1}

W = pE [ – { cosθ_{2} – cosθ_{1} } ]

**W = pE ( cosθ _{1} – cosθ_{2} ) **

**Conclusion : **

1 ) If the dipole is initially oriented parallel to the electric field and it is rotated through the angle θ, then the work done is given by

W = p E ( 1 – cosθ )

2 ) If the dipole is perpendicular to the electric field and it is rotated through angle θ, then the work done is given by

W = p E ( cos 90 – cosθ )

W = – p E cosθ

3 ) If the dipole makes the angle θ with the electric field and it is rotated through the angle 90 degree, then the work done is given by

W = p E cosθ

W = **p** . **E**