ELECTRIC DIPOLE, its MOMENT, INTENSITY.

Dipole :

Electric Dipole And Dipole Moment :

Two equal and opposite charges separated by a distance together constitute a dipole.

Dipole momentdefined as the simple product of magnitude of  either charge and the distance of separation between the two charges.      

Dipole moment always points from -q to +q . Its SI unit is coulomb metre (Cm).

Electric Field Due To A Dipole At A Point Lying On Axial Line :

Consider an electric dipole consisting of two point charges – q and +q separated by some distance 2a. Let P be an observation point on axial line such that its distance from centre of the dipole is r.

If  E_A is the electric field intensity at P due to charge +q then 

If E_B is electric field intensity at P due to charge – q then        

So, 

 

        

Since

                For r>>>>a  

Electric field due to dipole at a point lying on the equatorial line :

Consider an electric dipole consisting of two point charges +q and –q separated by distance 2a. Let P be an observation point on equatorial line such that its distance from mid-point O of the electric dipole is  r.

If  E_A is the electric field intensity at P due to charge +q 

E_A is represented both in magnitude and direction 

If E_A is electric field intensity at P due to -q, 

is represented both in magnitude and direction byclearly, Let us resolve  into two components in two mutually perpendicular directions components of  and  along the equatorial line cancel each other but the components perpendicular to equatorial line get added up because they act in same direction. So magnitude of resultant intensity at P

for r>>>>>a

Electric field intensity at a general point due to short electric dipole :

                Let P be the general point. Consider a short electric dipole of dipole moment placed in vacuum. Let O be the midpoint of the dipole. Let the line OP make an angle with Resolving along OP and perpendicular to OP, we get  and  respectively. Point P is on the axial line of dipole of dipole moment. Let be the electric field intensity at P due to 

then 

 along PA

Let be the electric field intensity at P due to

Then 

 along PB

If E is the magnitude of the resultant electric intensity , then 

or 

or 

If is the angle which  makes with  , then 

or 

Case I : When P  lies on the axial line of the dipole

or 

 

 

So, the electric field intensity is along the axial line.

Case II : When P  lies on the equatorial line of the dipole

  

 

or  

or 

So, the electric field intensity is perpendicular to equatorial line and hence parallel to axial line.

Field intensity at a distance r from a finite line of charge of length L and linear charge density , as shown in the figure.

Consider a small element of length dx at a distance x, as shown in figure. The magnitude of the contribution to the field at point P from this element is

To carry out the integration, we express the variables in terms of the angle theta,

From the figure 

and 

on differentiating, 

Using these expressions (i) may be modified as 

The components of dE are

On integrating 

and 

For any equatorial point,

Note : For an infinite line of charge,

Therefore, 

For any axial point,

  

Electric field intensity at an axial point at a distance r from the centre of the ring having charge Q which is uniformly distributed over the circumference of a ring.

The electric field at P due to the charge element dq of the ring is given by

Hence, the electric field at P due to the uniformly charged ring is given by

At the centre of the ring, (r = 0) , E = 0

The variation of E with r can be seen as

The electric field strength is maximum at a point where

Electric field strength due to a charged disc :-

The variation of E with r can be seen as

            On the surface of the disc